This is why when I'm programming I put in extra parentheses to make it crystal clear what the formula is supposed to do and not rely on a compiler to interpret it the way I intended.
@@Hclann1 compilers do indeed use the traditional "left to right", i.e. multiplication and division are considered equal order of precedence. So in that sense, compilers (at least c and c++, the python interpreter too) are "incorrect". You cannot ommit parenthesis.
As an engineering student, I never use the division operator. Every equation or solution is in fraction form. I believe this would clear up a lot of confusion. Thanks for the video!
Totally agree. The fraction bar is much clearer and is a grouping tool like parentheses. Solve all numerator stuff, solve all denominator stuff, the divide top by bottom. Division signs are dumb
They literally have the exact same meaning. The division operator signifies the exact same thing as the fraction bar. The symbol is even supposed to represent a fraction, with the equation to the left replacing the top dot and the equation to the right replacing the lower dot. The reason why it falls out of use in higher math is solely because certain countries decided to use the symbol for subtraction, so they dropped the symbol entirely to avoid confusion.
@@GremlinSciences the use of the division bar is confusing because the fraction bar is so clear. 6÷2+3 could be clarified so easily is the fraction bar was used because this problem is NOT supposed to read (6÷2) +3 but actually the numerator 6/(2+3) When written out without the division sign (which keeps everything on one line) it reads much more clearly because the bar serves as a grouping symbol
@@swedbp1 You only have confusion because you are giving two symbols with identical meaning and usage two separate meanings. there is as much difference between '/' and '÷' as between '*' and '×' and '⋅' 6÷2(1+2) is exactly equal to 6/2(1+2) The only difference is an arbitrary distinction you are imposing that is not part of either symbol's actual meaning.
This is something I explained to my wife, early in our relationship. She was under the impression that she "just isn't good at math". She gave me examples of things she didn't understand. I found that, to a last, the problem was that her teachers had never actually explained the mathematical concept. They had instead simply had the students memorize procedural algorithms like PEMDAS. They weren't teaching mathematics. They were basically teaching handwritten calculators, a list of steps you follow to "get the right answer". At no point did any of her instructors help her understand the language of mathematics. I was the first person she had ever met, who referred to mathematics as the language of patterns. Once I started explaining the actual patterns the equations describe, she immediately caught up. She was never "bad at math". She'd just never actually been taught about math.
I used to tutor prep classes for the GMAT and GRE. My examples always started with a simple problem students could follow all the way through, and then I'd show them the general form of the equation once they understood what it was for. One of my students' evals said that she'd taken Algebra in high school and college, but she'd never "gotten" it until I showed it to her my way. For example, I'd explain compound interest by showing an account through two two compounding periods, _then_ give them the formula with exponents and variables.
I mean, pemdas is just a tool to get students to the point where you can start discussing how the maths actually work. But, like, you have to teach the tools correctly or they'll never get to that point... I've known there were issues with how kids are being taught, but this is just BASIC. I can't even. I've run into your experience a lot and it makes me so mad. EVERY person I've ever talked to who professed to be "bad at math" were just taught poorly. Usually the basic concepts, and then they were rushed into stuff they had no chance of understanding and just had to struggle memorizing meaningless number combinations so they could pass tests. Every one of these people immediately "got" the math once it was properly explained to them - though I often had to work through years and years of conditioned response.
Math is like an onion, it's got layers (to quote Shrek). At the deepest levels, it becomes so abstract that even the brightest minds struggle to wrap their intuitions around it. In some ways, this is also where many advanced topics, such as abstract and linear algebra, topology, calculus, real and complex analysis,etc, tend to merge into each other. As the level of abstraction goes down, the fields drift apart and operations become more procedural. Reduce the abstractions further, and topics simply become impossible to express. Eventually, we have basic arithmetic presented merely as a few procedures to memorize. Precisely because of the increasing levels of abstraction, students tend to be taught from the outside in. For instance, you can teach complex numbers in two ways: 1) The common way, it to introduce i=sqrt(-1), and c=a+bi. This is the procedural way, introducing i totally without justifying it or explaining why its useful or how to build intuition about it. But it can be taught in an hour. 2) The intuition based approach, is to first teach group theory,then introduce the SO(2) group, represented by the matrix R=[a, -b | -b, a] and the relation to the O(1) rotation group. NOW you can introduce c=a+bi as just a shorthand for this, and work from there. But this takes the average student perhaps 1-2 more years of study, and before you reach c=a+bi, you lost half the students. Primary school teachers have the same dilemmas, just at a lower level. The smartest students tend to see most patterns without explicitly learning them, while the least talented would never understand those patterns. Those in the middle MIGHT gain a little from learning those patterns explicitly.
@@RWAKitty I was sitting in a statistics class when the prof put a formula for the line of best fit for a data regression on the board, then asked what the slope would be. I told him "Beta sub one" immediately. He asked if I could do the proof. I admitted I couldn't, but if we rearranged the formula slightly and renamed the terms it became Y = mx + b, which is just the slope/intercept formula from algebra 1. I can't depict the beta symbols, subscripts and carets properly in a YT comment, but the terms were in the order y = b + mx for some reason.
@haakoflo Your reference to primary school suggests to me that you were not educated in America. I think your statements are correct. But understand that in America, it is common for the teachers themselves to have little comprehension, of even basic algebra. They aren't executing pedagogy, to lay the groundwork for understanding. They are often just regurgitating what is written in their teaching materials, without really grasping it themselves. There is a reason this country ranks near last in the world, in mathematics education.
In engineering school we always wrote division operations as fractions, never in line. Not only did it keep things clear, but it shortened already lengthy equations. If we had to write things in line for whatever reason, copious parenthesis were used
"we always wrote division operations as fractions, never in line". Really? Have you never had to enter an expression into a computer programming language, or even Excel? There are a few programs out there, like Wolfram Alpha that will accept expressions in your favoured form, but those are relatively rare. For the great majority you do have to be prepared to enter them in line form.
@TheEulerID we only ever used Excell to makes charts and graphs from data. My college did try to heavily push Matlab while I was there, but that "program" was such hot garbage that it was easier to do it by hand.
I’m a retired astronomer, have been doing math for 60 years and would have always come up with the answer 1 for this equation. As someone mentioned the confusion is eliminated with the use of brackets. I just ran the equation twice in Excel 2007. Written as = 6/2*(1+2) Excel gives 9. Written as = 6/(2*(1+2)) Excel gives 1. In the old days I programmed in Algol and FORTRAN and always used the brackets. I was taught to religiously test equations in the program before finalizing.
There is only 1 set of brackets in the equation. You got the answer 1 by injecting a second set of brackets which is not in the equation. I do not agree with PEMDAS. In fact, I never heard of PEMDAS until I was around 70 years old. The proper way to solve is left to right. Starting from the left the first step is 6/2. Because 6 is the first number, you deal with the 6 first. That is 3. The second step is 3x ( ) because (1+2) is the next number. In this case, 2x3. that is 9.
@@jeromewaldemar6963 I haven't heard of PEMDAS either, mostly because I didn't grow up speaking English. Around here, the rule is "dot before dash" (because multiplication and division are written with dots). Yes, left to right, and that's also what PEMDAS seems to say, as explained in the video.
@@jeromewaldemar6963 _"I do not agree with PEMDAS. The proper way to solve is left to right."_ Well that's just plain wrong, whether you accept the priority of implicit multiplication by juxtaposition or not. Are you saying that you believe 1+3×5 = 20?
As an engineer, I agree 100% with you. Also it is up to the author of the equation to frame it with brackets etc where necessary to avoid any possible confusion of intent.
Comletely agree. If, as an engineer you write expessions like this one that are ambiguous, you are a moron. This kind of video is absolutely ridiculous.
Then you missed the slight of hand she pulled. She went from an OBELUS (which is NOT universally agreed upon with respect to its grouping or non-grouping conventions) to a slash which is literally defined in the sources she mentioned as functioning like a fraction bar. And yes, brackets should be used. And an OBELUS should NEVER be used, and it pisses me off that she equated an obelus with a slash, especially because in at least the last two sources they are clearly used in place of a fraction bar to save space.
@@eliteteamkiller319 There is no functional difference between an obelus and a solidus. Both of those division symbols do not have grouping properties. Only the vinculum or horizontal fraction bar has grouping properties.
Thank god I finally saw this. I've always said that needing to use PEMDAS is the sign of a sloppy equation. I absolutely hate those "trick" math problems on the internet.
Totally agreed, especially the part about the responsibility of the equation writer to be clear. If one uses the leaning slash “/“ as you would when typing, then make the effort to toss brackets on there to convey the grouped terms of the horizontal divide sign.
@@irrelevant_noobon paper you should clarify if 6/2 as a whole is multiplying or if 2 is and the equation is under 6 that’s just an issue with keyboards not Order of Operations
honestly the solidus (/) is really only used if an expression must be written inline - for absolute clarity the vinculum (--) is the notation to use - it's all I ever saw used from Algebra I onward
I was taught the pemdas model but I was also taught that when there was an implied multiplication like for instance 2(1+2) or xy, that there was also an implied paranthesies around that segment, ie 2(1+2) is (2*(1+2)) not 2*(1+2) and xy is (x*y) not x*y.
I'm discovering I learned math from mathematicians. Every teacher I had in middle school had their doctorate. The only bad math teacher I had after fifth grade was my geometry teacher who didn't know how to teach me. And give me enough but I somehow passed our state's very hard geometry assessment test with a 110%... A score that should have been technically impossible but because they say choose the best answer and then they had questions that actually had no correct answer I got the best answer correct so I got credit for the best answer and the question itself.
@@markprange2430 general substitution rules for integrating and differentiating by parts if f:(n) = a÷b(p+q) then let c= (p+q) therefore f: (n) = a÷bc if a=6, b=2, p=1 & q=2. a÷bc = 1 that MEANS , f:(n) = 1 The solution should be the same no matter HOW you work it. If I can work it 5 different ways to get 1, and only one way to get 9, then that way to get 9 must be wrong.
I'm a retired physicist with degrees in Mathematics and Engineering as well. It has always been a pet peeve of mine when someone is too lackadaisical to use parentheses to specify the order of operations explicitly when the order is open to multiple reasonable interpretations without them. In many applications, misinterpreting the order of operations in an expression can have dire consequences which could be avoided by simply putting in parentheses to explicitly show the order of operations. I also hate when people don't specify their units or mix unit systems, and when people don't label the axes in their graphs, or don't label multiple curves on a graph or don't provide a legend. All of these problems are far too common in the day-to-day practice of science and engineering for which paying attention to detail and being specific in technical communications are essential skills.
Well, there is some notation corner cutting, like when you write ds^2 = dx^2 + dy^2 + dz^2 . It's `understood' that ds^2 is (ds)^2 , instead of 2s ds. Same corner cutting for the denominator of the second derivative.
@@trumanburbank6899 However, ds is not two symbols like the product of d times s, it is a single symbol for the differential of s so that there is no ambiguity in the application of the exponentiation when writing ds^2 as there would be if one were intending to write the squaring of the product d*s for which ds^2 might mean (d*s)^2 or d*(s^2).
@@luminiferous1960 Actually it is the product of the operator d times the variable s. For example d(r^3) = 3r^2dr is the the product of the operator d with r^3. Also, a = dv/dt = (dv/dx)(dx/dt) = vdv/dx = (1/2) dv^2/dx . ds^2 = (ds)^2 is accepted by convention as an exception to the rule. I do agree that PEMDAS probably wasn't designed with operator algebra in mind, but it would be on par with Parenthesis.
@@trumanburbank6899 In your first example, d(r^3) = 3r^2dr, the operator is d/dr, i.e., the derivative with respect to the variable r. In the expression, d(r^3) is the differential element of r^3, not a product of d and r^3. In differential calculus, if s is a variable, then ds is the infinitesimal element or differential element of s, not an operator. The differential element is, as the name implies, elemental. Thus, ds is not a compound of d and s, i.e., it is not d times s, nor d of s. As such, no parentheses are needed around ds, and ds^2 = (ds)^2 is not an exception, and the parentheses are simply superfluous. Your example a = dv/dt = (dv/dx)(dx/dt) = vdv/dx = (1/2) dv^2/dx is an illustration of the fact that the derivative operator of one variable with respect to the other equals the differential element of that variable divided by the differential element of the other variable, and the fact that the differential elements are treated algebraically as variables. Thus, (d/dt)(v) = dv/dt, where d/dt is the derivative operator with respect to the variable t, dt is the differential element of t, v is a variable, and dv is the differential element of v. You also did not define your variables in this example, which is necessary since you use the fact that v = dx/dt when v = velocity, x = distance, and t = time. As a physicist, I recognize the equations relating acceleration a, to velocity v, distance x, and time t, but others may not.
I frame this as not a problem of PEMDAS, but of the ambiguity of inline equations, which you touched on in this. You're always following PEMDAS, but you have to take the denominator as a whole before you do the rest. Since the denominator itself is the ambiguous part, different people get different answers. Good video overall!
I agree, inline equations are sometimes interpreted this way. But, I have seen it both ways In the literature, it is important to pay attention to the context. PEMDAS is often not used in inline equations, but this changes when it isn't. I don't remember it not being used when it was a stand alone equation.
@@SpacePhys Implicit parenthesis after the division symbol is a bad convention. A math equation should not require any context. The context forms the equation, at which point it is pure math.
Yeah that's true, but the point she touches on is also true. When I see these kinds of questions being answered on reddit, yt etc, even adults who did not go deep into stem or use advanced maths they would use pemdas. So pemdas/order of operations is a tool for younger kids to have a better grasp on how to do comparatively difficult questions a bit easier. So the point she touches on is valid
If the person seeking an answer gave us the problem to be solved, they would tell you how to solve it. In the given example 2x/3y-1, PEDMAS could be used if you will equate ÷ with /. In the video, the presenter uses the Division Slash / to represent Division when doing the Order of Operations, but while reading the problem she said 2x OVER 3y-1. That would necessitate it being a fraction of 3(x) ÷ (3y-1) or 3(x) / (3y-1).
I've had this argument numerous times on facebook; it seems to come up in my feed at least three times per year. The most recent time, I had someone respond that they had been teaching math for 15 years, and that they always taught strict PEMDAS, and therefore I was wrong. My response to them was that they have been teaching it wrong for 15 years.
It's the same for me but on instagram. And it's always the replies of "You are wrong, you don't know how to do basic maths" or the "No, but PEMDAS" and I'm like: Dude, you probably don't even know why the priority of operations even exists and why is it like it is. Do not tell me to use something that you do not even understand
I tried to explain this in a twitter thread only to be bombarded by people telling me I'm an idiot who can't do "3rd grade math" despite being a maths grad, and stating it clearly both in the thread and my bio.
I think half the problem is trying to write out a statement on a single line using symbols like ÷ or /. For as long as I can remember, I always carried out multiplication by juxtaposition first. 1/2πi has always been in my eyes 1 over (2πi).
÷ and / are two different symbols entirely is the issue same as A*B and AB are different (those two are more similar though). ÷ means you are doing division of A divided by B. where / marks the right side OVER the left side, which can be broken up by a *. So A/BC and A/B*C are different, but A÷BC and A÷B*C are the same (in this case A/B*C is also the same). You almost never see A÷BC though because it's ugly and unintuitive.
@@NunoRVOliveira "s" is not a constant or a variable. It is a unit of measure. Also, according to the International Bureau of Weights & Measures (the BIPM), there is supposed to be a space between the number and the unit. So one second should be written as "1 s". Two seconds should be written as "2 s". And half a second, if written as an in-line fraction, should be written as "1/2 s". If the rules are followed, there is no confusion.
@@MilescoI would still argue it's not clear. It's not uncommon to write units in the denominator in certain fields such as Physics. The context matters.
So, I'm from croatia and never heard of pemdas. Our textbooks always had fractions written unambiguously, and now I understand why. I coldn't understand at first how you got anything but a 1 out of that equation because my brain automatically saw it as 6/[2*(1+2)]😅
Despite being from America, i also got 1, as i aways do the parentheses with any number next to it (mostly because thats also technically multiplication)
@woodendoorgarage I say you are correct. Those 2 equations are completely different and have different answers. That is how I was taught during my 7 years of honors math classes in the USA. It's exactly the same as when writing a sentence, a comma can change the entire meaning.
@@raridino600 Some people said that a number right outside parentheses should be considered as "quantity". And "quantity" binds tighter than explicit operators. So, 6÷2a is not "6 divided by 2 times a", but "6 divided by twice of a". The word "twice" obviously has no meaning if coupled with "6"; "6 divided by twice" is meaningless. In the problem's case, a = 1+2 = 3
when i learned the order of operations in algebra in eighth grade, we were told to NEVER use the ÷ and to ALWAYS use the fraction bar (vinculum) because that is a grouping symbol whereas the ÷ is not. And you can never use too many parentheses for clarification. Thank you Ms Goslee!!!!
You didn't learn the Order of Operations until 8th grade?? Wow. _______ 2(1+2) two grouping symbols (2(1+2)) two grouping symbols q 6÷(2(1+2))=1 6÷2(1+2) does not equal 1
@@suineitling there is no mathematical difference. The obelus ÷ and the solidus / are both equal signs of division and neither one serves as a grouping symbol. The solidus / is more prominent these days when writing inline infix notation mainly because the obelus ÷ is used in some countries to represent subtraction instead of division. Anther reason is that prior to 1917 some text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math that were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses when dealing with inline infix notation so the ERROR was corrected post 1917. This ERROR i.e. misuse of the obelus by the text book printing industry to try and save time and money was wrong and means that 1 is not and has never been the correct answer... 6÷2(1+2) and 6/2(1+2) are mathematically the same. Both equal 9
@@suineitling Practically there is no difference. As a rule of thumb just do what the equation tells you. 6/2(1+3). This equation says 6 OVER 2 times 1+3. Or it says divide 6 by the product of 2 times 3. Practically, if you see a division sign or slash then everything to the left is the numerator and everything to right is the denominator. Let parenthesis tell you anything different.
I remember that we were taught (70 years ago) that the horizontal line between dividend and divisor should be treated as parenthesis. This seems to make everything work
It's funny that you can actually show people how real mathematicians and people who use mathematics professionally understand these expressions and still have people arguing its wrong because it's not what their junior school math teacher told them. .
These are people who have over-reliance in rules. They see the rules as something absolute and not conventions. Ask them why is multiplication done before before addition and they can only say PEDMAS. The answer is simply. If you buy 5 apples à 2 € and 7 oranges à 3 € how much does it cost. It is nice if you can write 5*2 + 7*3.
I did not see any example showing ÷ symbol. Either because they don't know how to write/type the symbol or real mathematicians or people who use mathematics professionally don't use the ÷ symbol. The examples showed if there is / that means group numerator and then denominator. 6 ÷ 2(1+2) ≠ 6 / 2(1+2)
The video showed nothing. The original expression contained ÷, which is the source of ambiguity, and then immediately pivoted to using the fraction bar. They are not the same thing.
And then there are English teachers who say, "Never begin a sentence with a conjunction." But good writers aren't held to such silly limitations, are they?
@@ILoveCunnilingus that symbol ÷ (called the obelus) is no longer of use according to ISO 80000-2, only "/" is used (a consequence of it being used in most programming languages and of keyboards not having the obelus symbol I think), and it was rarely used by mathematicians and scientists either in the first place, it was mostly used in some printed books). But ÷ and "/" were always the same thing. 1÷ab was also 1/(a*b).
My initial instinct was to distribute the leading 2 through the expression in the parentheses (either before or after resolving the expression in parentheses), which gets 1 because the parentheses are still intact after doing so. I always interpret the number before parentheses as being "bonded" to the parentheses unless there is a multiplication symbol separating them.
@@nekogod I was told to only omit multiplication symbols if the values belong together Why are you omitting symbols if there is no relation between the values?
That's multiplication by juxtaposition. It's just tha juxtaposition also applies to numbers and not just variables. I always interpret 2(3) like 2x (but x=3).
Thank you for being a glimmer of sanity against the winds of smug RUclips math PEMDAS zealots ignoring implicit conventions pervasive among people actually using math for useful things like science, engineering, etc.
Simply put, don't rely on your systems algorithm to work the way you expect. Control the outcome and reduce the probability for logical errors by utilizing parenthesis whenever ambiguity in the order of operation may exist.
This was the way I was taught PEMDAS by my math teachers, at least starting in 7th grade algebra (we didn't use PEMDAS before that), that multiplication came before division. I was very confused when I started seeing people online trying to solve these math equations and they were doing multiplication and division in the order they came, and not giving multiplication priority. I should also note, that none of them wrote in-line equations when it came to division. It's just not how we were taught, but then again starting in 6th/7th grade is when we started algebra and then moved to geometry and trig in high school.
We were taught BODMAS (as you can see D comes before M) but still 6 ÷ 2(1+2) is obvious to do the juxtaposed multiplication first. Mainly because if it wasn't meant to be first then they would have written it in a way that doesn't suggest that it is first.
Same here. Which makes it seem kinda weird when she says PEMDAS is wrong and yet the right rule follows PEMDAS exactly as it was taught to me. Makes me think the thing of doing multiplication and division in the order written must be a newer thing I think and PEMDAS the way we learned it was older? So it's not so much that PEMDAS is wrong, but that it's being taught to mean different things.
I've learned it as implicit vs explicit multiplication. Implicit meaning the multiplier is a common factor in an expression and should be evaluated at the same time with its expression, whilst the explicit multiplication must be written out with an explicit multiplication symbol of " x " or " * ". And for cases with 2x / 3y -1 in order to include the "-1" into a single expression you'd use parentheses like 2x / (3y -1).
Unfortunately there is no central authority to make official rules. This implicit vs explicit rule that a lot of people try to insert into these kinds of meme internet problems don't work. If there was a difference between implicit and explicit multiplication then 2x and 2*x would be different. Likewise 2(1+2) would be different to 2*(1+2). And if they were different then contracting 2*x into 2x at any point would be a terrible idea and we would have a 'rule' not to do that etc.. Our 'rules' have ambiguity, they always have and likely always will because the 'rules' are often not rules and are actually just conventions that some people make up to try to make it work better. Even the video called these 'unofficial rules' that engineers and stuff use when writing and interpreting formulae. The reason we have parenthesis and why that is evaluated first is because this ambiguity was a known problem and the parenthesis was a way to address it, but it only works when the person writing the equation uses them correctly. If a person writing the original doesn't see how it could be interpreted differently then they might not be clear enough when writing it down, which can lead to different people getting different answers. But the only person who knows the 'correct' way of evaluating the formula is the person who originally wrote it. No amount of arguing on the internet about what the correct order of operations is will get you the 'only and only absolutely correct and everyone else is wrong' answer.
@@marian-gabriel9518 Of course. It is just a pretty common thing that comes up on these kinds of videos. Yours was the first comment that I came across which mentions different types of multiplication with different rules etc.. so I added my two cents. It is a public forum after all. I was in no way trying to straw man an argument or anything and if that was the impression I gave then I can only apologise.
@@Subjagator _"If there was a difference between implicit and explicit multiplication then 2x and 2*x would be different."_ And it is! _"Likewise 2(1+2) would be different to 2*(1+2)."_ And it is! 6 ÷ 2(1+2) = 1. 6 ÷ 2 × (1+2) = 9. The operator (the multiplication sign in these examples) serves to separate the coefficient from its variable so that it doesn't get the priority that implied multiplication gets. _"And if they were different, then contracting 2*x into 2x at any point would be a terrible idea and we would have a 'rule' not to do that, etc."_ In real life, engineers and, well, just about everyone I've ever come across, rarely if ever use actual multiplication signs, so the idea of "contracting 2*x into 2x" isn't an issue. If it were ever necessary to insert a multiplication sign to avoid the priority that implicit multiplication by juxtaposition creates (e.g. "2 × x" or "2 · x"), most likely the person writing the expression would use parentheses in that case. But contrary to your assertion, the universal and ancient convention of the priority of implicit multiplication by juxtaposition is so, well, universal and ancient, I believe it can fairly be considered a "rule". Indeed, the creator of this video has referenced two authoritative sources that say so. I don't think there's a mathematician or engineer or similar professional in the world who would say that a/bc is anything other than "a" divided by the product of b and c.
As others have commented - ALWAYS be explicit in how you write equations so that there is no need for interpretation. Use brackets & parentheses, write clear fractions, etc. As an example, put parentheses around exponents with multiple terms. Parentheses may also be necessary for proper order of (multiply - divide) or (add - subtract).
I somewhat disagree. Overusing brackets can make an equation a lot less readable, which is why in papers and textbooks in fields utilising a lot of maths you will find xy/zw rather than something like (xy)/(zw). As far as the maths itself is concerned there is no real "proper order". So long as everyone agrees with the notations and conventions used in a field, it's all good.
@@adamlindstrom5750 there is no such thing as "overusing" brackets. you either need them or not. they solve a problem. left to right is the proper order is brackets are not used.
@@adamlindstrom5750 unfortunately, the key problem is no one can agree! Everyone was hot different methods, and I don’t think we should call anyone method wrong. I may have been taught pemdas, but people might’ve been taught different. The issue comes with the fact that since people aren’t at the same things, we need to be more clear about what the problem wants using parentheses/brackets/braces/and an actual fraction, not just a slash cause that gets confusing too. Because the slash was sometimes taught to people as a division, but for other people it was taught as a fraction (division using grouping).
@@adamlindstrom5750 You're essentially saying it saves time by not writing extra brackets. However, what if you factor a fraction? Let's say you have 20 and want to factor 1/2 from it. You would now have to write (1/2)40, instead of 1/2(40). Either way you do it it'll require the use of brackets somewhere, so this doesn't save any time, or readability.
@@mariuspuiu9555 brackets are not needed when you have juxtaposition, so using them would be an overuse. If you use brackets to communicate to people who were not taught the correct order, you should also consider that your audience might not understand brackets.
Exactly. Anybody's engineering boss would have their red pen all over that submission. If your submitted equation can be interpreted >1 way, you're doing it wrong.
@@Ignoranceisbliss-i2e I'm a senior engineer and have been doing that job for 24 years, I think I'll manage. Disagreeing with you makes somebody an automatic moron? Nothing arrogant about that. You're only proving my point.
It's definitely a consistency issue across the board, but it is often related to presentation, not function. "Context is important", but beyond context, there still should be some standardization throughout. What you learn in primary school, what you learn in college, and what you use in the fields of math, science, and engineering for the rest of your life should all be the same so as to remove this ambiguity. In these fields, it is often critical, because a simple misunderstanding or misrepresentation of context can cause someone's pacemaker to fail, or send a spacecraft smashing into a planet uncontrolled. That said, PEMDAS is still very common in computer programming languages, because they have to explicitly and uniformly handle the translation of syntax to function. Since the vast majority of languages use simple human-readable text to represent the source code, they have to have a common order of evaluation, which often includes PEMDAS as a basis. The printing resource excuse may still be necessary for dead tree publishing, but a great majority of knowledge is moving or has moved to digital, so that excuse is now pretty much obsolete. We have graphical expression languages now, so we can properly display formulas and do away with the problematic shortcuts.
in the sciences the brackets that should be there are removed for laziness and the assumption that everyone who reads it should know that it is implied.
parentheses, exponentiation, adjacency, (multiplication | division), (addition | subtraction ) isn't even really different than what we generally teach primary school students. merely an extension thereof. primary school students, if my memory serves, simply never use adjacency to mean multiplication. Elementary students always use operators between terms. Adjacency doesn't hit till algebra for students since it makes no sense till you have variables. The algebra teachers merely should mention that algebra extends PEMDAS with the knowledge that the adjacency of coefficients binds tighter than operators, while any exponentiation, which isn't an operator, will still apply only to the exponentiated term.
Computer languages don't use juxtaposition because it changes the name of a variable. If you have variables x and y, then xy is not x*y, it is a compiler error caused by an attempt to reference the non-existent variable "xy".
Something that I picked up over the years which is not obviously stated or taught very often is that the fraction line actually implicitly groups both sides whereas normal division “÷” does not have that grouping property. The order of operations are completely consistent with these textbooks since they use the fraction line for their equations.
@@drednaught608 the text book printing industry tried to push the obelus as a grouping symbol to be used like the vinculum because the vinculum took up to much vertical page space for the printing methods at that time...BUT this was in direct conflict with the Order of Operations just as parenthetical implicit multiplication is in direct conflict with the Order of Operations. The obelus ÷ and the solidus / are equal signs of division and nothing more... Parenthetical implicit multiplication does NOT have priority over division despite the misquided and confused notion of many people.... Math is not based on opinion or popularity it's based on rules and the rules do not support parenthetical implicit multiplication having a higher precedence...
As one that taught engineers, we did teach BEDMAS as a rule of thumb. But we also put the onus on the student to make sure it was clear to the guy giving the marks what your math meant. Like I read below, you can never have too many brackets to group terms and processes in your solution.
@@FrostSpike machine code and assembly language.. Fast and efficient. no brackets.. lol just a pain in the ass professor that made sure there were no mistakes so my engineers understood everyone has to understand there work./
The 2 is linked with the parentheses, and is reinforced by the distributive property. The 2 is still connected even after adding. This is proven by distributing it out which forces the expression to become 6÷[(2×1)+(2×2)]. In the end, using proper notation when developing problems is important. Also knowing ir stating the conditions placed on expressions for the format in use. To quote Pirates of the Caribbean - "It is more what you'd call guidelines than actual rules" 😊
*This* 100%. The expression 2(3) has not yet been resolved as a parenthetical operation, and is not equal to 2*(3). Just because people misinterpret the rules, does not in any way make the rules wrong... just improperly taught/understood. Ignorance of the Law is no excuse.
have seen a lot of this misinterpretation of PEMDAS and was extremely shocked to find out that even Google algorithms use the misinterpretation Its probably necessary to overload any equation or expression with parentheses to avoid ambiguity Strangely enough when I learnt BODMAS i learnt that the O stood for both Order (Exponetial) as well as Of which meant evaluate completely any expression within or adjacent to parenthesis so in this case 2(3) has to be evaluated before the division. Reading the expression would be 6 divided by 2 of 3. 2 of 3 means you have two sets of 3 but these guys have turned this completely around and decided that this statement should read 6 divided by 2 times 3 which is not quite the same thing
@@drift_ah1518Yep. This is why I used the Distributive Property in my example. This is an actual mathematical "law" where as PEDMAS is only a procedural ranking based on a set of conditions and in this case based on the "grammar" of writing it in one line rather than long form on paper.
@@drift_ah1518 adding in additional (and needlessly complicated and completely irrelevant) parentheses into a mathematical expression, only to hold the hand of people who never understood the concept loses some of the elegance math is capable of. When leading Trolls and Ogres through a Magical Forest, they are bound to mash some flowers into the mud. Better to teach them the *right* way to do it, as some of these RUclipsrs try to, with the assistance of others. And maybe fix the education systems that pass people ho don't actually know what they ere taught.
Brilliant! As a math tutor for elementary, high school, and college students, I've been fighting against the tide of PEMDAS forever. Allowing the use of the ÷ and / symbols has been the greatest detriment! Thanks again for your presentation here!
@@drinkoldcokeCommon Core is what educators came up with when they analyzed what all the countries that had high math scores and successful math programs, had in common.
I think all calculators in schools should be required to feature something like Natural V.P.A.M, Writeview, Textbook Display, or MathPrint. These are pretty-print input methods from different manufacturers for equations and expressions which allow for a more natural visualization of the syntax by utilizing a dot-matrix display or other high resolution display to allow the user to select the function and navigate empty fields with the directional keys to fill it in, automatically adjusting the visual representation as necessary. This allows students to type in an expression or equation the way they would see it in their textbook or how they would write it instead of needing to figure out how to write the in-line notation so the syntax is unambiguous. My favorite will always be the Casio FX-115ES series of calculators due to their impressively cheap price and eminently versatile feature set. You can 100% use one of those to get a Bachelors in a math field, no problem.
In engineering we'd read it as 1, which is why I was a bit tilted at the start. PEMDAS does apply, but this question is not framed properly, if you input this into your calculator, it will auto correct the input to 6÷(2(3)) which gives 1. So if the question is framed incorrectly, it makes it hard to understand what to do. In my years of study I came accross this issue maybe once or twice, it's very rare that someone does not frame a question in such a way that makes it easily understandable, especially since these days we calculate our equations using Matlab, so naturally you need to us parenthese very precise for the computer to know what to do.
Is it a casio calculator? Because I know casios do that autocorrect thing, but it'd be very interesting if other manufacturers are adopting that behaviour.
@@THaWoM All proper calculators follow the PEMDAS (etc.) rules these days - these rules are built into them, and they use a stack to make it possible to delay intermediary results until they are needed. What you have to watch out for is old and / or very cheap calculators without these features, which can only handle two operanda and one operation at any one time.
Yeah, I'm not very well-versed in math but wouldn't PEMDAS literally make the answer 1? Parentheses first (3), then Multiplication (6), then Division (1). I don't see how PEMDAS could possibly lead to the answer being 9. It seems like willful wrongness.
Thank you for figuring this out. My integral calculus instructor, who also taught nuclear engineering labs, made this basic algebra. He used to say “Calculus is easy. It’s the algebra that trips most people up”.
It's not that people struggle with algebra. It's that there are inconsistent conventions. Writing something in an ambiguous manner and blaming others for misinterpreting it is the writers fault.
I'm 55 and Canadian and was taught BEDMAS, but like other commenters was always instructed that parentheses/brackets were not solved until they were eliminated, thus the order (in this case) would be what's inside the bracket, then the multiplier next to the bracket, then the divisor, leaving the answer of 1. But I can definitely see where it could go wrong with other equations laid out differently....
The implied multiplication is not connected to the brackets. Sure brackets allow its use: you can write 3(2) but not 32 to mean 3*2. You also can use implied multiplication with variables. If you had x=3 and did 6/2x you still would get the difference even though parenthesis are nowhere.
@@okaro6595 you actually can't write 3(2), implicit multiplication must always contain a symbol (both from usage, as you can see in the examples she provides, there are always symbols/variables in implicit multiplications and also from the ISO 80000-2 which defines the rules for mathematical and scientific writings). This internet meme is just improper writing and the answer should be "no value" :). But of course if somehow you *have* to decide what value it should have, any mathematician would reluctantly interpret the 2(2+3) as an implicit multiplication and answer 1.
As a physicist that's what I was taught as well. Later, I think in high school, it was explained that juxtaposition meant the juxtaposed terms were to be thought of essentially as a single term, essentially an implied parenthesis. The other thing I learned that to avoid confusion, use appropriate parentheses when needed rather than order of operations to ensure proper grouping. One other thing I learned in life, this time from a linguist, is that many of the "rules" taught in school, not just PEMDAS, are merely teaching tools to make the job easier for teaching elementary students and weren't meant to be used for higher levels.
Not to put a fine point on it, but maybe lazy instruction at the starting level is part of why our education system sucks. It's easy to learn something when you're a child but unlearning something or relearning something is very difficult. IMO, math equations are a precise set of instructions for calculations and should never be a question about it. As you said, use appropriate parentheses.
Juxtaposition does NOT indicate grouping! Juxtaposition simply indicates multiplication between the two quatities being juxtaposed, as if a multiplication symbol (e.g. "*") had been explicitly written between them. So 3x means "3*x" not "(3x)" and a/bc means a÷b*c = (a/b)*c not "a/(b*c). However, the horizontal fraction line, called a "vinculum," DOES indicate grouping, both above and below the horizontal line. Thus, mn/ab simply means m*n÷a*b = m(n/a)b. This is NOT the same thing as writing a horizontal fraction line (vinculum) with mn in the numerator and ab in the denominator, which means (mn)/(ab). The essential misunderstanding is that the slash symbol used for division in command line interfaces is NOT equivalent of the horizontal fraction line (vinculum) symbol commonly used when students do math with paper and pencil: The former is NOT a grouping symbol; the latter IS a grouping symbol. It was the widespread adoption of command line computer interfaces - wherein vinculums are not available as symbols (as they are when doing math with paper and pencil) - that screwed everything up. Graphing calculators just solved this problem, recently. Newer editions of the Ti-84 graphing calculator operation system, using "Math Print," now allow the on-screen display of vinculum symbols to indicate division (and fractions) exactly as students have traditionally done using paper and pencil. Doing so has completely removed the ambiguity, confusion, and errors caused by non-equivalent use of "/" in place of a vinculum to indicate division. en.wikipedia.org/wiki/Vinculum_(symbol)
@@chrisborland4972 While that may be the case in the area you're from or the circles you frequent, that is not necessarily the case everywhere, as the multiple examples provided in this video illustrate. Among multiple professionals, the usage is obviously different from the one you describe. I do personally agree with your interpretation, but the only proper answer is: the usage of the symbols is not uniquely determined, and therefore the "instruction" is a priori ambiguous. As with any text, the meaning is contextual, and you must learn to become acquainted with the author's meaning from the surrounding text. But, as you mention, an author who is not certain of their readers' abilities should therefore use parentheses or the horizontal fraction line to avoid ambiguities of interpretation and thus confusion for uninitiated readers (such as elementary or secondary students).
@Flutesrock9800 Thanks for your thoughts. However, I strongly disagree, and I think I can prove you're in error. Without a universal standard as precisely what mathematical symbols do and do not mean, how are we to cooperate on anything mathematical? How are we to relate mathematical ideas and results? How are we to understand each other and collaborate? There would simply be no way to do so. In fact, mathematical symbols (like “/“) and conventions (like juxtaposition and the order of arithmetic operations) ARE standardized - and those who do not conform to these standard and well-established usages of math symbols and conventions are just doing their work incorrectly - whether or not they happen to be “professionals.” The simplest way to test this is to use any computer programming language to assign values to the variables in the expressions I listed, ask the computer to evaluate those expressions with the given values, and see what value the computer returns. In each case, the computer will return the value that I have stated is the correct value. For instance: If m=2, n=3, a=4, and b=5, mn/ab (or the equivalent m*n/a*b) will be calculated by any computer language as m*(a/b)*b and the value returned will be 7.5. It will NOT be calculated as (mn)/(ab), which would instead return 0.3. If it weren't for agreed-upon standards defining correct usage, it would be impossible for different computing machines programmed according to different "usages" of mathematical symbology to share calculations productively (since each would return different values under identical conditions), and the information age would never have arrived. Those who do not employ standard usage of symbols and conventions are being unrigorous, unprofessional (even if they happen to be “professionals”), and certainly unhelpful. At best, non-standard usage exposes a habit of sloppiness and laziness on the part of the writer. Mathematicians and computer scientists have decided upon standard conventions and clear definitions for mathematical symbols. These standards define correct usage. All other usages are incorrect, as demonstrated by the simple programming experiment I’ve suggested above. The verdict is in. There is one - and only one - correct way to employ mathematical symbols and conventions. No matter what “multiple professionals” may or may not do, “alternative” usages of mathematical symbols and conventions are just wrong. Full stop.
We did BODMAS in school but I think I remember learning to always aim to get rid of brackets before solving the whole problem, which in this instance gives the correct answer. I don't think I've ever heard the idea that we should deal with multiplication or division in order of left to right.
You have to deal with it from left to right with DM because we read from left to read and if you don’t then you end up with different answers, this is not the case for AS because the order doesn’t change the overall answer, it’s the same regardless of whether you do A first or S first.
I learned multiplication and division are in the same hierarchy so you do left to right, not multiplication then division, just left to right. think about 4/6*3, it wouldn't be 4/18, it would be 2, so the rules do infact matter. also it was only for 6/x, where x is 2(1+2), and the new rule states "when slashing a fraction, this is the accepted order of operations: raising to a power, multiplication, division, addition and subtraction" it's far different than the way we would do other equations without slashing fractions because pemdas/bodmas always do parenthesis, exponents, multiplication and division, addition and subtraction. left to right also only apply for multiplication and division, and addition and subtraction.
So with 6 ÷ 2(1+2) we get rid of brackets (parentheses) to get. 6 + 21 + 2 = ? Correct? Which, accordung to, PEDMAS, would give 6 ÷ 21 = 0.2857, then + 2 = 2.2857
@@aspenrebel no wrong, you are not using BODMAS/PEDMAS, you need to follow the order. I have no idea where you get 6 / 21 + 2 from 6 / 2(1+2) = 6 / 2(3) = 3 x (3) = 9
I was taught PEMDAS but I was also taught to multiply by juxtaposition before moving on to division. For reference I was in elementary school in the 1990s. When 6÷2(1+2) came around I thought the world had gone insane or that I was being gaslit. Thank you for making this video.
In 1979 I was taught this in 7th grade and I guess I had a good teacher, for it was ever so simple! First 'in between the brackets', then what's 'attached to the brackets'. Then the rest according to EMDAS (we of course already used EMDAS within the 'P' part of the brackets too).
@@leighz1962 Incorrect. Guess you never heard of the Distributive Law. The Distributive LAW states: A(B+C) = (AB + AC). This is actually one of the fundamental laws of Math. So the facts are the number attached to the parenthesis should be moved into the parenthesis since that is actually a LAW in Math.
It's 100% NOT a law. It's called the distributive PROPERTY of multiplication and its no more powerful than the operation next to it. There is no division or subtraction really. All of it can be reduced to multiplication and addition (of inverses) so that they are closed under the respective operations. I'm actually a little annoyed that this presenter referenced the laziness of book writers to cite that Pemdas means nothing. There IS an order of operations and it's not helpful to have something like this go viral.
Except 6÷2(1+2) would = 9. If you wanted the multiply first you'd simply add the bracket or right it as a fraction properly. Her whole explanation is bullshit. Paraphrased as "people don't write things properly so order of operations shouldn't apply". Theirs absolutely no ambiguity with pemdas. People who argue ambiguity just don't follow the order. And people that write the equation and claim to want a different order should write it properly
I really think this whole confusion comes from typewriters, keyboards, etc. not being able to type out the actual notation you would see in a textbook. I think after seeing it written in the textbooks so many times, I just kind of knew what they meant when they tried to type it out in a paragraph. My TI-89 Calculator does Pretty Print notation so you can better understand what is going on.
I agree completely. I graduated high school in 2007 in the increasingly digital age, all our tests and exams were written on PC and printed out but everything in math was formatted correctly to avoid this problem such as 1 ------ = y 2x but I always understood 1/2x=y to be the same thing because otherwise it would be written as x/2=y
In mathematics you avoid things like "a/bc". Not a correct way of writing down a mathematical expression! This is creating confusion because in the early days printing of the correct way was often not possible and took too much space. So pemdas etc. is mathematical correct and so you write a ------ b c
Exactly. You have to follow what is meant by using the parentheses (brackets as we called them) in the first place. I went to school in the 1950s and we were taught that a number touching the brackets and the numbers inside the brackets were all part of a function which had to be carried out as a separate operation. And we were taught to do that first before doing anything further. So a(b+d) was (ab+ad) and not to be translated as a X (b+d) in a longer expression. This is logical and avoids confusion.
@@anchorskid in the UK we use BODMAS - Brackets, Orders, Division, Multiplication, Addition, Subtraction. So they're definitely brackets here, and the wikipedia article for Bracket agrees. Even though BODMAS specifies division before multiplication, and juxtaposition is not included, we always multiplied juxtaposed symbols as a priority.
So glad you posted this as I think the same. I remember reading Donald Knuth’s argument that multiplication written as juxtaposition binds more tightly than PEMDAS suggests, whereas expressions using multiplicatio and division signs would tend to proceed left to right. I remember presenting a view about the viral problem, playing devil’s advocate, and seeing both sides but preferring to the the juxtaposition multiplication first, and I had so many rude comments, one person telling me I should hand back my maths degree certificate as I didn’t deserve it!
@@pandoorloki1232 the textbook to one his programs, METAFONT. It’s basically a programming language for building fonts, but the rules for building expressions include multiplication by juxtaposition, not generally seen in either typical high level languages or Excel. And multiplication by juxtaposition binds more tightly than multiplication by *. It’s in his METAFONTbook.
Mathematician: mn is mathematically equivalent to m*n! Programmer: That's a totally different variable. You could do mn = m*n. Mathematician: That's illegal! Programmer: You telling me dy/dx is three different variables?
As a programmer, I apologize for our abuse of the equals sign. As annoying as it (sort of) is to write := or similar for assignment in the languages that use it, it's probably a saner choice.
Absolutely. I've been an engineer for 50 years and your presentation exactly matches what I, and my colleagues, have been doing during my entire university studies and engineering career. If in there is any possibility of misinterpretation / doubt, parentheses (brackets) are utilised.
@@aspenrebel The Guiness Book of World Records used to list the most expensive typographic error as a multi-hundred million-dollar launch that failed for similar reasons. Parentheses are cheaper.
@@cliffordschaffer5289 yup!! Did I ask this question, somewhere? What is 6 ÷ ab? Where a=3 and b=2. The Answer: 1. But the "woke", New World Math, Globalist will tell you it is 4. "ab" is one complete term. Just as "2(1+2)" is one complete term. I cannot see anyway to write 6 ÷ 2(1+2) =, under the PEDMAS-er's Rules, to make the answer 1. Yet under centuries old pure math, the answer is 1.
I agree that in professional mathematics and physics papers, that's the typical usage. On the other hand, the typical usage in *programming languages* that accept algebraic expressions is really strict PEMDAS (a convention probably inherited from FORTRAN many years ago). Scientific calculators are all over the place. And all of these expressions are using a slash / , which suggests a fraction bar, rather than the primary-school division sign ÷ . I tend to think that if you're inlining a mathematical expression you should probably add parentheses to disambiguate these things, but often papers don't.
Different languages have different rules of precedence for example C vs. Pascal. In anyway PEMDAS does not even begin to address the complexity to operations in programming languages.-
FORTRAN and other languages supporting mathematical expressions in a similar way will not accept an implicit multiplier anywhere. You must put in a multiplication operator (typically *) or the example will not be accepted. Once you do, it becomes easy to see how the expression is evaluated left to right with a "PEMDAS" precedence. If you intend different grouping, you need to show it with additional parentheses. FORTRAN furnished a method of stating any explicit calculation unambiguously, which made the engineers happy. It did not provide for a mix of implicit and explicit calculation operators and would give you an error diagnostic if you tried to compile that into a program, so did not make lazy algebraists (intending a nod to blackboard style notation) happy. Having a masters degree in computer science, my take on this is "What's the big fuss. Write what you mean, using the rules of the language, and if someone gives you an incomplete expression, tell them so and don't proceed without clarification." This is not being passive aggressive or obstructionist. This is being a computer, a machine, which promises nothing but to be a machine.
A well formatted paper will arrange to format a blackboard style equation when intended, even if it has to incorporate extra space to do so. The viral example is simply an unhappy mix of the two.
PEMDAS is a subset of PEJMDAS. Computers use PEJMDAS just like all the professional bodies she cited, but because they can't use juxtaposition PEMDAS works too.
I learned that you were supposed to keep multiplication values where the multiplication operator isn't visible together as if they were a whole like in "1/2x". Here "2x" is grouped together by a hidden multiplication operator and should not be dealt with separately. This would be the same as "1/(2*x)". So you first multiply "2" by "x", then you divive "1" by the value you just got (the result of "2*x".
this is correct. Am just shocked but not entirely surprised that Google algoriths also follow the wrong interpretation of PEMDAS as "explained" in the video. I will never trust Google or Android calculators and will have to insert unnecessary parentheses to avoid wrong answers
Mathematicians don't write "1/2x" like that for precisely that reason. It's not ambiguous, but it's too easy to misinterpret. As written you have 1/2 of x, not the reciprocal of 2x, despite what you may have been taughtt.
@@herbie_the_hillbillie_goat Technically, if you can interpret it a different way, a different interpretation exists. Whether it's misinterpreted can only be determined by the author of the problem, perhaps they're the ones who miswrote it. I find 1/(2x) to be a more natural representation, because implicit multiplication seems like scaling. A scaled value is still a single value, not a multiplication. It's a "thing that's twice the size of x", but not "two times x". If it was the latter, it should be explicitly written as "2*x". But then again, everyone uses the fraction representation to avoid misinterpretations. It's ambiguous, neither is right or wrong, it depends on consensus and there is no such thing in this case.
@@Ignas_ The debate around the expression 6÷2(1+2) often boils down to the interpretation of implicit multiplication. According to the widely accepted oerder of operations, multiplication and division should be performed from left to right. Therefore, 6÷2(1+2) would be evaluated as 3(1+2), which equals 9. I understand the temptation to view implicit multiplication as a form of scaling. You suggest that 2x feels like a single, scaled unit rather than 'two times x.' However, this subjective interpretation doesn't override the existing mathematical framework. In this framework, implicit multiplication is no different from explicit multiplication: 2x and 2*x are the same. Moreover, you question why 2 should be grouped with x but not 1/2. In both cases, the numbers are being used to scale x. From a mathematical standpoint, there's no reason to treat 2 and 1/2 differently. They are both numbers that scale x, whether implicitly or explicitly. It's true that ambiguity can be resolved by using fraction representation or parentheses, but the need for such workarounds highlights a lack of clarity rather than a genuine ambiguity in the mathematical rules themselves.
I'm an engineering student, and it is almost always beneficial in the field to represent division with a fraction. We use fractions A LOT in calculations, so using it as a division symbol is kinda killing two birds with one stone most of the time. There were many times in calculus where I needed to cancel out terms in a fraction to make an equation more doable. Having the numerator be physically on top of the denominator lets me better see if there is something that I can cancel out. There are also advanced mathematical concepts like Laplace Transforms that are, most of the time, completely reliant on fractions to rewrite tough equations into something more doable. In addition, using a fraction eliminates the need to put more parentheses in an equation to clarify what a division symbol is dividing from what. It just makes things more clear in my opinion.
not the best thing to dumb it down. Example 5:5+5=? vs 5/5+5=? If dumbed down to "use fractions" one can interpret as 5 divided by the result of 5+5 when 5 divided by 5 then add 5 needed.
Absolutely. Sometimes we try to write, or type equations out on one line, and it confounds things. I definitely found writing the equation out long form on multiple lines for fractions made this make much more sense.
@@AudriusN For the question 5/5+5=?, the fraction would be with the 5 in the numerator and the other 5 in the denominator, and then the +5 would be completely outside of the fraction. That or if you did want to do 5/(5+5)=?, putting the 5+5 in the denominator would also send the message that you wanted the 5+5 to be done first before the division. It is super useful and concise to use fractions in engineering math problems because sometimes you need to divide a number by the result of a bunch of variables that need to be added and/or subtracted together and it's nice to show it on 2 lines, like 10/(1+b(60/a)).
I’m an engineer and I would say the answer is 1. I think the simplest way of explaining it is that multiplication by juxtaposition comes first. In this case the 2 is also effectively part of the bracketed term which should be executed first under Bodmas / Pedmas. I would argue that under bodmas the precedent is of the arithmetic symbols used in primary education, when juxtaposition is not typically taught. When juxtaposition is used there is either no symbol or as in this case there is a bracket which you could say comes first in bodmas. That is how most people would do it in algebra as well (imagine substituting 1 and 2 with x and y). Which gives: 6 / 2(1+2) = 6 / (2+4) = 6 / 6 = 1
I’m a software engineer. We don’t have that problem in programming. To avoid confusion we use parenthesis. But new programmers do have that problem. That’s why they should start teaching in schools the importance of using parenthesis in math expressions.
This is why I hate PEMDAS. Such an awful thing that in order to have multiple factors in a denominator we need parens. If it were up to me, I would just go with prefix or postfix notation. If, for example, you have an expression that is definitely the sum of two smaller expressions, then prefix is good because you can just slap down a plus sign before jotting down the smaller expressions. If you have a bunch of small expressions that you’ll be somehow combining into one large expression, postfix allows you to jot down the small expressions first and then tie them up with the necessary plus/minus/multiply/divide symbols at the end. Prefix is also natural for programming because we call functions in prefix style. However, I’m also partial to postfix because postfix math naturally evaluates from left to right while prefix evaluates naturally from right to left. To explain evaluation direction, if you tried to evaluate prefix notation from the left, you would need a stack of operators and operands as you scan right, waiting until there’s enough operands ahead of an operator to perform an operation. If you evaluate prefix from right to left, you only maintain a stack of operands, and every time you run into an operator you immediately perform the operation, as if there aren’t enough operands available for the operation, you just have an incomplete expression. So the evaluation order for post fix makes more sense if you want to evaluate left to right. If you’re lisp, you’ll still need parentheses because of functions with an arbitrary number of args, but if you require functions to have a fixed number of args, you don’t need parens.
when I deal with math equations in programming, I will do it across several lines. Each line doing an assignment that will be used in subsequent lines. I take a different approach in Hastell. Within each line of the calculations, I try to make everything as unambiguous and succinct as possible. Indeed, to avoid using too many parens, I may resort to throwing in a few extra lines.
@@J7Handle I also like postfix in a calculator. But really, you're explicitly defining the order of evaluation : you're doing the parsing mentally. It's not the use of postfix that gives the right answer, it's that you're parsing the expression rather than the calculator.
@@theelmonk I can trivially write code to evaluate a postfix expression. Read from left to right, putting operands on a stack. If you encounter an operator instead of an operand, apply the operator to the operands on top of the stack (note that the operator must have a fixed number of operands, so the binary minus sign and unary negative sign can't use the same character), then replace the operands on the stack with the result of the operation. So 1 2 3 + - would evaluate as 1, 2, 3, then + takes the 2 and 3 and replaces them with 5, so 1 5 - follows, then the - takes the 1 and 5 and makes -4. Prefix is evaluated in the opposite direction, that is, starting from the right side of the expression, which is a little weird, but is standard for functions in both math and programming languages (we do f(x) and print("Hello world!") instead of (x)f or ("Hello world!")print). This is because reading prefix from left to right, while a bad idea from an evaluation perspective, is really good for giving us the big picture of the expression rather than having to dig through the nitty gritty details first. Infix is awful in so many ways. It only works when you have the superscript exponents, multiplication by juxtaposition, proper divisions that physically put the numerator and denominator over each other, and subscripts to allow more variable names without running out of symbols to use (since juxtaposition kind of prevents variables with more than one character in their name). In simple text with none of those features, prefix and postfix are absolutely superior to infix.
The video nicely documents that there are various interpretations. Whether some person has learned it one way or like as missionary wants to convince others that their way is the right one does not change the fact there are different answers which can be given and justified and that even programming languages disagree. This fact can not be changed and it can be experimentally tested that there are disagreements. Already watching a few youtube videos for example does this already. There are cases, where even different models of calculators gave different answers to such PEMDAS problems like 1/2x (the TI-82 disagrees with TI-83). Florian Cajori, still the best authority in this matters wrote in his book on mathematical notation: "If an arithmetical or algebraic terms contains both division and multiplication signs, there is at present no agreement as to which sign shall be used first. It is best to avoid such expressions." Already the comments in this video show that this assessment has not changed and that Cajori's advise should still be followed.
I've been thinking of doing a follow-up video on the various ways that calculators and computer algebra programs deal with expressions like this, because the variety of rules they use is quite interesting. Some don't allow implicit multiplication, some put it at the same precedence as explicit multiplication, some put it higher, and some give implicit multiplication a higher precedence in some situations but not others (wolfram alpha for instance treats 1/2x differently to 1/xy). I've found that the Casio and HP calculators available to me locally evaluate implicit multiplication before division, but the TI one doesn't, and I'd be interested to find out why TI switched over given that they state on their website that implied multiplication has a higher priority when writing on paper (see epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 ). I'm familiar with Cajori's statement, but note that he was only talking about expressions that involve the × symbol, not ones involving implied multiplication by juxtaposition. I know your stance is that it's ambiguous and I'm forced to agree with you because there is debate around the answer to 6/2(1+2), but the only reason there is ambiguity is that primary school teachers are pushing an oversimplified version of the rules widely used by those in STEM, and the real rules are rarely stated explicitly. We don't have to stop at asking "How is it interpreted?" but can also ask "How should it be interpreted?" and then adjust teacher training and textbooks accordingly.
@Oliver Knill I have no problem agreeing with you that Florian Cajori is the best authority in the HISTORY of Mathematical Notations. Too bad he is not the highest authority in setting standards in mathematics in the USA where the confusion as to whether implied multiplication should be given a higher priority in calculation than division has become an epidemic. As far as I know in the tradition of science and engineering calculations, implied multiplication by juxtaposition has always been given a higher priority than division. That means in the expression 6÷2(2+1), 2(2+1) should be considered as a product with the factors of 2 and 3. That also means in the expression 6÷ 2(3), the number 6 should be divided by either the product of 2 and 3, i.e. 6, or by both factors 2 and 3 instead of divided by 2 only. Therefore, all following calculations are correct: 6÷2(2+1) = 6 ÷ 2(3) = 6 ÷ 6 = 1 6÷2(2+1) = 6 ÷ 2(3) =6 ÷ 2 ÷ 3 = 3 ÷ 3 = 1 6÷2(2+1) = 6 ÷ 2(3) =6 ÷ 3 ÷ 2 = 2 ÷ 2 = 1 because a ÷ b(c) = a ÷ (b*C) = a ÷ b ÷ c = a ÷ c ÷ b In other words 2(2+1) is always equal to (2 (2+1)) However 6 ÷ 2(3) ≠ 6 ÷2*3 In 6 ÷ 2(3), 2(3) is the divisor. In 6 ÷2x3 , only 2 is the divisor. In other words, 2(3) means more than multiplication of 2 and 3. The use of parentheses implies the 2(3) must be treated as a product that would become a divisor when there is a division sign before it. Remember 2(3) is always equal to (2(3)). But 2*3 or 2x3 does not equal to 2(3) or (2(3)) when there is a division sign before each one of them. *It is a wrong idea to think that parentheses is for representing multiplication only when it also has the function of grouping.* In the expression 1÷2π, 2π is the divisor instead of the number 2 only. When the Common Core teachers do not follow the long tradition of giving higher priority to implied multiplication by juxtaposition than division, they treat 1÷2π as 1÷ 2 x π, their result would be in conflict with the order of operations being used in science and engineering. 2π is always equal to (2π) or 2(π) but 2*π or 2xπ does not equal to 2π, (2π) or 2(π) when there is a division sign before each one of them. Again implied multiplication by juxtaposition like 2π has the function of grouping as well as multiplication. Of course, implied multiplication by juxtaposition without the use of parentheses cannot be used on two numbers like 2 and 5 or 2 and 3.1416 because 2(5) is not equal to 25 an 2π is not equal to 23.1416. When an engineering project is worked on by different engineers, some uses SI units but some use the British units, the result would be very confusing. However, as long as each engineer specifies the units being used instead of just mentioning the numbers, they can still convert the units to the same ones agreed on in the final phrase. However, when a crooked leader tried to sabotage a project has caused some engineers to disobey the traditional rule of giving higher priority to implied multiplication by juxtaposition than division in interpreting the data given and in calculation while some other engineers follow the traditional rule of giving priority to implied multiplication by juxtaposition, there is no way to ensure that no mistakes will be made because there is no way to bring their works together when we don't know which order of operations they had used. Even if each engineer states which system of operations he has used in his calculations, we still need to know if the data supplied to the engineers were from someone who follows or disobey the traditional order of operations. For example, if one of the data for calculation is 1÷2π, we don't know if it means 1÷(2π) according to the traditional order of operation or it means 1÷2 x π according to the new instruction of the crooked leader. That means we cannot be sure if our calculation is correct according to our preferred order of operations if the preferred order of operations of the person who gave that expression of 1÷2π is not known. The fact that different models of calculators, even from the same manufacturer, may be programmed with different order of operations, without specifying the order of operations being used in the calculator makes the bad situation even worse. Since almost all science and engineering documents mandate giving higher priority to implied multiplication by juxtaposition in calculations, the only way to correct the present confusing situation is to ban the common core nonsense of not giving a higher priority to implied multiplication by juxtaposition than division because there is no way we can change tons and tons of science and engineering documents existed before the common core nonsense to conform to the stupid and wrong PEMDAS ONLY operation that ignores the rule of giving a higher priority to implied multiplication by juxtaposition than division. The leftist politicians who hate the USA know that a unified country is a strong country and a divisive country is a weak country. That's why they tried to force the the American children to use PEMDAS without teaching them about giving a higher priority to implied multiplication by juxtaposition than division, which is still being used in engineering and science, in order to create confusion and conflicts among the different generations and different groups of people for the purpose of weakening our strength and unity. PEMDAS without juxtaposition should be history just as Obama's other Common Core nonsense should be history in the USA.
Earlier I found out that Google Chrome has partially hidden my comment on Oliver Knill's thread but it is not censored on Firefox.The Common Core cohorts promoting the PEMDAS ONLY nonsense who have also hijacked my connection to Google are scared of the truth being exposed.
@@THaWoM Here is a note from Wolfram alpha, you should read it, it's about the division symbol when used in 'in-line math' : mathworld.wolfram.com/Solidus.html "Special care is needed when interpreting the meaning of a solidus in in-line math because of the notational ambiguity in expressions such as a/bc. Whereas in many textbooks, "a/bc" is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages such as the Wolfram Language, it means (a/b)×c. For clarity, parentheses should therefore always be used when delineating compound denominators." So even Wolfram alpha say that YOU should check if parenthesis are needed for the expression to be calculated properly. It also say that they know that for such expression the result they give may be wrong !! You may read too that document from a Berkeley professor math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html It is a document about the order of operation and the expression 48/2(9+3) On a french site, I found out a reference to this document (discussion in french writen by a french scientific researcher, from the CNRS [ french acronym for National Center of Scientific Research] about the need about a new convention of calcul or 'Does the mathematician still have a say in the evolution of the calculation conventions of the secondary school?') images.math.cnrs.fr/Une-nouvelle-convention-de-calcul.html In that page, the author talk about his exchange with Professor Bergman, who said that prof. Alan Schoenfeld, « who is the intersection of the Math Department and the School of Education here, also agrees that such expressions are simply ambiguous ». And Alan Schoenfeld is a math professor at Berkeley gse.berkeley.edu/alan-h-schoenfeld Concerning the calculators, you may find some online casio calculator manual. In them you may find their inner 'order of operation'. You'll find that for some implied multiplication is to be evaluated at step 7 and general multiplication/division at step 10 but in other, they are both evaluated at the same step... I know that when I was at school (in France), one of the check to have an 'school approuved calculator' was if 1/2Pi was interpreted as 1/(2*Pi) (Good) or (1/2)*Pi (Wrong) But mainly a special course is present in the math book (still now) to check how 'YOUR' calculator respond to several cases, just to be sure HOW to enter YOUR math expression in it later to have the result that YOU want. (and not re-interpreted differenly by your calculator)
I must have actually been taught this as my mind has been yelling "implied multiplication first" every time I've seen these "problem'" on the net. I've known the "order of operations" since the 1960s, but don't remember ever having been taught the PEMDAS [or equivalent] acronym nor any mnemonic phrase.
Me too. Only came across the pemdas/bodmas mnemonics and their fanatics as an adult when I stumbled on these questions. I was horribly shocked at how terrible math education is. And how they seem to think this childish oversimplification is the top "law" of math 🤦♀️
@@kurtshaw229 Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial Factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9
@@TheJocelynrae BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....
I always understood it as variables. 6 is one variable in the equation. It is separated out from other variables by the division symbol. 2(3) is the other variable in the equation. The lack of a multiplication symbol implies the 2 and 3 are part of the same variable. However to solve for that variable the parentheses implies the 2 and 3 are to be multiplied. So, 6 / 2(3) = 1
2(3) implies that it is a factorisation such that you can factor an ammount of 2 from an unknown ammount, which you can find out to be 6 if you multiply the factor in again. same thing if you factor for instance a from a^2-a which you would get a(a-1) the original ammount was a^2-a which you should take care of before anything else since it is in brackets only then the division
To be more precise, it's woefully inadequate and simplified such that it breaks with algebra and most more advanced math classes. x(stuff inside here) is one such exception that it does not cover correctly. So people and teachers.. well, maybe the teacher certification tests need to be harder. lol.
There are no ambiguities in math. If you ever find an ambiguity it's because you did something wrong. 1 will always equal 1, it will never sometimes equal 2.
This is great! The comment about "mathematicians being lazy" is good. There are actually implicit parenthesis all over calculations that PEMDAS does not account for. The viral solution of "9" falls apart when the terms are variables. If PEMDAS was used by engineers we would probably have more bridge disasters.
PEMDAS is a mnemonic, it is no more of a "Lie" as is E=MC^2 because obviously, PEMDAS doesn't include the exceptions and neither does E=MC^2 include resting mass. So, of course there are a lot of exceptions depending on how a person decides to handle implicit multiplication and implied parenthesis for fractions vs pure left to right... A*B / C*D is handled differently when it appears as a fraction because there is an implied parenthesis for the denominator, as opposed to A*B÷C*D where there is no implied fraction and order of precedence is simply left to right. And depending on the teacher, some include that in their lessons and some don't. I don't really see this as a disaster for PEMDAS or any other mnemonic, it simply means that teachers and students need to be aware that there are subtleties in the rules and depending on the level of math (ie the scope) of knowledge, they might have to adjust the order of operation. The smart move is to ensure that parenthesis are in places where you want your precedence to be highest, but basic math questions leave a lot of that as part of the question to gauge the student's level of understanding for the basic concepts. I think this video points out some flaws with PEMDAS, but the bigger issue is integrity... the thumbnail claims PEMDAS is a lie... as usual in order to boost views everyone shouts "This is a lie"... no, it's not a lie, it's click bait to the rescue.
This is nonsense. I am an engineer and PEMDAS does still apply. The problem is when people convert papers with mathematical equations to single line type using / for division instead of a proper numerator above the denominator separated by a horizontal line. When converting to single line equations they should use parenthesis where needed to avoid mistakes and confusion or ambiguity. If they don't it is either the printer that does not know or it is pure laziness. The most important thing is to avoid ambiguity so costly errors are not made.
I always assumed that because the 1+2 is inside the parentheses, you would either factor the 2 into the 2(1+2) to get 2+4=6 or just prioritize the 2(3) over 6/2 because the 3 is inside the parentheses
@@shiijei26389 is INCORRECT!! Brackets are not necessary, there are akready parentheses. 6 ÷ 2(1+2) is different than 6 ÷ 2 x (1+2). In the first, the 2 is part of the parentheses expression, so 2(1+2) MUST be completed first. In the 2nd, the 2 is not part of (1+2). I'm sorry that you are wrong, and I'm sorry that you were taught wrong. And I'm sorry that you have been brainwashed. But that is your problem not mine/ours. 6/2 x (1+2) = 9 is diff than 6/2(1+2) = 1. That is same as 6/[2(1+2)]. I hope you don't work in some important engineering, medical, or scientific field, and use your kind of math. Because if you do, I'm very scared right now.
Even in college calculus and engineering classes (and computer programming) we were never *told* that PEMDAS was a limited case -- but we very quickly absorbed by example this juxtaposition precedence and learned to add parentheses if needed for clarity when linearizing an equation or formula.
It seems that if you have to 'add' parentheses to an already stated question you actually change the question. But if you actually use PEMDAS as written and not worry about the left to right issue, then in this question presented the answer still comes to 1(I seem to remember learning both ways in HS and then the traditional left to right order in college)
Look up PEMDAS rules. Ties(MD or AS) are done left to right. The answer is 9 and other answers are not correctly done using PEMDAS. NYT has an article from 2019 covering this. 8÷2(2+2) = 16.. not 1.
PEMDAS is a mnemonic for the order of operations. The order of operations must always be equal otherwise two people would get two different answers. 8+2*4 = 8 + 8 = 16 using the order of operations. 10 * 4 = 40 just going left to right. Which is correct? The order of operations is not just some random process someone came up with. Why the order of operations? P - Parentheses - groupings. If someone emphasized a group then that should be done first. a + (b - c), b-c should come before adding a. E - Exponents - exponents are just a group of multiplications. 3^2 = (3*3) M - Multiplication is just a group of additions, 2*4 = 2 + 2 + 2 + 2, thus 8 + 2 * 4 = 8 + (2 + 2 + 2 + 2) = 16. D - Division is just a group of subtractions, 12 / 4 = 12 (- 4 - 4 - 4) = 3 -Multiplication and Division are inverses of each other thus they are of the same precedence, 12/4 = 3, 3*4 = 12. Addition and Subtraction are just the last operations, the most basic so they're saved for last. In this issue 6/2(1+2), the 1+2 are first, then it comes down to 6/2(3) which is only division and multiplication, the same precedence, so they're done from left to right, 6/2 = 3, 3(3) = 9. This could also be written as 6 * 1/2 * (1+2). Multiplication can be done in any order thanks to the communitive property, this 1/2 of 3 is 1.5, 6 * 1.5 is 9. The only answer for 6/2(1+2) is 9.
@@leighz1962 Damn bro never knew the new york times, a bunch of journalists, know more than actual mathematicians! Might as well remove mathematics as a major and replace it with more courses in journalism as they clearly know more!
The way I read it is that when you have a parenthesis that is preceded by a number, this means you are using x times the amount in the parenthesis for the rest of the calculation. So although it is a multiplication you are doing, it is seen as it’s own group that needs calculated first before it is calculated with the rest of the equation. It does look weird when the equation is just using numbers, but when you start using variables in equations, it makes complete sense.
Actually when you see a parenthesis preceded by a number; It mean this number has been removed from the parenthesis and to properly evaluate the parenthesis it must first be distributed back. 2(1+2)=(2+4)=(6)=6 Or 2(1+2)=2(3)=(2*3)=(6)=6
Nowhere in PEMDAS order of operations is outer multiplication of parentheses stated as being done with P. It is the the MD portion. 6÷2(1+2) = 6÷2×(1+2) = 9 6÷(2×(1+2)) = 1
There are two ways that I would solve this equation. One way is to use the distributive property to eliminate the ambiguity of whether to follow multiplication/division from left to right when simplifying the set in parenthesis. 6÷2(1+2)=6÷(2+4)=6÷(6)=1 The other way is to simplify the set inside the parenthesis first then, you are left with 6÷2(3) and realize that the parenthesis are still there and still must be solved before multiplication/division from left to right. 6÷2(3)=6÷(6)=1 Either way, the answer is still 1.
Nein.. there are technically hidden parentheses around everything. 6÷2(1+2) is also ((6)÷(2)((1)+(2))) If you are solving using PEMDAS, the answer is 9. 6÷(2(1+2)) would be 1.
@@J-kd2qc you've not handled juxtaposition in that solution. As the video explained, PEMDAS cannot be used when an equation uses juxtaposition - you need PEJMDAS.
Often grateful for my 70-80s Canadian education. This is another of those times because I learned this as you described. I'd never heard of PEMDAS until an argument about the example or a very similar equation making its rounds online.
Thank you!! I graduated from the Naval Academy. This is how we were taught in our math, engineering, and science classes as well. This issue isn't just a fun brain teaser, it's important. If you are designing a bridge or a building, or worse - a more complex dynamic system, math matters.
And if you attempt to use a computer or calculator to assist computation for such endeavor you had better be aware that that device is likely to use PEMDAS to parse expressions. Simply do not write ambiguous expressions -- use parens to coerce an exact parsing.
It is a "fun little brain teaser" though because NO ONE writes an elementary expression like that. It's completely academic. Furthermore, I hope you meant to say you were taught PEMDAS in the Naval Academy. "If you are designing a bridge or a building, or worse - a more complex dynamic system, math matters." I agree. That's why mathematicians, scientists, and engineers don't write stupid little "trick" problems like this. Nevertheless, even as written, the expression is NOT ambiguous. It's 9.
That is an answer by computation alone. The correct answer is 1. If we read the equation plainly, it is simple to see how the answer is derived. 6 is being divided by twice the sum of 1 and 2.
Has grade school math changed this much? Because this is how I was taught in grade school in the 80's.... I never even heard of PEMDAS until I was an adult.
@@antonioliles5027 as someone who went through 2000s grades school, math at an elementary level has been slowly made easier with different dumb and lengthy methods. Nowadays time tables arent even memorized because it takes memorization and teachers don't spend enough time worrying about helping that be memorized. From what I see what most grade school teachers use to teach math is some weird method where you put numbers into boxes and somehow multiply them/divide them. Math on an elementary level has basically become something entirely based on how a teacher teaches instead of actual formulas or methods.
I remember getting taught this ruling in primary school(UK) during the 90s. This was when we were learning BODMAS. It was explained at the time that you had to do the multiplication at the same time as removing the brackets as it was considered higher on the order of operations than regular multiplication. I also remember very distinctly as a ruling because they purposely gave us questions with stuff like 2( and 2*(, just to make sure we were paying attention.
Could we call this an inconstancy in notation? Even to a beginner like myself, it is clear that the numerator and denominator are to be treated as isolated expressions. That means that, for the sake of clarity, if that is the intended effect, the vertical-stacking notation of a fraction is preferred. A horizontal-linear notation immediately invokes the PEMDAS association, however.
I seem to remember learning (in the 1970s) MIDAS, which stood for Multiply Immediately, Divide, Add, Subtract. (This was probably early enough that parentheses/exponents weren't yet covered.) That would seem to emphasize that multiplication has priority over division, though it's less clear whether order or operation take precedence when you're down to addition and subtraction.
Excellent explanation. Now I understand why elementary school teachers keep telling me "you're doing it wrong" when I knew this is how we did it at school (in a fairly math heavy program). I mean, it's all just convention (as long as both the person writing the expression and calculating the expression follow the same rules, it's all good), but still, the gap was disturbing to me.
To see something that I have been doing intuitively for all my life being formerly explained is mind-blowing! Using the textbook examples was what made me realize that I had done it this way all my life.
The issue with this is that no one uses the "a/b" notation. Engineers, physicists, and mathematicians use FRACTIONS. The original problem is meant to be confusing. PEMDAS is correct, you just have to write equations properly instead of writing them in ways that are intentionally misleading.
The real problem seems to be that mathematicians will automatically assume that everyone will interpret things exactly the same way they do and fail to see that other interpretations are possible. So when in doubt, use all the parentheses needed to make absolutely sure that people do the operations in the proper order.
Nonsense. Mathematicians are communicating with likewise educated (at least college grad in the sciences) people using a convenient standardized notation, and that notation has a lot more symbols and conventions than the few you learned in elementary school. Even if mathematicians tediously parenthesized everything, making it far less readable, mathematically uneducated people still wouldn't be able to make heads or tails of it.
Other interpretations are only PLAUSIBLE when rules aren't CLEAR! I don't see how the fuck this kind of shit is so hard for people to understand! But when influencers do stupid shit and convince morons that the correct way is wrong everything goes to hell!
@@ihadtochoosethisuser a*(b+c) = ab + ac True? If True then: a*(b+c) = a(b+c) True? If True then: 2x/a*(b+c) = 2x/a(b+c) Still true? Or do we need to start adding in more brackets at some point to clear up some ambiguity? 2*x = 2x? True? If True then: 2x/a*(b+c) = 2*x/a(b+c) = 2x/a(b+c) = 2*x/a*(b+c) True? Unfortunately our 'rules' have ambiguity, especially when the person initially writing the problem is sure there is only a single way of interpreting it. Anyone who claims that there is only one true and correct way of evaluating an ambiguous statement needs to stop because those are the types of people who will end up causing ambiguity by failing to see different potential interpretations when writing down their own formulae.
@@SubjagatorYou're intentionally breaking up perfectly implied multiplication in order to introduce unnecessary ambiguity. No one who is trying to actually do arithmetic or algebra would ever do this; you're just trying to win points on the Internet. Do you need a cookie that badly?
I remember being sucked into several of those Presh Talwaker videos where he posted arithmetic expressions like this, and explained the debate about PEMDAS and the ambiguous answers to simple math problems. The way I was taught math, in 5th-6th grades in the early 1980s, was as explained in the video above: That the expression 6 / 2(1+2) would be equal to 1. However, "New Math" and the way students are taught today, (and the way Presh Talwaker explains it) would evaluate that expression as 9. My primary-school math teachers explained "Order of operations" as that any number directly adjacent to a parenthetical expression, in this case 2(1+2), would always be evaluated first as multiplying the number outside the parenthses by the value inside it. So, that answer to the above problem would be 1, but to obtain the answer 9, (according to 1970s/early 80's math) you would write it as (6/2)(1+2).
I was taught that the divisor symbol is a relic of the type-writer era. The fact is that we didn't have tools like LaTeX so we needed an easy way show above the line and below the line for division. Feels like this point is keeps getting skipped over by everyone explaining the division symbol on youtube. Also, the example at 5:30 is given by a teacher that shouldnt be teaching. 2x/3y should be written as 2xy/3 if the answer has any hope to be 11.
@@RaspKNo it is not, because there are no variables, and the parentheses' purpose (to state a quantity) is fulfilled after resolving the quantity within them. Therefore, they are to disappear and no juxtaposition is applicable.
For me, when I was taught PEMDAS, I was taught that multiplication came before division, and addition came before subtraction. I suppose I could have mis-remembered, or that the teacher was mistaken, but is that the way it should really be taught instead? Or is it only correct in the specific case of multiplication by juxtaposition? Actually though, what I was taught left out Exponents: "Perhaps Math Does A Sequence."
It would certainly be wrong to do addition before subtraction - subtraction is literally just addition of a negative value. So if you had 10 - 5 + 8 then doing addition first would be 10 - 13 = -3 while treating them as equal you get 13. Substitute a = -5 you get 10 + a + 8 and you see that now by just expressing it with a variable we'd have _changed the answer_ if addition came first! ;) In practice it's often just juxtaposition that explicitly precedes division, but to avoid ambiguity there are a number of journals that have a specific style guide specifying multiplication in general comes first. So it sort of depends on context tbh, and usually you can know from context the intention of a equation and which interpretation was intended. In practice I don't think I've ever seen a real world instance where this was actually an issue because any time there's ambiguity most peeps in maths and related fields will simply write it down in a way that avoids that ambiguity instead - a good example from the video is the case of x/2. If I saw written 1 ÷ 2 × c or 1/2 ⋅ c then the inclusion of the explicit multiplication symbol would make me think it was intended as a separate operation - ie, equivalent to ½c or c/2. The spacing too can make things more or less ambiguous, so if the 1/2 is tight then the intention is _clear_ that it actually means ½c as opposed to 1/2c which would _always_ be interpreted as 1 / (2c). Maths is a language, and like any other language, people take shortcuts, and usually those shortcuts are clear and obvious to anyone that speaks that language and don't really need to be clarified because context and intention usually eliminate ambiguity for those that know the language well. So people violating rules for brevity when intention is clear, or conventions that are not explicitly spoken but become ubiquitous through their usage are just things that happen.
@@Phidaissi Thank you for your reply, that made a lot of sense and I feel like I understand it now. And I suppose if Math is a language, PEMDAS could be described as similar to someone coming up with the English "i before e" rule that people learn, but it turns out there are far more exceptions in actual practice.
@@Phidaissi With 10 - 5 + 8 doing addition first is perfectly fine. It would be 10 + 3 = 13 or 18 - 5 = 13 though. What you did was 10 - (5 + 8) which isn't the same question. You just added incorrectly. A and S have equal priority so the order doesn't matter.
@@WooperSlim Since M and D have equal priority the order doesn't matter. Same with A and S. You can always do M before D and A before S and always get the same correct answer as someone who does D before M and S before A. Like with 10 - 4 + 7 S first: 6 + 7 = 13 A first: 10 + 3 = 13 or 17 - 4 = 13 and with 16/8×4/2 (this notation isn't great though. You shouldn't have multiplication or division directly after division on one line without brackets to remove ambiguity) M first: 16/2/2 (again, bad notation) = 8/2 = 4 D first: 2×4/2 = 8/2 = 4 See? Same answer regardless of the order. The ambiguity here though is with the notation, not the rule. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. I.e. does a(b) = (a×b) or a×b? Both are widely used but lead to different answers when used directly after division. if you have 6÷2(3) one interpretation of the notation is 6÷(2×3) using PEMDAS now gives 1 The other interpretation is 6÷2×3 using the same PEMDAS now gives 9. I do think the juxtaposition interpretation of implied brackets is the more common academic interpretation though.
I came up with 1 but I used distribution and that process eliminating the parentheses isn't finished until you complete the whole of 2(1+2) then you divide the resulting 6 by 6
Since the majority of people believe the correct answer is 1, let me prove to you that's wrong. We all agree we get to this point 6÷2(3). If the answer is 1 then the equation will be 6÷2(3)=1. Now replace 3 with "a". 6÷2(a)=1. Solve for "a". I guarantee you "a" will not be 3. But "a" will be 3 if the equation =9 instead of 1. The correct answer is 9.
@@ri3m4nn 1÷3 or 1/3 does not equal 3. 3÷1 or 3/1 does equal 3. Why are you grouping anything? The math problem is straight division and multiplication. In that case you solve from left to right.
I've always used the idea that non notated multiplication was always considered tight and always done first (it would be included in the parentheses step). Division would be notated with a slash at this level and would be secondary and controled by a large horizontal line to be clear if needed. Using actual division and mulitplication symbols, then yeah, you do them in order, but that is very elementary mathematics and is a slightly different system. Maybe I'm wrong I don't know.
The whole problem is that one even has such a term People should not use or need any memory rules for these kind of things. They should just learn them.
I think that the notation for writing multiplication and division are taught but not clarified well enough to instill how division/fractions relate and interact. Pemdas isn't wrong but the concepts aren't clarified enough to ensure pemdas can be used properly as a tool. How we write and interpret multiplication and distributive property is one example. If a number should be in the numerator or denominator and if it should be written as a seperate number instead is another. Many who discuss this likely understand but those that don't may not be able to guess where their issue with understanding is making it difficult if not impossible to fix the gap in understanding without aid. An inconsistent foundation makes further building an unstable and hazardous endeavor.
Thank you! It's been 50 years since I finished engineering school and have never worked in my field, so when I saw this problem with the "new" solution I thought maybe I had lost my mind. It turns out that just maybe it is not me who is crazy -- just everybody else.
No, it's you and the lady in this video... The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y. This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2 All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. There are no coefficients in this expression... 6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b is 3 not 2 Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property, when FULLY applied, is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property, when FULLY applied, REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2 no parentheses required 3×1+3×2= 3+6= 9 You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression..... When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses. TERMS are separated by addition and subtraction not multiplication or division. 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 6÷2(1+2)= 3(1+2) no rules have been broken 2×2×4(a+b) partial Distribution 2×2(4a+4b) However the TERM outside the parentheses when simplified equals 16 and 16(a+b) 2×2(4a+4b)= 2(8a+8b)= 16a+16b which is the same as 16(a+b) 2×2×4(a+b) when fully Distributed is 2×2×4×a+2×2×4×b and the LIKE TERMS can be factored out of the expanded expression. The LIKE TERMS being 2×2×4 So... 2×2×4(a+b) 6÷2×1+6÷2×2+6÷2×3-6÷2×4= 6÷2(1+2+3-4) as the LIKE TERM 6÷2 was factored out of the expanded expression... Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board...
@@forgottenpeopleplacesandol4258 Convention when it comes to directly prefixed coefficients and variables... 2y there are no parentheses by which you can Distribute... I can explain it to you but I can't understand it for you...
@@RS-fg5mf Somehow I managed to get through an engineering program and also managed those skills to achieve substantial retirement benefits while never drawing a paycheck from anyone and I am enjoying my accomplishments just fine despite my lack of understanding. I don't know how I made it without your abilities. But I did.
@@forgottenpeopleplacesandol4258 🤣🤣🤣🤣🤣🤣 if you say so... LMAO You never worked in your field and never recieved a paycheck?? Hmmm. And this supports your wrong answer, how exactly?? And FYI.. Any career field that relies heavily on math would NOT use the obelus by rather the vinculum so you're argument is lacking...
The problem is the equation can be written more than one way, and there should really be a second set of parenthesis. A way to test it is to write it as 6 OVER 2(1+2). Then it becomes clear.
To a degree yes you do need PEMDAS but you also have got to remember you need law of mathematics supersedes PEMDAS first so in like this equation is the distributive law will be applied first and then PEMDAS then the answer is still one. Nobody ever remembers that..
I teach math at the middle school level. There’s nothing wrong with PEMDAS as long as you teach its caveats. Addition and subtraction are inverse operations, so all subtraction can be rewritten as addition of negatives. Multiplication and division are inverse operations, so division can be rewritten as the multiplication of its inverse-or vice versa. That’s why sometimes we see one-half times something or that something over two. Regarding multiplication, these are implied as a group, even without written brackets: 2(3) 2x xy A traditional division symbol is in-line and resolved from left to right. A slash / is the same as a fraction bar, so the denominator that follows is implied to be grouped with its numerator. It’s resolved before other multiplications preceding it. If the denominator includes addition or subtraction, those must be grouped in brackets. Grouping symbols aren’t necessary in these cases, but can be helpful.
I WAS GOING INSANE! I have a physics degree and I am an acoustician and I got in an argument with my entire family and extended family because I was helping a cousin with a question like this and kept telling him he is doing the math wrong, and I had everyone telling me IM DOING THE MATH WRONG. I literally use advance mathematics every day and EVERYONE is telling me the wrong answers to all of these 6th grade math problems, and im sitting here pulling my hair out and solving them in moments time and verifying my work and getting the right answers over and over and over again. I'm glad to see I'm not insane. My understanding of math is accurate and the rest of my family is just not at a level of understanding this stuff that I should even care or let it get to me.
Did you get nine as the answer? As a physicist you should be able to solve this very easy, if you got one you made a math error and I can show you what went wrong.
@@DoseofScienceDoS The whole point is whether you acknowledge the "hidden" parenthesis or not. Did you watch the video? Elementary school PEMDAS says it should be 6 / 2 * (1+2), which is 9, but physics generally uses "hidden" parenthesis when using division, so the question becomes 6/(2*(1+2)), which is 1. Programming also typically uses the PEMDAS way, and I constantly have to put in an abundance of parenthesis just to calculate the correct function I actually need. This video discusses this, as it is not possible to nicely write a divider in a single line, single line equations are fairly ambiguous. One of those being a/bc, would you regard that as a/b * c, the same as (ac)/b? For most engineering, physcis, maths post elementary school, a/bc means a/(b*c)
@@Of_UnCommon_Sense Let’s do a simple rearranging 6/2(1+2)= 6(1+2)/2= (1+2)/2*6= Rearranging is a way to check you got the right answer as outlined by Charles Hutton. Answer all three with the same answer and it will be correct. Now you know 9 is correct.
i'm 72. i ALWAYS get these internet math quizzes wrong. one of the things modern people gloss over, is that the BAR separating numerator from denominator was at one time seen almost as a parenthetical. also, using the distributive rule to simplify what the expression means takes precedence over an 'earlier' operation. thanks, i was beginning to suspect early dementia.
That's pretty funny. I studied physics and game design in college and when I went back to get my teaching certifications I had to take an algebra exam, and even though I had been doing calculus with analytical geometry for years in college I pretty much failed the exam because my understanding of PEMDAS needed to be "corrected" 😅
I’m totally agree. All this “so intelligent” youtuber trying to explain in the youtube are wrong because they never go to higher level in mathematic. Im glad you say it
@@RS-fg5mf I’ve read a lot of your comments and agree with you. I was surprised by how many people were ignorant of the topic (It seems like it’s the entire comment section). Thank you for taking the time to correct people (lord knows someone has to say it).
I'm not sure I'd even consider this "higher level" math. I would've gotten 1 as an answer by grade 7, at least the way i learned basic mathenatics and order of operations back in the 70s (born in '64). I don't know where she went to school or how they teach it these days, but it scares me that anyone could get 9 out of that, much less that - as someone mentioned - an app like Excel wouldn't accept it without making 2(3) into 2*(3), adding what i think is a confusing level to it. I was taught that implied operations - 2(3) or bc in her examples - always took precedence.
@@captainz9 The key here is that math doesn't define a rule that says implicit operations take precedence. That's only a convention, and it's not universal. So without any context this operation is ambiguous, as you can't tell if you have to apply that convention (resulting 1) or not (resulting 9). The convention exists pretty much to reduce the number of required parenthesis in common formulas. And as a side effect, to confuse people.
I've been pointing this out for years. The real issue, in my opinion, is that PEMDAS ignores distribution completely, when distribution is the only reason that the parentheses can be used as multiplication in the first place. We have a perfectly good multiplication symbol that we can use to represent 2 x 3. We don't need 2(3) to exist at all. However, in distribution, we get: ab + ac = a(b + c). Now, the parentheses function with multiplication, but strictly speaking you're multiplying through the entire term. In other words, using the 2(1 + 2) part of the equation, because of distribution we should get the same answer whether we distribute the 2 into (1 + 2) and then add, or if we add and then multiply. And we can see 2(1 + 2) = (2 + 4) = 6, just the same as 2(1 + 2) = 2(3) = 6. That's the whole point of distribution, and PEMDAS destroys this concept, which is critical when you're looking at 2(a + b), since you can't add a + b, but you can still consider (2a + 2b) if you need to. Also, PEMDAS says nothing about when to do things like factorials, trigonometric functions, etc. so it's not even useful after elementary mathematics anyway. But that's a different issue.
If you do it properly with distribution, 6 div 2 x (1 + 2) = 3 x (1 + 2) = 3 + 6 = 9, matching PEMDAS. It's the ambiguity of 6/2(1+3) when written inline, is it (6/2)(1+3) or 6/(2(1+3)).
Thank you for the interesting explanation. I would have answered 1, and if asked why not 6, I would say BODMAS, but I’ve always instinctively seen multiplication by juxtaposition as somehow “more tightly binding” than operations using a sign.
I’ve done plenty of physics/engineering labs where the measurements don’t match the calculations unless you really mind your brackets. 100% agree no one outside of k12 math does it the Pemdas method.
PEMDAS not a "method". It's a reminder intended to help beginners remember some of the basic concepts of order of operations. And everyone at every level uses order of operations to help evaluate mathematical expressions. However, "PEMDAS" represents a beginners' version of order of operations, intended for use with elementary arithmetic. As such it's not wrong. Blindly using only an elementary set of rules in a more advanced context, and then calling them wrong because they're not adequate to the task, misrepresents "PEMDAS".
Isn't there also an element of (sorry for a none technical term here) "do everything under the line first"? For example, a square root sign only applies to the first number after it, but in may equations there's often a long line above a bunch of numbers / letters which I believe is meant to be the same as a bracket. So, as an example, the equation to determine the roots of a quadratic equation is [-b +/- SQRT b^2-2bc /2a ]. The rule I thought also applies to to division when written by "free hand": yc/rb would mean rb has a big line above it and that should be interpreted as having to be done first. That is, it's y times c divided by (r times b) and not y times c divided by r then the result times by b. An interest thought then is that it is NOT the same to write yc/rb like yc rb as one would have to add the implied brackets as well. And, of course, anyone who has tried to use a typewritter or computer line text knows it's really hard to draw out lines so the conventions you mention just make sense when writing a paper too!
The line is called a vinculum, and yes, it acts the same as brackets. I've read a lot of comments saying that whether the obelus (÷) or solidus (/) is used should affect the precedence rules, but I don't think there's much official support or consistency to this. Many authors use yc÷rb to mean yc÷(rb) and the same with the solidus. See my latest video ruclips.net/video/4x-BcYCiKCk/видео.html for examples. And some calculators treat 1/2x as (1/2)x (wolfram alpha for instance) so the solidus is no guarantee of grouping.
@@THaWoM The biggest problem we have here is the confusion and misconception that a convention given to variables in an algebraic expression/equation applies to parenthetical implicit multiplication..... 6÷2a the 2 is the coefficient of a by algebraic convention NO parentheses (grouping symbols) necessary. It forms a coefficient/variable compound quantity... 2(3) is not a coefficient/variable compound quantity. Real numbers (constants) can be coefficients but real numbers do not have coefficients.... Grouping symbols only give priority to (OPERATIONS INSIDE) the symbol not outside.... Do you people lack the intelligence to know the difference between inside and outside?? 2(3) is not a parenthetical priority and is not a coefficient/variable compound quantity... It is called implicit multiplication not implicit grouping... Implicit as in the physical operator is implied as if it were there when a constant, variable or TERM is juxstaposed next to the parentheses without a physical operator.... TERMS are seperated by addition and subtraction. When you have an algebraic expression like 6÷2a when you replace a with a constant or value such as (1+2)=3 we know the value is 3 so we would then write 6÷(2×3) NOT 6÷2(3) OR we could write 6÷(2(1+2))... When you replace a variable with a constant value proper grouping symbols are required to maintain the grouping that WAS the coefficient/variable compound quantity... To give priority to parenthetical implicit multiplication over division is to break the Order of Operations and the various properties of math. NOT to be confused with the simplistic use of PEMDAS. BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a parenthetical priority and is exactly the same as 2×3 M not P or B in PEMDAS/BODMAS. Brackets/Parentheses only give priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property states to multiply all TERMS inside the parentheses with the TERM outside the parentheses not just the factor next to it.... 6÷2 is one TERM juxstaposed to the parentheses... Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator... Parenthetical implicit multiplication is nothing more than multiplication without the need for a physical multiplication sign and the only correct answer is 9 not 1 Algebraic proof against this BS.... The reciprocal of a is a^-1 6a(1+2) = 6÷a^-1(1+2) Operational inverse by the reciprocal. 6a(1+2)=9 6a(3)=9 6a=9÷3 6a=3 3÷6=a a=1/2 or 0.5 0.5^-1= 2 6÷0.5^-1(1+2)= 6÷0.5^-1(3)= 6÷2(3)= 3(3)= 9 6÷(2(1+2))= 6÷(2×1+2×2)= 6÷(2+4)= 6÷6= Distribution is a method of eliminating the need for parentheses by bringing the terms INSIDE the parentheses to the outside.... You can NOT take 6÷2(1+2) and write 6÷2×1+2×2 You can write 6 --------- 2×1+2×2 because the vinculum is a grouping symbol and is the same as writing 6÷(2×1+2×2) I'm saddened by the fact that so many people share the same misconceptions BUT I'm even more sad and concerned with those who want to blame Common Core or politics.... Common Core has absolutely nothing to do with this debate. It is a teaching method and does NOT change the rules or the answer.... Those who think that this debate has anything to do with Common Core or a political conspiracy are just flat out CLUELESS and STUPID
@@THaWoM : First off there should be absolutely no reason for a grade school kid to ever be introduced to PEDMAS. Why are they be given ambiguous equations in the first place? This just means whoever is supplying the equations is lazy or sloppy and don't know what they are doing. An equation is essentially a sentence and as with all sentences you should employ correct mathematical grammar and punctuation. They should be taught the exhaustive use of parentheses and brackets. Their understanding of what they want an equation to say increases immeasurably. There should be no ambiguity. Secondly this cavalier use of the obelus wreaks havok with trying to perform algebra. I was taught that everything to the left of the obelus is the numerator and everything to the right is the denominator. 2x/3y-1 SAYS Two ex DIVIDED BY Three why minus 1. That is (2x)/(3y-1) And that is how it should be carried out. Any variations should be indicated by parentheses.
The part of the quadratic formula under the square root symbol has a name. It is called the discriminant. And sorry to say, you have it written incorrectly. It is b squared minus four ac. Not 2 bc. It is for finding the roots, or factors, of an expression which has the form y = ax^2 + bx + c I can still remember exactly how it was always written and said out loud, and I learned it over 43 years ago. Minus b plus or minus the square root of b squared minus 4 ac, all over two a. The plus or minus means that there are two solutions, also called roots, and the sign of the discriminant tells whether the roots are real or not real. Another way to say it is it says how many zeroes there are, zeroes being the places on a Cartesian coordinate where the parabola crosses the x-axis. If the discriminant is zero, there is only one solution, or root, or zero, or factor, (all mean the same thing, basically IIRC) meaning the two factors are the same, and the parabola just touches the x-axis in exactly one place. IOW Something like (x +2)(x +2). If the discriminant is positive, there are two real roots. Physically this means the parabola has it's vertex below the x-axis and there are two y intercepts, or zeroes, or factors. If negative, there are no real roots, as negative numbers do not have a real square root, only an imaginary one, some multiple of i, which is the square root of negative 1, which does not exist. These are called complex numbers. c is where the parabola intercepts the y-axis. All of this has meaning that applies to the physical world. When you point out that the expression yc/rb is taken to mean the y and c are multiplied together, as are the r and the b, and then the division is done, is exactly correct in real world usage, at least in the world of science. Because if the order of operations rule that is at issue here applied, this would be the same as saying yc/r multiplied by b. But that could be very clearly written as ycb/r. And no one ever writes ycb/r as yc/rb. It simply makes no sense. There is no logical reason to write it out that way if what is meant is to have the b in the numerator.
@@williejohnson5172 well to begin with 2x implies that 2 has been multiplied to x Equivalent to 10 implies 5 had been been multiplied to 2 Next 2x/3y-1 is an ambiguous if presented without context. Based on the problem you solve 3y-1 may or may not make sense and with a some spacing it will make sense in both Interpretation. Here's another 1/5x²+6/5x-6y Equation is ambitious and solely depends on context of what you are solving. With a bit of brackets, it will make sense in bodmas for anyone without context.
These abstract, ambiguous math problems need to go away. There's no application for these numbers so you can have multiple correct answers, depending on where you decide to drop a parenthetical expression. The math problems we did in school weren't like these. Because the textbook writers and teachers know that ambiguously notated expressions can have multiple solutions. These kinds of problems exist only to aggravate people on Facebook.
PEMDAS or not, the correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended is 9 not 1. The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y. This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2 All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. There are no coefficients in this expression... 6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b is 3 not 2 Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property, when FULLY applied, is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property, when FULLY applied, REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2 no parentheses required 3×1+3×2= 3+6= 9 You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression..... When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses. TERMS are separated by addition and subtraction not multiplication or division. 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 6÷2(1+2)= 3(1+2) no rules have been broken 2×2×4(a+b) partial Distribution 2×2(4a+4b) However the TERM outside the parentheses when simplified equals 16 and 16(a+b) 2×2(4a+4b)= 2(8a+8b)= 16a+16b which is the same as 16(a+b) 2×2×4(a+b) when fully Distributed is 2×2×4×a+2×2×4×b and the LIKE TERMS can be factored out of the expanded expression. The LIKE TERMS being 2×2×4 So... 2×2×4(a+b) 6÷2×1+6÷2×2+6÷2×3-6÷2×4= 6÷2(1+2+3-4) as the LIKE TERM 6÷2 was factored out of the expanded expression... Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board...
Many years ago in grade school, multiply superseded division and not equal - as an engineer, I still have my reference! For equations written in line, I always provide appropriate parentheses, brackets and braces to force a particular mathematical procedure!
You are absolutely right, I was taught to do multiplication and division operations from left to right, but then, at college, I was kind of implicitly taught that when the operator does not appear, then it gets precedence, so, A/BxC is (A/B)xC, but A/BC is A/(BxC)
@@grantofat6438: Why? I bet you were taught at school that "i before e except after c" and then you should remember the "anCIEnt soCIEty of sCIEnce" and that you should "NEIther be FEIsty nor foREIgn", so you were also taught wrong. As with everything, like language, math is a construct, a set of rules, and if everyone understands each other, it finds a purpose. It is VERY true that at school we all are vehemently taught that we must operate from left to right when the signs have no precedence (division and multiplication, sum and subtraction, etc.), but when you go to college, it is never said but implicitly used time and time again that a non explicit multiplication has precedence. Just try it, give a problem of the type Y=A/BC to college graduates (engineers, physicists, chemists, etc. -maybe not mathematicians-) and they will most commonly operate the multiplication first, BUT, give them the same problem written as Y=A/BxC and most will divide first. Short conclusion: ALWAYS USE PARENTHESIS so there is no doubt at all of what needs to be operated first.
I know from my many calculations for structural engineering, that Pemdas is wrong. Of course, I will never see 90 again, and in my time we used paper and pencil, mechanical adding machines and the most important tool of all, a slide rule. There were no computers or internet.
PENDAS isn’t wrong, but there are two different things happening here. The first is the multiplication before the bracket, from memory it was called implicit multiplication, it comes first because a number outside a bracket means it had been initially factored out elsewhere. The second is the way we try typing something into a computer whereas in the day of writing it down by hand it was easy to put in the correct numerators and denominators but by PC you get a long string and as per other comments here, you should add extra parentheses to clarify.
Thank you! This perplexes me today when I see these "what is the answer" on basic algebra and math questions. I'm almost 63 now. I graduated from a USA high school in 1977. I think that I was taught to solve inside parenthesis first and multiply (or divide if that's the case) by what's attached next to it, which gives me 6 divided by 6 equals 1. On social media and other math videos they try to convince you the answer is 9 using a twisted order of operations which differs from the way I was taught like some form of alien math. This has me so bugged out that I'm searching for what changed over the decades, and what in life is really true or false, especially when it comes to math which should be absolute!
The TERM outside the parentheses is juxstaposed to the parentheses as a whole not just the numeral next to it. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2. 6÷2(1+2)= 3(1+2) NOT 6÷(2×3)
I agree with what you've said but it's been consistently reconfirmed that math is by no means absolute. Perhaps in it's form as a commonly used construct we all agree on certain things but it can be surprising how many things are decided on from necessity to decide rather than on a naturally grounded explanation.
@@RS-fg5mf that is wrong , since you divided before removing the parenthesis . the "problem" , has do to with the way Americans interpret parenthesis . They "think" it means, solving what is on the inside. That is incorrect. to rationalize an equation with parenthesis , you need to remove them completely before moving forward with any other function. b÷a(x+y) = b÷(ax+ay) | when b=6,a=2,x=1,y=2 then b÷a(x+y) = 1
@@avibhagan you're wrong. That's exactly what grouping symbols do. Give priority to operations INSIDE the symbol not outside the symbol. There is no rule in math that says you have to open, clear, remove, take off, eliminate or get rid of parentheses. The RULE is to evaluate operations WITHIN the symbol of INCLUSION as a priority and nothing more... 6÷2(1+2)= 3(1+2) No rules have been broken because it does not literally mean you have to evaluate the parentheses first and foremost... Even if you decide to get rid of the parentheses... 6÷2(1+2)= 6÷2×1+6÷2×2 by the Distributive Property. Parentheses REMOVED...
Yep. I have exactly the same issue with PEMDAS. It's the juxtaposition, the implied multiplication, the algegraic grouping that has been neglected by PEMDAS. (But you don't have to make Multiplication always higher order than Division if you respect algebraic grouping.) I wouldn't say PEMDAS is wrong, I'd say it is incomplete. Add grouping like PEJMDAS it would fix it.
Sounds like an issue in how PEMDAS is explained (though, I learned it as BIDMAS). When I was told P/B, I was told to complete or solve them - as you put, as algebraic grouping. The first 2 in 2(1+2) is very much part of the P. 2(1+2) = (2×1+2×2) = (2+4) = 6. Then, it becomes 6÷6=1. But, if all you get told or all you remember is to simply calculate the internals of the P and then leave it at that, you get this ridiculous scenario. Whilst I can agree that change in name might help with that, I'd rather see the maths of it taught and explained properly.
I would agree the letters of the term PEMDAS might be misleading/incomplete, this video's interpretation of it was never how I was actually taught though in the 70s, i was always taught the implicit multiplication of "bc" is interpreted first/differently than "b*c" - though taken standalone like that they may be the same, as part of an equation with other terms to evaluate they may not be. As you say, adding the J in might be appropriate for people with a blindly simplified understanding, but it never would have flown in even Albegra1 class. This perhaps show us the flaw in blindly following acronyms rather than engaging in critical thinking and actually learning a subject rather than just blindly following along and misremembering things years later based on a stupid acronym.
@@temisu_namisu The problem with having implied multiplication and explicit multiplication as separate things is that is just adds a different type of ambiguity. Suddenly 2(1+2) is different to 2*(1+2). That difference means either a) you can never contract a formulae in the form x*(y+z) to the form x(y+z) because those are now different things, or b) you are allowed to change x*(y+z) to x(y+z) to clean up but then you risk adding ambiguity if someone sees the simplified version and doesn't know where it came from originally. All you have done with the implicit > explicit multiplication is moved the ambiguity to a different place. People know we have this ambiguity problem with rules, parentheses is the solution to make sure people evaluate it correctly.
I learned the multiplication by juxtaposition rule when I was in seventh or eighth grade during the mid-fifties. For the past few years I have been a volunteer in our local grade school and have argued that the workbooks the students used were wrong but to no avail. I have, however, showed the better students (those who might advance past primary arithmetic) how the rule will be applied in the future.
Absolutely. Implied multiplication is different from explicit multiplication. Juxtaposition makes it a single term. You must treat it as a single term in all cases. You cannot just arbitrarily change terms. For example, the number 12 can be rewritten as 3 × 4. So if you had a problem like, what is 24 ÷ 12, you can't then just rewrite 12 as 3 × 4 and come up with: 24 ÷ 12 = 24 ÷ 3 × 4 = 8 × 4 = 32. You have to resolve all the terms. Terms are separated by explicit operators. Implicit multiplication is a part of the single term.
As a practicing physicist with degrees in physics, mathematics, and engineering, here's my opinion: 1. I never learned PEMDAS or BODMAS or any other such mnemonic devices. I was simply taught "multiplication and division is done before addition and subtraction". 2. Parentheses come first, then exponents, then multiplications and divisions IN ANY ORDER, then additions and subtractions IN ANY ORDER. You certain CAN evaluate them left to right, but it isn't necessary. You'll get the same result using ANY order. 3. Nobody uses the obelus (÷) symbol after elementary school, except for a few specialized uses such as calculator keyboards and the APL programming language. 4. As for a/bc -- I'd say that would usually be interpreted as a/(bc), because otherwise it would be written ac/b. But I would consider it technically wrong and should be rigorously written as a/(bc). Writing a/bc is informal usage, and generally used only when you're constrained to write on one line. In real life, one would write a horizontal bar with "a" on top and "bc" below, to make clear what is a dividend and what is a divisor. 5. In real life, nobody would write an expression like 6 ÷ 2(1+2), precisely because it may be interpreted differently by different people. Most everyone would write the division using a horizontal bar to make it clear what part of the expression is a dividend and what part is a divisor.
The order does matter in subtraction and division. 10-20=(-210) and 2/3=⅔ but 20-10=10 and 3/2=³⁄₂ even though division has the same priority as itself and subtraction has the same priority as itself.
Order of operations does matter. In programming, you have to write your opperation in a linear fascion and the computer will execute them from left to right. So something like 8/2*(2+2) might not be out of place.
@@aryan_kumar - We're saying two different things. Of course 2/3 is not the same as 3/2, and 20-10 is not the same as 10-20. Nobody is saying otherwise. But subtractions and divisions can be done in ANY order -- left to right, right to left, or any other order. For example: 10 - 3 - 4 - 6, left to right: 10 - 3 = 7, 7 - 4 = 3, 3 - 6 = -3. Right to left: -6 - 4 = -10, -10 - 3 = -13, -13 + 10 = -3. Same answer. Likewise with division: 2 ÷ 3 ÷ 6 ÷ 4, left to right: 2 ÷ 3 = 2/3, (2/3) ÷ 6 = 1/9, (1/9) ÷ 4 = 1/36. Right to left: (1/4)(1/6) = 1/24, (1/24)(1/3) = 1/72, 2 / 72 = 1/36. Same answer.
The ending of the video is how I learned in school (also always going from left to right). When I have a good math day, the expressions and equations I’ve evaluated always prove to be the right answer. I think this “dilemma” has more to do with lazy / forgetful teachers and the problems of transcribing inline equations.
There really is no problem with PEMDAS! It is a question of notation at different levels of math. PEMDAS works at the foundational level of math where expressions are non-composite. As the level of math advances, mn/qs "implies" taking mn and qs together. It can no longer just simplify as a × n/q × s ! There are many more operators that cannot follow PEMDAS (factorials, summations, even simple trigs --> sinAb is not bsinA, etc.). In short PEMDAS for basic order of operations, like LIATE for integration by part are general and convenient ways of manipulating tasks. They do not work for every situation. As complexity of a task increases, mathematical notations take over to guide the order of operation.
It's not that PEMDAS is wrong or a lie, it's simply that it's not designed to handle expressions such as this. PEMDAS and others are used to teach the order of evaluation for the BASIC operations it refers to. Grouping values together is not one of these operations any more than factorials, moduli, trigonometry or tetration are. As such, it's pointless trying to apply it to such expressions. There is no acronym that can cover all mathematics in this way. We learn the rules of interpretation as we learn the more advanced functions. When there's any doubt we should use additional brackets for clarification.
IT does not matter. The problem is that people are using PEDMAS wrongly. Some are coming up with a 6/2*3 . There is no 3. It is 6/2(4+2) or 6/2(2+4). In 2(1+2) the 1+2 is commutative and can be swapped to 2(2+1).The 2 gets distributed to the 1 and the 2. It is 6 /2(4+2) or 6/6
![Why you didn't learn tetration in school[Tetration]](/img/1.gif)
This is why when I'm programming I put in extra parentheses to make it crystal clear what the formula is supposed to do and not rely on a compiler to interpret it the way I intended.
Are you saying compilers don't interpret mathematical e pressions correctly
@@Hclann1 They certainly do, if you make sure to tell them how.
@@QuesoCookies Currently extending Java code. PMD keeps telling me I have superfluous brackets in my equations. Err, no I don't.
@@Hclann1 compilers do indeed use the traditional "left to right", i.e. multiplication and division are considered equal order of precedence. So in that sense, compilers (at least c and c++, the python interpreter too) are "incorrect". You cannot ommit parenthesis.
@@fernandogiongo I was not aware of this, thanks for letting me know.
As an engineering student, I never use the division operator. Every equation or solution is in fraction form. I believe this would clear up a lot of confusion. Thanks for the video!
Nobody uses the division operator
Totally agree. The fraction bar is much clearer and is a grouping tool like parentheses.
Solve all numerator stuff, solve all denominator stuff, the divide top by bottom.
Division signs are dumb
They literally have the exact same meaning. The division operator signifies the exact same thing as the fraction bar. The symbol is even supposed to represent a fraction, with the equation to the left replacing the top dot and the equation to the right replacing the lower dot. The reason why it falls out of use in higher math is solely because certain countries decided to use the symbol for subtraction, so they dropped the symbol entirely to avoid confusion.
@@GremlinSciences the use of the division bar is confusing because the fraction bar is so clear. 6÷2+3 could be clarified so easily is the fraction bar was used because this problem is NOT supposed to read (6÷2) +3 but actually the numerator 6/(2+3)
When written out without the division sign (which keeps everything on one line) it reads much more clearly because the bar serves as a grouping symbol
@@swedbp1 You only have confusion because you are giving two symbols with identical meaning and usage two separate meanings. there is as much difference between '/' and '÷' as between '*' and '×' and '⋅'
6÷2(1+2) is exactly equal to 6/2(1+2)
The only difference is an arbitrary distinction you are imposing that is not part of either symbol's actual meaning.
This is something I explained to my wife, early in our relationship. She was under the impression that she "just isn't good at math". She gave me examples of things she didn't understand. I found that, to a last, the problem was that her teachers had never actually explained the mathematical concept. They had instead simply had the students memorize procedural algorithms like PEMDAS. They weren't teaching mathematics. They were basically teaching handwritten calculators, a list of steps you follow to "get the right answer". At no point did any of her instructors help her understand the language of mathematics. I was the first person she had ever met, who referred to mathematics as the language of patterns. Once I started explaining the actual patterns the equations describe, she immediately caught up. She was never "bad at math". She'd just never actually been taught about math.
I used to tutor prep classes for the GMAT and GRE. My examples always started with a simple problem students could follow all the way through, and then I'd show them the general form of the equation once they understood what it was for. One of my students' evals said that she'd taken Algebra in high school and college, but she'd never "gotten" it until I showed it to her my way. For example, I'd explain compound interest by showing an account through two two compounding periods, _then_ give them the formula with exponents and variables.
I mean, pemdas is just a tool to get students to the point where you can start discussing how the maths actually work. But, like, you have to teach the tools correctly or they'll never get to that point...
I've known there were issues with how kids are being taught, but this is just BASIC. I can't even.
I've run into your experience a lot and it makes me so mad. EVERY person I've ever talked to who professed to be "bad at math" were just taught poorly. Usually the basic concepts, and then they were rushed into stuff they had no chance of understanding and just had to struggle memorizing meaningless number combinations so they could pass tests. Every one of these people immediately "got" the math once it was properly explained to them - though I often had to work through years and years of conditioned response.
Math is like an onion, it's got layers (to quote Shrek). At the deepest levels, it becomes so abstract that even the brightest minds struggle to wrap their intuitions around it. In some ways, this is also where many advanced topics, such as abstract and linear algebra, topology, calculus, real and complex analysis,etc, tend to merge into each other.
As the level of abstraction goes down, the fields drift apart and operations become more procedural.
Reduce the abstractions further, and topics simply become impossible to express.
Eventually, we have basic arithmetic presented merely as a few procedures to memorize.
Precisely because of the increasing levels of abstraction, students tend to be taught from the outside in. For instance, you can teach complex numbers in two ways:
1) The common way, it to introduce i=sqrt(-1), and c=a+bi. This is the procedural way, introducing i totally without justifying it or explaining why its useful or how to build intuition about it. But it can be taught in an hour.
2) The intuition based approach, is to first teach group theory,then introduce the SO(2) group, represented by the matrix R=[a, -b | -b, a] and the relation to the O(1) rotation group. NOW you can introduce c=a+bi as just a shorthand for this, and work from there. But this takes the average student perhaps 1-2 more years of study, and before you reach c=a+bi, you lost half the students.
Primary school teachers have the same dilemmas, just at a lower level. The smartest students tend to see most patterns without explicitly learning them, while the least talented would never understand those patterns. Those in the middle MIGHT gain a little from learning those patterns explicitly.
@@RWAKitty I was sitting in a statistics class when the prof put a formula for the line of best fit for a data regression on the board, then asked what the slope would be. I told him "Beta sub one" immediately. He asked if I could do the proof. I admitted I couldn't, but if we rearranged the formula slightly and renamed the terms it became Y = mx + b, which is just the slope/intercept formula from algebra 1. I can't depict the beta symbols, subscripts and carets properly in a YT comment, but the terms were in the order y = b + mx for some reason.
@haakoflo Your reference to primary school suggests to me that you were not educated in America. I think your statements are correct. But understand that in America, it is common for the teachers themselves to have little comprehension, of even basic algebra. They aren't executing pedagogy, to lay the groundwork for understanding. They are often just regurgitating what is written in their teaching materials, without really grasping it themselves. There is a reason this country ranks near last in the world, in mathematics education.
In engineering school we always wrote division operations as fractions, never in line. Not only did it keep things clear, but it shortened already lengthy equations. If we had to write things in line for whatever reason, copious parenthesis were used
"we always wrote division operations as fractions, never in line". Really? Have you never had to enter an expression into a computer programming language, or even Excel? There are a few programs out there, like Wolfram Alpha that will accept expressions in your favoured form, but those are relatively rare. For the great majority you do have to be prepared to enter them in line form.
Same in physics.
@TheEulerID we only ever used Excell to makes charts and graphs from data. My college did try to heavily push Matlab while I was there, but that "program" was such hot garbage that it was easier to do it by hand.
Until you start writing entropy units.
It isn't about engineering schools.
I’m a retired astronomer, have been doing math for 60 years and would have always come up with the answer 1 for this equation. As someone mentioned the confusion is eliminated with the use of brackets. I just ran the equation twice in Excel 2007. Written as = 6/2*(1+2) Excel gives 9. Written as = 6/(2*(1+2)) Excel gives 1. In the old days I programmed in Algol and FORTRAN and always used the brackets. I was taught to religiously test equations in the program before finalizing.
There is only 1 set of brackets in the equation. You got the answer 1 by injecting a second set of brackets which is not in the equation. I do not agree with PEMDAS. In fact, I never heard of PEMDAS until I was around 70 years old. The proper way to solve is left to right. Starting from the left the first step is 6/2. Because 6 is the first number, you deal with the 6 first. That is 3. The second step is 3x ( ) because (1+2) is the next number. In this case, 2x3. that is 9.
What if you typed in 6/2(1+2) without the Times symbol? Excel doesn’t even recognize it as a valid formula
@@jeromewaldemar6963 I haven't heard of PEMDAS either, mostly because I didn't grow up speaking English. Around here, the rule is "dot before dash" (because multiplication and division are written with dots). Yes, left to right, and that's also what PEMDAS seems to say, as explained in the video.
I got 1 also
@@jeromewaldemar6963 _"I do not agree with PEMDAS. The proper way to solve is left to right."_
Well that's just plain wrong, whether you accept the priority of implicit multiplication by juxtaposition or not. Are you saying that you believe 1+3×5 = 20?
As an engineer, I agree 100% with you. Also it is up to the author of the equation to frame it with brackets etc where necessary to avoid any possible confusion of intent.
Comletely agree.
If, as an engineer you write expessions like this one that are ambiguous, you are a moron.
This kind of video is absolutely ridiculous.
Then you missed the slight of hand she pulled. She went from an OBELUS (which is NOT universally agreed upon with respect to its grouping or non-grouping conventions) to a slash which is literally defined in the sources she mentioned as functioning like a fraction bar. And yes, brackets should be used. And an OBELUS should NEVER be used, and it pisses me off that she equated an obelus with a slash, especially because in at least the last two sources they are clearly used in place of a fraction bar to save space.
@@eliteteamkiller319 There is no functional difference between an obelus and a solidus. Both of those division symbols do not have grouping properties. Only the vinculum or horizontal fraction bar has grouping properties.
Thank god I finally saw this. I've always said that needing to use PEMDAS is the sign of a sloppy equation. I absolutely hate those "trick" math problems on the internet.
Totally agreed, especially the part about the responsibility of the equation writer to be clear. If one uses the leaning slash “/“ as you would when typing, then make the effort to toss brackets on there to convey the grouped terms of the horizontal divide sign.
As a retired engineer, I never saw any engineers use "÷". We always used "/" to keep equations as fractions. I agree with what you said.
But there is no "÷" on the keyboard. I had to copy and paste yours.
And it doesn't matter anyway. The same issue remains with 6/2(1+2), or with the later story which is actually using / instead of ÷ in the 2x/3y-1.
@@irrelevant_noobon paper you should clarify if 6/2 as a whole is multiplying or if 2 is and the equation is under 6 that’s just an issue with keyboards not Order of Operations
ALGEBRA WILL KICK PEMDAS ASS ANY TIME ANY WHERE ANY EQUATION!
honestly the solidus (/) is really only used if an expression must be written inline - for absolute clarity the vinculum (--) is the notation to use - it's all I ever saw used from Algebra I onward
I was taught the pemdas model but I was also taught that when there was an implied multiplication like for instance 2(1+2) or xy, that there was also an implied paranthesies around that segment, ie 2(1+2) is (2*(1+2)) not 2*(1+2) and xy is (x*y) not x*y.
....were implied brackets around the parenthesis...
I'm discovering I learned math from mathematicians. Every teacher I had in middle school had their doctorate. The only bad math teacher I had after fifth grade was my geometry teacher who didn't know how to teach me. And give me enough but I somehow passed our state's very hard geometry assessment test with a 110%... A score that should have been technically impossible but because they say choose the best answer and then they had questions that actually had no correct answer I got the best answer correct so I got credit for the best answer and the question itself.
Yes, 2(1 +2) is a grouping. Sin 2(1 + 2) ≠ Sin 2 × (1 + 2).
Just as 2pi and 2theta and 3.14 are grouped.
@@markprange2430 general substitution rules for integrating and differentiating by parts
if f:(n) = a÷b(p+q) then let c= (p+q) therefore
f: (n) = a÷bc
if a=6, b=2, p=1 & q=2.
a÷bc = 1
that MEANS , f:(n) = 1
The solution should be the same no matter HOW you work it.
If I can work it 5 different ways to get 1, and only one way to get 9, then that way to get 9 must be wrong.
@@markprange2430 AND yes, when you have something like
2x, or 2y , if x = 2+1 , and y = 2+1
6÷2x = 1
6÷2y = 1
6÷2x does NOT mean (6÷2)*x
I'm a retired physicist with degrees in Mathematics and Engineering as well. It has always been a pet peeve of mine when someone is too lackadaisical to use parentheses to specify the order of operations explicitly when the order is open to multiple reasonable interpretations without them.
In many applications, misinterpreting the order of operations in an expression can have dire consequences which could be avoided by simply putting in parentheses to explicitly show the order of operations.
I also hate when people don't specify their units or mix unit systems, and when people don't label the axes in their graphs, or don't label multiple curves on a graph or don't provide a legend.
All of these problems are far too common in the day-to-day practice of science and engineering for which paying attention to detail and being specific in technical communications are essential skills.
Well, there is some notation corner cutting, like when you write ds^2 = dx^2 + dy^2 + dz^2 . It's `understood' that ds^2 is (ds)^2 , instead of 2s ds. Same corner cutting for the denominator of the second derivative.
@@trumanburbank6899 However, ds is not two symbols like the product of d times s, it is a single symbol for the differential of s so that there is no ambiguity in the application of the exponentiation when writing ds^2 as there would be if one were intending to write the squaring of the product d*s for which ds^2 might mean (d*s)^2 or d*(s^2).
@@luminiferous1960 Actually it is the product of the operator d times the variable s.
For example d(r^3) = 3r^2dr is the the product of the operator d with r^3. Also,
a = dv/dt = (dv/dx)(dx/dt) = vdv/dx = (1/2) dv^2/dx .
ds^2 = (ds)^2 is accepted by convention as an exception to the rule.
I do agree that PEMDAS probably wasn't designed with operator algebra in mind, but it would be on par with Parenthesis.
@@trumanburbank6899 In your first example, d(r^3) = 3r^2dr, the operator is d/dr, i.e., the derivative with respect to the variable r. In the expression, d(r^3) is the differential element of r^3, not a product of d and r^3.
In differential calculus, if s is a variable, then ds is the infinitesimal element or differential element of s, not an operator.
The differential element is, as the name implies, elemental. Thus, ds is not a compound of d and s, i.e., it is not d times s, nor d of s. As such, no parentheses are needed around ds, and ds^2 = (ds)^2 is not an exception, and the parentheses are simply superfluous.
Your example a = dv/dt = (dv/dx)(dx/dt) = vdv/dx = (1/2) dv^2/dx is an illustration of the fact that the derivative operator of one variable with respect to the other equals the differential element of that variable divided by the differential element of the other variable, and the fact that the differential elements are treated algebraically as variables.
Thus, (d/dt)(v) = dv/dt, where d/dt is the derivative operator with respect to the variable t, dt is the differential element of t, v is a variable, and dv is the differential element of v.
You also did not define your variables in this example, which is necessary since you use the fact that v = dx/dt when v = velocity, x = distance, and t = time. As a physicist, I recognize the equations relating acceleration a, to velocity v, distance x, and time t, but others may not.
@@luminiferous1960 There's just a `d` operator tho when doing implicit differentiation.
I frame this as not a problem of PEMDAS, but of the ambiguity of inline equations, which you touched on in this. You're always following PEMDAS, but you have to take the denominator as a whole before you do the rest. Since the denominator itself is the ambiguous part, different people get different answers. Good video overall!
I agree, inline equations are sometimes interpreted this way. But, I have seen it both ways In the literature, it is important to pay attention to the context. PEMDAS is often not used in inline equations, but this changes when it isn't. I don't remember it not being used when it was a stand alone equation.
@@SpacePhys Implicit parenthesis after the division symbol is a bad convention. A math equation should not require any context. The context forms the equation, at which point it is pure math.
Hihi I have always enjoyed explanation and argument for maths and stuff ❤
Yeah that's true, but the point she touches on is also true. When I see these kinds of questions being answered on reddit, yt etc, even adults who did not go deep into stem or use advanced maths they would use pemdas. So pemdas/order of operations is a tool for younger kids to have a better grasp on how to do comparatively difficult questions a bit easier. So the point she touches on is valid
If the person seeking an answer gave us the problem to be solved, they would tell you how to solve it. In the given example 2x/3y-1, PEDMAS could be used if you will equate ÷ with /.
In the video, the presenter uses the Division Slash / to represent Division when doing the Order of Operations, but while reading the problem she said 2x OVER 3y-1. That would necessitate it being a fraction of 3(x) ÷ (3y-1) or 3(x) / (3y-1).
I've had this argument numerous times on facebook; it seems to come up in my feed at least three times per year. The most recent time, I had someone respond that they had been teaching math for 15 years, and that they always taught strict PEMDAS, and therefore I was wrong. My response to them was that they have been teaching it wrong for 15 years.
It's the same for me but on instagram. And it's always the replies of "You are wrong, you don't know how to do basic maths" or the "No, but PEMDAS" and I'm like: Dude, you probably don't even know why the priority of operations even exists and why is it like it is. Do not tell me to use something that you do not even understand
I tried to explain this in a twitter thread only to be bombarded by people telling me I'm an idiot who can't do "3rd grade math" despite being a maths grad, and stating it clearly both in the thread and my bio.
These threads always attract a lot of order-of-operations trolls. It's weird, I didn't even know it was a thing.
And reportedly even Nobel laureate scientists have almost come to blows arguing about the Monty Hall problem.
Yeah, it's a bit like explaining tides and eclipses to flat earthers - pretty much pointless 🤣
@@Skank_and_Gutterboy It is a thing, and it should be utilized in linguistics as well.
@@justanotherguy469
Great, I'll go find some linguistics videos and start trolling their comments sections. Consider it done!
I think half the problem is trying to write out a statement on a single line using symbols like ÷ or /. For as long as I can remember, I always carried out multiplication by juxtaposition first. 1/2πi has always been in my eyes 1 over (2πi).
*_Exactly._*
÷ and / are two different symbols entirely is the issue same as A*B and AB are different (those two are more similar though). ÷ means you are doing division of A divided by B. where / marks the right side OVER the left side, which can be broken up by a *. So A/BC and A/B*C are different, but A÷BC and A÷B*C are the same (in this case A/B*C is also the same).
You almost never see A÷BC though because it's ugly and unintuitive.
1s is one second
2s is 2 seconds
Are you telling me that 1/2s is not half a second but 0.5hertz?????
@@NunoRVOliveira "s" is not a constant or a variable. It is a unit of measure.
Also, according to the International Bureau of Weights & Measures (the BIPM), there is supposed to be a space between the number and the unit. So one second should be written as
"1 s". Two seconds should be written as "2 s". And half a second, if written as an in-line fraction, should be written as "1/2 s".
If the rules are followed, there is no confusion.
@@MilescoI would still argue it's not clear. It's not uncommon to write units in the denominator in certain fields such as Physics. The context matters.
So, I'm from croatia and never heard of pemdas. Our textbooks always had fractions written unambiguously, and now I understand why. I coldn't understand at first how you got anything but a 1 out of that equation because my brain automatically saw it as 6/[2*(1+2)]😅
According to the pemdas thing it should have been (6/2)(2+1)
Nope. The number adjacent to the brackets is part of the brackets sub-equation. PakShuMan read it as pemdas/bodmas reads. This is so weird!
Despite being from America, i also got 1, as i aways do the parentheses with any number next to it (mostly because thats also technically multiplication)
@woodendoorgarage I say you are correct. Those 2 equations are completely different and have different answers. That is how I was taught during my 7 years of honors math classes in the USA. It's exactly the same as when writing a sentence, a comma can change the entire meaning.
@@raridino600 Some people said that a number right outside parentheses should be considered as "quantity". And "quantity" binds tighter than explicit operators.
So, 6÷2a is not "6 divided by 2 times a",
but "6 divided by twice of a".
The word "twice" obviously has no meaning if coupled with "6"; "6 divided by twice" is meaningless.
In the problem's case, a = 1+2 = 3
when i learned the order of operations in algebra in eighth grade, we were told to NEVER use the ÷ and to ALWAYS use the fraction bar (vinculum) because that is a grouping symbol whereas the ÷ is not. And you can never use too many parentheses for clarification. Thank you Ms Goslee!!!!
You didn't learn the Order of Operations until 8th grade?? Wow.
_______
2(1+2) two grouping symbols
(2(1+2)) two grouping symbols q
6÷(2(1+2))=1
6÷2(1+2) does not equal 1
@@RS-fg5mf Can you please tell me the difference between ÷ and /
@@RS-fg5mf please
@@suineitling there is no mathematical difference. The obelus ÷ and the solidus / are both equal signs of division and neither one serves as a grouping symbol. The solidus / is more prominent these days when writing inline infix notation mainly because the obelus ÷ is used in some countries to represent subtraction instead of division.
Anther reason is that prior to 1917 some text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math that were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses when dealing with inline infix notation so the ERROR was corrected post 1917.
This ERROR i.e. misuse of the obelus by the text book printing industry to try and save time and money was wrong and means that 1 is not and has never been the correct answer...
6÷2(1+2) and 6/2(1+2) are mathematically the same. Both equal 9
@@suineitling Practically there is no difference. As a rule of thumb just do what the equation tells you.
6/2(1+3). This equation says 6 OVER 2 times 1+3. Or it says divide 6 by the product of 2 times 3. Practically, if you see a division sign or slash then everything to the left is the numerator and everything to right is the denominator. Let parenthesis tell you anything different.
I remember that we were taught (70 years ago) that the horizontal line between dividend and divisor should be treated as parenthesis. This seems to make everything work
I might need an example.
which you cannot express in a computer language. Only parens provide grouping in computer languages.
@@garymartin9777 You could probably make a language that it works for, though that doesn't mean it exists.
@user-ij3we9jp5j Well that won't work for the story at 4:29... since it doesn't mark where the parenthesis end. :-|
?
It's funny that you can actually show people how real mathematicians and people who use mathematics professionally understand these expressions and still have people arguing its wrong because it's not what their junior school math teacher told them.
.
These are people who have over-reliance in rules. They see the rules as something absolute and not conventions. Ask them why is multiplication done before before addition and they can only say PEDMAS.
The answer is simply. If you buy 5 apples à 2 € and 7 oranges à 3 € how much does it cost. It is nice if you can write 5*2 + 7*3.
I did not see any example showing ÷ symbol. Either because they don't know how to write/type the symbol or real mathematicians or people who use mathematics professionally don't use the ÷ symbol. The examples showed if there is / that means group numerator and then denominator.
6 ÷ 2(1+2) ≠ 6 / 2(1+2)
The video showed nothing. The original expression contained ÷, which is the source of ambiguity, and then immediately pivoted to using the fraction bar. They are not the same thing.
And then there are English teachers who say, "Never begin a sentence with a conjunction." But good writers aren't held to such silly limitations, are they?
@@ILoveCunnilingus that symbol ÷ (called the obelus) is no longer of use according to ISO 80000-2, only "/" is used (a consequence of it being used in most programming languages and of keyboards not having the obelus symbol I think), and it was rarely used by mathematicians and scientists either in the first place, it was mostly used in some printed books). But ÷ and "/" were always the same thing. 1÷ab was also 1/(a*b).
My initial instinct was to distribute the leading 2 through the expression in the parentheses (either before or after resolving the expression in parentheses), which gets 1 because the parentheses are still intact after doing so. I always interpret the number before parentheses as being "bonded" to the parentheses unless there is a multiplication symbol separating them.
I agree, perfectly good logic.
I always saw the (x) as a way to show that the number within the brackets is seperate from the others.
There is a multiplication symbol between them it's just implied rather than written.
@@nekogod I was told to only omit multiplication symbols if the values belong together
Why are you omitting symbols if there is no relation between the values?
That's multiplication by juxtaposition. It's just tha juxtaposition also applies to numbers and not just variables. I always interpret 2(3) like 2x (but x=3).
Thank you for being a glimmer of sanity against the winds of smug RUclips math PEMDAS zealots ignoring implicit conventions pervasive among people actually using math for useful things like science, engineering, etc.
Simply put, don't rely on your systems algorithm to work the way you expect. Control the outcome and reduce the probability for logical errors by utilizing parenthesis whenever ambiguity in the order of operation may exist.
This was the way I was taught PEMDAS by my math teachers, at least starting in 7th grade algebra (we didn't use PEMDAS before that), that multiplication came before division. I was very confused when I started seeing people online trying to solve these math equations and they were doing multiplication and division in the order they came, and not giving multiplication priority. I should also note, that none of them wrote in-line equations when it came to division. It's just not how we were taught, but then again starting in 6th/7th grade is when we started algebra and then moved to geometry and trig in high school.
I was also taught this method
We were taught BODMAS (as you can see D comes before M) but still 6 ÷ 2(1+2) is obvious to do the juxtaposed multiplication first. Mainly because if it wasn't meant to be first then they would have written it in a way that doesn't suggest that it is first.
I was also taught multiplication first
Same here. Which makes it seem kinda weird when she says PEMDAS is wrong and yet the right rule follows PEMDAS exactly as it was taught to me. Makes me think the thing of doing multiplication and division in the order written must be a newer thing I think and PEMDAS the way we learned it was older? So it's not so much that PEMDAS is wrong, but that it's being taught to mean different things.
Same. I never knew people did the blindly following the order of division and multiplication until the internet.
I've learned it as implicit vs explicit multiplication. Implicit meaning the multiplier is a common factor in an expression and should be evaluated at the same time with its expression, whilst the explicit multiplication must be written out with an explicit multiplication symbol of " x " or " * ". And for cases with 2x / 3y -1 in order to include the "-1" into a single expression you'd use parentheses like 2x / (3y -1).
Unfortunately there is no central authority to make official rules. This implicit vs explicit rule that a lot of people try to insert into these kinds of meme internet problems don't work. If there was a difference between implicit and explicit multiplication then 2x and 2*x would be different. Likewise 2(1+2) would be different to 2*(1+2). And if they were different then contracting 2*x into 2x at any point would be a terrible idea and we would have a 'rule' not to do that etc..
Our 'rules' have ambiguity, they always have and likely always will because the 'rules' are often not rules and are actually just conventions that some people make up to try to make it work better. Even the video called these 'unofficial rules' that engineers and stuff use when writing and interpreting formulae. The reason we have parenthesis and why that is evaluated first is because this ambiguity was a known problem and the parenthesis was a way to address it, but it only works when the person writing the equation uses them correctly.
If a person writing the original doesn't see how it could be interpreted differently then they might not be clear enough when writing it down, which can lead to different people getting different answers. But the only person who knows the 'correct' way of evaluating the formula is the person who originally wrote it. No amount of arguing on the internet about what the correct order of operations is will get you the 'only and only absolutely correct and everyone else is wrong' answer.
@@Subjagator Let me quote myself:
"I've learned it as [...]".
I did not claim it's "more betterer" or in any way shape or form the LAW.
@@marian-gabriel9518
Of course. It is just a pretty common thing that comes up on these kinds of videos. Yours was the first comment that I came across which mentions different types of multiplication with different rules etc.. so I added my two cents. It is a public forum after all.
I was in no way trying to straw man an argument or anything and if that was the impression I gave then I can only apologise.
@@Subjagator 2x is one expression, 2*x is an operation. Imagine 2pi would be read as 2 x pi everywhere. Then we should effectively use “tau” in stead.
@@Subjagator _"If there was a difference between implicit and explicit multiplication then 2x and 2*x would be different."_
And it is!
_"Likewise 2(1+2) would be different to 2*(1+2)."_
And it is! 6 ÷ 2(1+2) = 1. 6 ÷ 2 × (1+2) = 9.
The operator (the multiplication sign in these examples) serves to separate the coefficient from its variable so that it doesn't get the priority that implied multiplication gets.
_"And if they were different, then contracting 2*x into 2x at any point would be a terrible idea and we would have a 'rule' not to do that, etc."_
In real life, engineers and, well, just about everyone I've ever come across, rarely if ever use actual multiplication signs, so the idea of "contracting 2*x into 2x" isn't an issue. If it were ever necessary to insert a multiplication sign to avoid the priority that implicit multiplication by juxtaposition creates (e.g. "2 × x" or "2 · x"), most likely the person writing the expression would use parentheses in that case.
But contrary to your assertion, the universal and ancient convention of the priority of implicit multiplication by juxtaposition is so, well, universal and ancient, I believe it can fairly be considered a "rule". Indeed, the creator of this video has referenced two authoritative sources that say so. I don't think there's a mathematician or engineer or similar professional in the world who would say that a/bc is anything other than "a" divided by the product of b and c.
As others have commented - ALWAYS be explicit in how you write equations so that there is no need for interpretation. Use brackets & parentheses, write clear fractions, etc. As an example, put parentheses around exponents with multiple terms.
Parentheses may also be necessary for proper order of (multiply - divide) or (add - subtract).
I somewhat disagree. Overusing brackets can make an equation a lot less readable, which is why in papers and textbooks in fields utilising a lot of maths you will find xy/zw rather than something like (xy)/(zw). As far as the maths itself is concerned there is no real "proper order". So long as everyone agrees with the notations and conventions used in a field, it's all good.
@@adamlindstrom5750 there is no such thing as "overusing" brackets. you either need them or not. they solve a problem. left to right is the proper order is brackets are not used.
@@adamlindstrom5750 unfortunately, the key problem is no one can agree! Everyone was hot different methods, and I don’t think we should call anyone method wrong. I may have been taught pemdas, but people might’ve been taught different. The issue comes with the fact that since people aren’t at the same things, we need to be more clear about what the problem wants using parentheses/brackets/braces/and an actual fraction, not just a slash cause that gets confusing too. Because the slash was sometimes taught to people as a division, but for other people it was taught as a fraction (division using grouping).
@@adamlindstrom5750 You're essentially saying it saves time by not writing extra brackets. However, what if you factor a fraction? Let's say you have 20 and want to factor 1/2 from it. You would now have to write (1/2)40, instead of 1/2(40). Either way you do it it'll require the use of brackets somewhere, so this doesn't save any time, or readability.
@@mariuspuiu9555 brackets are not needed when you have juxtaposition, so using them would be an overuse. If you use brackets to communicate to people who were not taught the correct order, you should also consider that your audience might not understand brackets.
As I'm an Engineer, you are so right. I would never write an equation that way. There is a big confusion factor with that method.
if you are an engineer you'd know this is not an equation, it's an arithmetic expression...
@@Ignoranceisbliss-i2e
If you have to do crap like this to flex how smart you are, you're not.
Exactly. Anybody's engineering boss would have their red pen all over that submission. If your submitted equation can be interpreted >1 way, you're doing it wrong.
@@Skank_and_Gutterboy don't show the world how thick you are...could affect your job prospects, assuming of course you're worth employing...
@@Ignoranceisbliss-i2e
I'm a senior engineer and have been doing that job for 24 years, I think I'll manage. Disagreeing with you makes somebody an automatic moron? Nothing arrogant about that. You're only proving my point.
It's definitely a consistency issue across the board, but it is often related to presentation, not function. "Context is important", but beyond context, there still should be some standardization throughout. What you learn in primary school, what you learn in college, and what you use in the fields of math, science, and engineering for the rest of your life should all be the same so as to remove this ambiguity. In these fields, it is often critical, because a simple misunderstanding or misrepresentation of context can cause someone's pacemaker to fail, or send a spacecraft smashing into a planet uncontrolled.
That said, PEMDAS is still very common in computer programming languages, because they have to explicitly and uniformly handle the translation of syntax to function. Since the vast majority of languages use simple human-readable text to represent the source code, they have to have a common order of evaluation, which often includes PEMDAS as a basis.
The printing resource excuse may still be necessary for dead tree publishing, but a great majority of knowledge is moving or has moved to digital, so that excuse is now pretty much obsolete. We have graphical expression languages now, so we can properly display formulas and do away with the problematic shortcuts.
in the sciences the brackets that should be there are removed for laziness and the assumption that everyone who reads it should know that it is implied.
parentheses, exponentiation, adjacency, (multiplication | division), (addition | subtraction ) isn't even really different than what we generally teach primary school students. merely an extension thereof. primary school students, if my memory serves, simply never use adjacency to mean multiplication. Elementary students always use operators between terms. Adjacency doesn't hit till algebra for students since it makes no sense till you have variables. The algebra teachers merely should mention that algebra extends PEMDAS with the knowledge that the adjacency of coefficients binds tighter than operators, while any exponentiation, which isn't an operator, will still apply only to the exponentiated term.
Computer languages don't use juxtaposition because it changes the name of a variable. If you have variables x and y, then xy is not x*y, it is a compiler error caused by an attempt to reference the non-existent variable "xy".
Something that I picked up over the years which is not obviously stated or taught very often is that the fraction line actually implicitly groups both sides whereas normal division “÷” does not have that grouping property. The order of operations are completely consistent with these textbooks since they use the fraction line for their equations.
Look into the origins of the obelus as a symbol for division-- it worked via line division.
The “÷” symbol used to mean the same thing as line division? Does that mean it used to group both sides of the operation?
@@drednaught608 the text book printing industry tried to push the obelus as a grouping symbol to be used like the vinculum because the vinculum took up to much vertical page space for the printing methods at that time...BUT this was in direct conflict with the Order of Operations just as parenthetical implicit multiplication is in direct conflict with the Order of Operations.
The obelus ÷ and the solidus / are equal signs of division and nothing more...
Parenthetical implicit multiplication does NOT have priority over division despite the misquided and confused notion of many people....
Math is not based on opinion or popularity it's based on rules and the rules do not support parenthetical implicit multiplication having a higher precedence...
Every equation came from fraction in case u don't know.
That is true
As one that taught engineers, we did teach BEDMAS as a rule of thumb. But we also put the onus on the student to make sure it was clear to the guy giving the marks what your math meant. Like I read below, you can never have too many brackets to group terms and processes in your solution.
>> you can never have too many brackets
@@FrostSpike machine code and assembly language.. Fast and efficient. no brackets.. lol just a pain in the ass professor that made sure there were no mistakes so my engineers understood everyone has to understand there work./
The 2 is linked with the parentheses, and is reinforced by the distributive property. The 2 is still connected even after adding. This is proven by distributing it out which forces the expression to become 6÷[(2×1)+(2×2)].
In the end, using proper notation when developing problems is important. Also knowing ir stating the conditions placed on expressions for the format in use.
To quote Pirates of the Caribbean - "It is more what you'd call guidelines than actual rules" 😊
*This* 100%.
The expression 2(3) has not yet been resolved as a parenthetical operation, and is not equal to 2*(3).
Just because people misinterpret the rules, does not in any way make the rules wrong... just improperly taught/understood.
Ignorance of the Law is no excuse.
have seen a lot of this misinterpretation of PEMDAS and was extremely shocked to find out that even Google algorithms use the misinterpretation
Its probably necessary to overload any equation or expression with parentheses to avoid ambiguity
Strangely enough when I learnt BODMAS i learnt that the O stood for both Order (Exponetial) as well as Of which meant evaluate completely any expression within or adjacent to parenthesis
so in this case 2(3) has to be evaluated before the division. Reading the expression would be 6 divided by 2 of 3.
2 of 3 means you have two sets of 3 but these guys have turned this completely around and decided that this statement should read 6 divided by 2 times 3 which is not quite the same thing
@@drift_ah1518Yep. This is why I used the Distributive Property in my example. This is an actual mathematical "law" where as PEDMAS is only a procedural ranking based on a set of conditions and in this case based on the "grammar" of writing it in one line rather than long form on paper.
@@drift_ah1518 adding in additional (and needlessly complicated and completely irrelevant) parentheses into a mathematical expression, only to hold the hand of people who never understood the concept loses some of the elegance math is capable of.
When leading Trolls and Ogres through a Magical Forest, they are bound to mash some flowers into the mud.
Better to teach them the *right* way to do it, as some of these RUclipsrs try to, with the assistance of others. And maybe fix the education systems that pass people ho don't actually know what they ere taught.
@@corwyncorey3703 lol - am having quite the debate elsewhere with someone that just wont see the logic
Brilliant! As a math tutor for elementary, high school, and college students, I've been fighting against the tide of PEMDAS forever. Allowing the use of the ÷ and / symbols has been the greatest detriment! Thanks again for your presentation here!
What is really stupid is the new CORE math my kids had to learn, what a dumb confusing mess of nonsense !
@@drinkoldcokeIs that the same one that required us to do black out poetry during my Physics class back in High School?
@@drinkoldcokeCommon Core is what educators came up with when they analyzed what all the countries that had high math scores and successful math programs, had in common.
I think all calculators in schools should be required to feature something like Natural V.P.A.M, Writeview, Textbook Display, or MathPrint. These are pretty-print input methods from different manufacturers for equations and expressions which allow for a more natural visualization of the syntax by utilizing a dot-matrix display or other high resolution display to allow the user to select the function and navigate empty fields with the directional keys to fill it in, automatically adjusting the visual representation as necessary. This allows students to type in an expression or equation the way they would see it in their textbook or how they would write it instead of needing to figure out how to write the in-line notation so the syntax is unambiguous.
My favorite will always be the Casio FX-115ES series of calculators due to their impressively cheap price and eminently versatile feature set. You can 100% use one of those to get a Bachelors in a math field, no problem.
Is anyone else as impressed as I am that she can write that well with a mouse/pointer/clicker???
I'd love to claim those mad skills, but I'm using a stylus on a yoga 260 thinkpad :)
@@THaWoM ohhh I see! 😝 Still very satisfying handwriting nonetheless! Also love your voice 🤭
No one asked
EZc1apz Rude
I'm a bit concerned that she writes the letter "x" by using two mirrored "c" characters.
In engineering we'd read it as 1, which is why I was a bit tilted at the start. PEMDAS does apply, but this question is not framed properly, if you input this into your calculator, it will auto correct the input to 6÷(2(3)) which gives 1. So if the question is framed incorrectly, it makes it hard to understand what to do. In my years of study I came accross this issue maybe once or twice, it's very rare that someone does not frame a question in such a way that makes it easily understandable, especially since these days we calculate our equations using Matlab, so naturally you need to us parenthese very precise for the computer to know what to do.
Is it a casio calculator? Because I know casios do that autocorrect thing, but it'd be very interesting if other manufacturers are adopting that behaviour.
@@THaWoM yes it is. Would be interesting to know if the other manuefacturers do it too as well as the programmable models, like Texas.
@@THaWoM
All proper calculators follow the PEMDAS (etc.) rules these days - these rules are built into them, and they use a stack to make it possible to delay intermediary results until they are needed.
What you have to watch out for is old and / or very cheap calculators without these features, which can only handle two operanda and one operation at any one time.
I did the same.. I just assumed I'd make an easy mistake.
Yeah, I'm not very well-versed in math but wouldn't PEMDAS literally make the answer 1? Parentheses first (3), then Multiplication (6), then Division (1). I don't see how PEMDAS could possibly lead to the answer being 9. It seems like willful wrongness.
Thank you for figuring this out. My integral calculus instructor, who also taught nuclear engineering labs, made this basic algebra.
He used to say “Calculus is easy. It’s the algebra that trips most people up”.
It's not that people struggle with algebra. It's that there are inconsistent conventions. Writing something in an ambiguous manner and blaming others for misinterpreting it is the writers fault.
@@Andrew-it7fbNo, he saw errors in algebra.
@@jimbarrett5930 I'm sorry that you don't understand the difference between math and convention.
@@Andrew-it7fb I do. Intimately.
@@jimbarrett5930 If you did, we wouldn't be having this conversation.
I'm 55 and Canadian and was taught BEDMAS, but like other commenters was always instructed that parentheses/brackets were not solved until they were eliminated, thus the order (in this case) would be what's inside the bracket, then the multiplier next to the bracket, then the divisor, leaving the answer of 1. But I can definitely see where it could go wrong with other equations laid out differently....
The implied multiplication is not connected to the brackets. Sure brackets allow its use: you can write 3(2) but not 32 to mean 3*2. You also can use implied multiplication with variables. If you had x=3 and did 6/2x you still would get the difference even though parenthesis are nowhere.
@@okaro6595 you actually can't write 3(2), implicit multiplication must always contain a symbol (both from usage, as you can see in the examples she provides, there are always symbols/variables in implicit multiplications and also from the ISO 80000-2 which defines the rules for mathematical and scientific writings). This internet meme is just improper writing and the answer should be "no value" :). But of course if somehow you *have* to decide what value it should have, any mathematician would reluctantly interpret the 2(2+3) as an implicit multiplication and answer 1.
Thank You ! 53 here and this was junior high math class for me as well !
I've had same rule implemented, PEMDAS but not until you have parenthesis eliminated and that pretty much solved the problem
As a physicist that's what I was taught as well. Later, I think in high school, it was explained that juxtaposition meant the juxtaposed terms were to be thought of essentially as a single term, essentially an implied parenthesis. The other thing I learned that to avoid confusion, use appropriate parentheses when needed rather than order of operations to ensure proper grouping. One other thing I learned in life, this time from a linguist, is that many of the "rules" taught in school, not just PEMDAS, are merely teaching tools to make the job easier for teaching elementary students and weren't meant to be used for higher levels.
Not to put a fine point on it, but maybe lazy instruction at the starting level is part of why our education system sucks. It's easy to learn something when you're a child but unlearning something or relearning something is very difficult. IMO, math equations are a precise set of instructions for calculations and should never be a question about it. As you said, use appropriate parentheses.
Juxtaposition does NOT indicate grouping! Juxtaposition simply indicates multiplication between the two quatities being juxtaposed, as if a multiplication symbol (e.g. "*") had been explicitly written between them. So 3x means "3*x" not "(3x)" and a/bc means a÷b*c = (a/b)*c not "a/(b*c).
However, the horizontal fraction line, called a "vinculum," DOES indicate grouping, both above and below the horizontal line. Thus, mn/ab simply means m*n÷a*b = m(n/a)b. This is NOT the same thing as writing a horizontal fraction line (vinculum) with mn in the numerator and ab in the denominator, which means (mn)/(ab).
The essential misunderstanding is that the slash symbol used for division in command line interfaces is NOT equivalent of the horizontal fraction line (vinculum) symbol commonly used when students do math with paper and pencil: The former is NOT a grouping symbol; the latter IS a grouping symbol.
It was the widespread adoption of command line computer interfaces - wherein vinculums are not available as symbols (as they are when doing math with paper and pencil) - that screwed everything up.
Graphing calculators just solved this problem, recently. Newer editions of the Ti-84 graphing calculator operation system, using "Math Print," now allow the on-screen display of vinculum symbols to indicate division (and fractions) exactly as students have traditionally done using paper and pencil. Doing so has completely removed the ambiguity, confusion, and errors caused by non-equivalent use of "/" in place of a vinculum to indicate division.
en.wikipedia.org/wiki/Vinculum_(symbol)
@@chrisborland4972 While that may be the case in the area you're from or the circles you frequent, that is not necessarily the case everywhere, as the multiple examples provided in this video illustrate. Among multiple professionals, the usage is obviously different from the one you describe.
I do personally agree with your interpretation, but the only proper answer is: the usage of the symbols is not uniquely determined, and therefore the "instruction" is a priori ambiguous.
As with any text, the meaning is contextual, and you must learn to become acquainted with the author's meaning from the surrounding text.
But, as you mention, an author who is not certain of their readers' abilities should therefore use parentheses or the horizontal fraction line to avoid ambiguities of interpretation and thus confusion for uninitiated readers (such as elementary or secondary students).
@Flutesrock9800 Thanks for your thoughts. However, I strongly disagree, and I think I can prove you're in error.
Without a universal standard as precisely what mathematical symbols do and do not mean, how are we to cooperate on anything mathematical? How are we to relate mathematical ideas and results? How are we to understand each other and collaborate?
There would simply be no way to do so.
In fact, mathematical symbols (like “/“) and conventions (like juxtaposition and the order of arithmetic operations) ARE standardized - and those who do not conform to these standard and well-established usages of math symbols and conventions are just doing their work incorrectly - whether or not they happen to be “professionals.”
The simplest way to test this is to use any computer programming language to assign values to the variables in the expressions I listed, ask the computer to evaluate those expressions with the given values, and see what value the computer returns. In each case, the computer will return the value that I have stated is the correct value.
For instance: If m=2, n=3, a=4, and b=5, mn/ab (or the equivalent m*n/a*b) will be calculated by any computer language as m*(a/b)*b and the value returned will be 7.5. It will NOT be calculated as (mn)/(ab), which would instead return 0.3.
If it weren't for agreed-upon standards defining correct usage, it would be impossible for different computing machines programmed according to different "usages" of mathematical symbology to share calculations productively (since each would return different values under identical conditions), and the information age would never have arrived.
Those who do not employ standard usage of symbols and conventions are being unrigorous, unprofessional (even if they happen to be “professionals”), and certainly unhelpful. At best, non-standard usage exposes a habit of sloppiness and laziness on the part of the writer.
Mathematicians and computer scientists have decided upon standard conventions and clear definitions for mathematical symbols. These standards define correct usage. All other usages are incorrect, as demonstrated by the simple programming experiment I’ve suggested above.
The verdict is in.
There is one - and only one - correct way to employ mathematical symbols and conventions. No matter what “multiple professionals” may or may not do, “alternative” usages of mathematical symbols and conventions are just wrong. Full stop.
@@chrisborland4972 As an engineer with a PhD, mn/ab≠m*n/a*b. mn/ab=(m*n)/(a*b).
We did BODMAS in school but I think I remember learning to always aim to get rid of brackets before solving the whole problem, which in this instance gives the correct answer. I don't think I've ever heard the idea that we should deal with multiplication or division in order of left to right.
It is still wrong! DM can be MD
You have to deal with it from left to right with DM because we read from left to read and if you don’t then you end up with different answers, this is not the case for AS because the order doesn’t change the overall answer, it’s the same regardless of whether you do A first or S first.
I learned multiplication and division are in the same hierarchy so you do left to right, not multiplication then division, just left to right. think about 4/6*3, it wouldn't be 4/18, it would be 2, so the rules do infact matter. also it was only for 6/x, where x is 2(1+2), and the new rule states "when slashing a fraction, this is the accepted order of operations: raising to a power, multiplication, division, addition and subtraction" it's far different than the way we would do other equations without slashing fractions because pemdas/bodmas always do parenthesis, exponents, multiplication and division, addition and subtraction. left to right also only apply for multiplication and division, and addition and subtraction.
So with 6 ÷ 2(1+2) we get rid of brackets (parentheses) to get. 6 + 21 + 2 = ? Correct? Which, accordung to, PEDMAS, would give 6 ÷ 21 = 0.2857, then + 2 = 2.2857
@@aspenrebel no wrong, you are not using BODMAS/PEDMAS, you need to follow the order.
I have no idea where you get 6 / 21 + 2 from
6 / 2(1+2) = 6 / 2(3) = 3 x (3) = 9
I was taught PEMDAS but I was also taught to multiply by juxtaposition before moving on to division. For reference I was in elementary school in the 1990s. When 6÷2(1+2) came around I thought the world had gone insane or that I was being gaslit.
Thank you for making this video.
In 1979 I was taught this in 7th grade and I guess I had a good teacher, for it was ever so simple! First 'in between the brackets', then what's 'attached to the brackets'. Then the rest according to EMDAS (we of course already used EMDAS within the 'P' part of the brackets too).
A(B+C) is A * (B+C)
The A is not in parentheses and does not solve during P, in PEMDAS.
@@leighz1962 Incorrect. Guess you never heard of the Distributive Law. The Distributive LAW states: A(B+C) = (AB + AC). This is actually one of the fundamental laws of Math.
So the facts are the number attached to the parenthesis should be moved into the parenthesis since that is actually a LAW in Math.
It's 100% NOT a law. It's called the distributive PROPERTY of multiplication and its no more powerful than the operation next to it. There is no division or subtraction really. All of it can be reduced to multiplication and addition (of inverses) so that they are closed under the respective operations. I'm actually a little annoyed that this presenter referenced the laziness of book writers to cite that Pemdas means nothing. There IS an order of operations and it's not helpful to have something like this go viral.
Except 6÷2(1+2) would = 9. If you wanted the multiply first you'd simply add the bracket or right it as a fraction properly. Her whole explanation is bullshit. Paraphrased as "people don't write things properly so order of operations shouldn't apply". Theirs absolutely no ambiguity with pemdas. People who argue ambiguity just don't follow the order. And people that write the equation and claim to want a different order should write it properly
I really think this whole confusion comes from typewriters, keyboards, etc. not being able to type out the actual notation you would see in a textbook. I think after seeing it written in the textbooks so many times, I just kind of knew what they meant when they tried to type it out in a paragraph. My TI-89 Calculator does Pretty Print notation so you can better understand what is going on.
I agree completely. I graduated high school in 2007 in the increasingly digital age, all our tests and exams were written on PC and printed out but everything in math was formatted correctly to avoid this problem such as
1
------ = y
2x
but I always understood 1/2x=y to be the same thing because otherwise it would be written as x/2=y
In mathematics you avoid things like "a/bc". Not a correct way of writing down a mathematical expression!
This is creating confusion because in the early days printing of the correct way was often not possible and took too much space.
So pemdas etc. is mathematical correct and so you write
a
------
b c
OK, but isn't 5 (6 + 3 x 4 / 2) reallly PEMDAS instead. It gets much worse. How about
5 (6 + 3 x 14 / 2 (3x5) / 5)
@@jeanpauldelauw7716It's incorrect only in the eyes of the undereducated.
Get lost!
@@sqlexp
Exactly. You have to follow what is meant by using the parentheses (brackets as we called them) in the first place. I went to school in the 1950s and we were taught that a number touching the brackets and the numbers inside the brackets were all part of a function which had to be carried out as a separate operation. And we were taught to do that first before doing anything further. So a(b+d) was (ab+ad) and not to be translated as a X (b+d) in a longer expression. This is logical and avoids confusion.
On a keyboard, curved lines ( ) are parentheses. Cornered lines [ ] are brackets.
@@anchorskid We didn't have computers when I went to school in the 1950s here in the UK. And we called them brackets at the times.
@@anchorskid in the UK we use BODMAS - Brackets, Orders, Division, Multiplication, Addition, Subtraction. So they're definitely brackets here, and the wikipedia article for Bracket agrees. Even though BODMAS specifies division before multiplication, and juxtaposition is not included, we always multiplied juxtaposed symbols as a priority.
@@shaunpatrick8345 We use parentheses for these symbols ( ) wherever they're used. These [ ] or these { } are brackets.
So glad you posted this as I think the same. I remember reading Donald Knuth’s argument that multiplication written as juxtaposition binds more tightly than PEMDAS suggests, whereas expressions using multiplicatio and division signs would tend to proceed left to right.
I remember presenting a view about the viral problem, playing devil’s advocate, and seeing both sides but preferring to the the juxtaposition multiplication first, and I had so many rude comments, one person telling me I should hand back my maths degree certificate as I didn’t deserve it!
Where did Donald Knuth argue this? (Just asking for the source, not disputing it--it is certainly true.)
@@pandoorloki1232 the textbook to one his programs, METAFONT. It’s basically a programming language for building fonts, but the rules for building expressions include multiplication by juxtaposition, not generally seen in either typical high level languages or Excel. And multiplication by juxtaposition binds more tightly than multiplication by *. It’s in his METAFONTbook.
that complicates parsing and was probably rejected early on for that reason.
Mathematician: mn is mathematically equivalent to m*n!
Programmer: That's a totally different variable. You could do mn = m*n.
Mathematician: That's illegal!
Programmer: You telling me dy/dx is three different variables?
As a programmer, I apologize for our abuse of the equals sign.
As annoying as it (sort of) is to write := or similar for assignment in the languages that use it, it's probably a saner choice.
Absolutely. I've been an engineer for 50 years and your presentation exactly matches what I, and my colleagues, have been doing during my entire university studies and engineering career. If in there is any possibility of misinterpretation / doubt, parentheses (brackets) are utilised.
YES!!!! No matter how many you have to use. Otherwise that spaceship might be lost.
@@aspenrebel The Guiness Book of World Records used to list the most expensive typographic error as a multi-hundred million-dollar launch that failed for similar reasons. Parentheses are cheaper.
@@cliffordschaffer5289 yup!! Did I ask this question, somewhere? What is 6 ÷ ab? Where a=3 and b=2. The Answer: 1. But the "woke", New World Math, Globalist will tell you it is 4. "ab" is one complete term. Just as "2(1+2)" is one complete term. I cannot see anyway to write 6 ÷ 2(1+2) =, under the PEDMAS-er's Rules, to make the answer 1. Yet under centuries old pure math, the answer is 1.
I agree that in professional mathematics and physics papers, that's the typical usage.
On the other hand, the typical usage in *programming languages* that accept algebraic expressions is really strict PEMDAS (a convention probably inherited from FORTRAN many years ago).
Scientific calculators are all over the place.
And all of these expressions are using a slash / , which suggests a fraction bar, rather than the primary-school division sign ÷ .
I tend to think that if you're inlining a mathematical expression you should probably add parentheses to disambiguate these things, but often papers don't.
Different languages have different rules of precedence for example C vs. Pascal. In anyway PEMDAS does not even begin to address the complexity to operations in programming languages.-
The added set of parentheses really is the key to knocking off the ambiguity.
FORTRAN and other languages supporting mathematical expressions in a similar way will not accept an implicit multiplier anywhere. You must put in a multiplication operator (typically *) or the example will not be accepted. Once you do, it becomes easy to see how the expression is evaluated left to right with a "PEMDAS" precedence. If you intend different grouping, you need to show it with additional parentheses. FORTRAN furnished a method of stating any explicit calculation unambiguously, which made the engineers happy. It did not provide for a mix of implicit and explicit calculation operators and would give you an error diagnostic if you tried to compile that into a program, so did not make lazy algebraists (intending a nod to blackboard style notation) happy.
Having a masters degree in computer science, my take on this is "What's the big fuss. Write what you mean, using the rules of the language, and if someone gives you an incomplete expression, tell them so and don't proceed without clarification." This is not being passive aggressive or obstructionist. This is being a computer, a machine, which promises nothing but to be a machine.
A well formatted paper will arrange to format a blackboard style equation when intended, even if it has to incorporate extra space to do so. The viral example is simply an unhappy mix of the two.
PEMDAS is a subset of PEJMDAS. Computers use PEJMDAS just like all the professional bodies she cited, but because they can't use juxtaposition PEMDAS works too.
I learned that you were supposed to keep multiplication values where the multiplication operator isn't visible together as if they were a whole like in "1/2x". Here "2x" is grouped together by a hidden multiplication operator and should not be dealt with separately. This would be the same as "1/(2*x)". So you first multiply "2" by "x", then you divive "1" by the value you just got (the result of "2*x".
this is correct. Am just shocked but not entirely surprised that Google algoriths also follow the wrong interpretation of PEMDAS as "explained" in the video. I will never trust Google or Android calculators and will have to insert unnecessary parentheses to avoid wrong answers
Mathematicians don't write "1/2x" like that for precisely that reason. It's not ambiguous, but it's too easy to misinterpret. As written you have 1/2 of x, not the reciprocal of 2x, despite what you may have been taughtt.
@@drift_ah1518 The expression 1/2x is half of x, not the reciprocal of 2x. This is the only interpretaion of the order of operations.
@@herbie_the_hillbillie_goat Technically, if you can interpret it a different way, a different interpretation exists. Whether it's misinterpreted can only be determined by the author of the problem, perhaps they're the ones who miswrote it.
I find 1/(2x) to be a more natural representation, because implicit multiplication seems like scaling. A scaled value is still a single value, not a multiplication. It's a "thing that's twice the size of x", but not "two times x". If it was the latter, it should be explicitly written as "2*x".
But then again, everyone uses the fraction representation to avoid misinterpretations. It's ambiguous, neither is right or wrong, it depends on consensus and there is no such thing in this case.
@@Ignas_ The debate around the expression 6÷2(1+2) often boils down to the interpretation of implicit multiplication. According to the widely accepted oerder of operations, multiplication and division should be performed from left to right. Therefore, 6÷2(1+2) would be evaluated as 3(1+2), which equals 9.
I understand the temptation to view implicit multiplication as a form of scaling. You suggest that 2x feels like a single, scaled unit rather than 'two times x.' However, this subjective interpretation doesn't override the existing mathematical framework. In this framework, implicit multiplication is no different from explicit multiplication: 2x and 2*x are the same.
Moreover, you question why 2 should be grouped with x but not 1/2. In both cases, the numbers are being used to scale x. From a mathematical standpoint, there's no reason to treat 2 and 1/2 differently. They are both numbers that scale x, whether implicitly or explicitly.
It's true that ambiguity can be resolved by using fraction representation or parentheses, but the need for such workarounds highlights a lack of clarity rather than a genuine ambiguity in the mathematical rules themselves.
I'm an engineering student, and it is almost always beneficial in the field to represent division with a fraction. We use fractions A LOT in calculations, so using it as a division symbol is kinda killing two birds with one stone most of the time. There were many times in calculus where I needed to cancel out terms in a fraction to make an equation more doable. Having the numerator be physically on top of the denominator lets me better see if there is something that I can cancel out. There are also advanced mathematical concepts like Laplace Transforms that are, most of the time, completely reliant on fractions to rewrite tough equations into something more doable. In addition, using a fraction eliminates the need to put more parentheses in an equation to clarify what a division symbol is dividing from what. It just makes things more clear in my opinion.
not the best thing to dumb it down. Example 5:5+5=? vs 5/5+5=? If dumbed down to "use fractions" one can interpret as 5 divided by the result of 5+5 when 5 divided by 5 then add 5 needed.
Correct
Absolutely. Sometimes we try to write, or type equations out on one line, and it confounds things. I definitely found writing the equation out long form on multiple lines for fractions made this make much more sense.
@@AudriusN For the question 5/5+5=?, the fraction would be with the 5 in the numerator and the other 5 in the denominator, and then the +5 would be completely outside of the fraction. That or if you did want to do 5/(5+5)=?, putting the 5+5 in the denominator would also send the message that you wanted the 5+5 to be done first before the division. It is super useful and concise to use fractions in engineering math problems because sometimes you need to divide a number by the result of a bunch of variables that need to be added and/or subtracted together and it's nice to show it on 2 lines, like 10/(1+b(60/a)).
I’m an engineer and I would say the answer is 1.
I think the simplest way of explaining it is that multiplication by juxtaposition comes first.
In this case the 2 is also effectively part of the bracketed term which should be executed first under Bodmas / Pedmas.
I would argue that under bodmas the precedent is of the arithmetic symbols used in primary education, when juxtaposition is not typically taught.
When juxtaposition is used there is either no symbol or as in this case there is a bracket which you could say comes first in bodmas. That is how most people would do it in algebra as well (imagine substituting 1 and 2 with x and y).
Which gives:
6 / 2(1+2) = 6 / (2+4) = 6 / 6 = 1
As a programmer I would say that parentheses remove all doubt.
I’m a software engineer. We don’t have that problem in programming. To avoid confusion we use parenthesis. But new programmers do have that problem. That’s why they should start teaching in schools the importance of using parenthesis in math expressions.
This is why I hate PEMDAS. Such an awful thing that in order to have multiple factors in a denominator we need parens.
If it were up to me, I would just go with prefix or postfix notation.
If, for example, you have an expression that is definitely the sum of two smaller expressions, then prefix is good because you can just slap down a plus sign before jotting down the smaller expressions.
If you have a bunch of small expressions that you’ll be somehow combining into one large expression, postfix allows you to jot down the small expressions first and then tie them up with the necessary plus/minus/multiply/divide symbols at the end.
Prefix is also natural for programming because we call functions in prefix style. However, I’m also partial to postfix because postfix math naturally evaluates from left to right while prefix evaluates naturally from right to left.
To explain evaluation direction, if you tried to evaluate prefix notation from the left, you would need a stack of operators and operands as you scan right, waiting until there’s enough operands ahead of an operator to perform an operation.
If you evaluate prefix from right to left, you only maintain a stack of operands, and every time you run into an operator you immediately perform the operation, as if there aren’t enough operands available for the operation, you just have an incomplete expression. So the evaluation order for post fix makes more sense if you want to evaluate left to right.
If you’re lisp, you’ll still need parentheses because of functions with an arbitrary number of args, but if you require functions to have a fixed number of args, you don’t need parens.
I just use parenthesis, to avoid possibility of error and especially, method of documenting my code.
when I deal with math equations in programming, I will do it across several lines. Each line doing an assignment that will be used in subsequent lines.
I take a different approach in Hastell.
Within each line of the calculations, I try to make everything as unambiguous and succinct as possible.
Indeed, to avoid using too many parens, I may resort to throwing in a few extra lines.
@@J7Handle I also like postfix in a calculator. But really, you're explicitly defining the order of evaluation : you're doing the parsing mentally. It's not the use of postfix that gives the right answer, it's that you're parsing the expression rather than the calculator.
@@theelmonk I can trivially write code to evaluate a postfix expression. Read from left to right, putting operands on a stack. If you encounter an operator instead of an operand, apply the operator to the operands on top of the stack (note that the operator must have a fixed number of operands, so the binary minus sign and unary negative sign can't use the same character), then replace the operands on the stack with the result of the operation.
So 1 2 3 + - would evaluate as 1, 2, 3, then + takes the 2 and 3 and replaces them with 5, so 1 5 - follows, then the - takes the 1 and 5 and makes -4.
Prefix is evaluated in the opposite direction, that is, starting from the right side of the expression, which is a little weird, but is standard for functions in both math and programming languages (we do f(x) and print("Hello world!") instead of (x)f or ("Hello world!")print).
This is because reading prefix from left to right, while a bad idea from an evaluation perspective, is really good for giving us the big picture of the expression rather than having to dig through the nitty gritty details first.
Infix is awful in so many ways. It only works when you have the superscript exponents, multiplication by juxtaposition, proper divisions that physically put the numerator and denominator over each other, and subscripts to allow more variable names without running out of symbols to use (since juxtaposition kind of prevents variables with more than one character in their name).
In simple text with none of those features, prefix and postfix are absolutely superior to infix.
The video nicely documents that there are various interpretations. Whether some person has learned it one way or like as missionary wants to convince others that their way is the right one does not change the fact there are different answers which can be given and justified and that even programming languages disagree. This fact can not be changed and it can be experimentally tested that there are disagreements. Already watching a few youtube videos for example does this already. There are cases, where even different models of calculators gave different answers to such PEMDAS problems like 1/2x (the TI-82 disagrees with TI-83). Florian Cajori, still the best authority in this matters wrote in his book on mathematical notation: "If an arithmetical or algebraic terms contains both division and multiplication signs, there is at present no agreement as to which sign shall be used first. It is best to avoid such expressions." Already the comments in this video show that this assessment has not changed and that Cajori's advise should still be followed.
I've been thinking of doing a follow-up video on the various ways that calculators and computer algebra programs deal with expressions like this, because the variety of rules they use is quite interesting. Some don't allow implicit multiplication, some put it at the same precedence as explicit multiplication, some put it higher, and some give implicit multiplication a higher precedence in some situations but not others (wolfram alpha for instance treats 1/2x differently to 1/xy).
I've found that the Casio and HP calculators available to me locally evaluate implicit multiplication before division, but the TI one doesn't, and I'd be interested to find out why TI switched over given that they state on their website that implied multiplication has a higher priority when writing on paper (see epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 ).
I'm familiar with Cajori's statement, but note that he was only talking about expressions that involve the × symbol, not ones involving implied multiplication by juxtaposition. I know your stance is that it's ambiguous and I'm forced to agree with you because there is debate around the answer to 6/2(1+2), but the only reason there is ambiguity is that primary school teachers are pushing an oversimplified version of the rules widely used by those in STEM, and the real rules are rarely stated explicitly. We don't have to stop at asking "How is it interpreted?" but can also ask "How should it be interpreted?" and then adjust teacher training and textbooks accordingly.
That would be lovely. Can not wait to see that.
@Oliver Knill
I have no problem agreeing with you that Florian Cajori is the best authority in the HISTORY of Mathematical Notations. Too bad he is not the highest authority in setting standards in mathematics in the USA where the confusion as to whether implied multiplication should be given a higher priority in calculation than division has become an epidemic.
As far as I know in the tradition of science and engineering calculations, implied multiplication by juxtaposition has always been given a higher priority than division. That means in the expression 6÷2(2+1), 2(2+1) should be considered as a product with the factors of 2 and 3. That also means in the expression 6÷ 2(3), the number 6 should be divided by either the product of 2 and 3, i.e. 6, or by both factors 2 and 3 instead of divided by 2 only.
Therefore, all following calculations are correct:
6÷2(2+1) = 6 ÷ 2(3) = 6 ÷ 6 = 1
6÷2(2+1) = 6 ÷ 2(3) =6 ÷ 2 ÷ 3 = 3 ÷ 3 = 1
6÷2(2+1) = 6 ÷ 2(3) =6 ÷ 3 ÷ 2 = 2 ÷ 2 = 1
because a ÷ b(c) = a ÷ (b*C) = a ÷ b ÷ c = a ÷ c ÷ b
In other words 2(2+1) is always equal to (2 (2+1))
However
6 ÷ 2(3) ≠ 6 ÷2*3
In 6 ÷ 2(3), 2(3) is the divisor.
In 6 ÷2x3 , only 2 is the divisor.
In other words, 2(3) means more than multiplication of 2 and 3. The use of parentheses implies the 2(3) must be treated as a product that would become a divisor when there is a division sign before it.
Remember 2(3) is always equal to (2(3)).
But 2*3 or 2x3 does not equal to 2(3) or (2(3)) when there is a division sign before each one of them.
*It is a wrong idea to think that parentheses is for representing multiplication only when it also has the function of grouping.*
In the expression 1÷2π, 2π is the divisor instead of the number 2 only.
When the Common Core teachers do not follow the long tradition of giving higher priority to implied multiplication by juxtaposition than division, they treat 1÷2π as 1÷ 2 x π, their result would be in conflict with the order of operations being used in science and engineering.
2π is always equal to (2π) or 2(π) but 2*π or 2xπ does not equal to 2π, (2π) or 2(π) when there is a division sign before each one of them. Again implied multiplication by juxtaposition like 2π has the function of grouping as well as multiplication. Of course, implied multiplication by juxtaposition without the use of parentheses cannot be used on two numbers like 2 and 5 or 2 and 3.1416 because 2(5) is not equal to 25 an 2π is not equal to 23.1416.
When an engineering project is worked on by different engineers, some uses SI units but some use the British units, the result would be very confusing. However, as long as each engineer specifies the units being used instead of just mentioning the numbers, they can still convert the units to the same ones agreed on in the final phrase.
However, when a crooked leader tried to sabotage a project has caused some engineers to disobey the traditional rule of giving higher priority to implied multiplication by juxtaposition than division in interpreting the data given and in calculation while some other engineers follow the traditional rule of giving priority to implied multiplication by juxtaposition, there is no way to ensure that no mistakes will be made because there is no way to bring their works together when we don't know which order of operations they had used. Even if each engineer states which system of operations he has used in his calculations, we still need to know if the data supplied to the engineers were from someone who follows or disobey the traditional order of operations.
For example, if one of the data for calculation is 1÷2π, we don't know if it means 1÷(2π) according to the traditional order of operation or it means 1÷2 x π according to the new instruction of the crooked leader. That means we cannot be sure if our calculation is correct according to our preferred order of operations if the preferred order of operations of the person who gave that expression of 1÷2π is not known.
The fact that different models of calculators, even from the same manufacturer, may be programmed with different order of operations, without specifying the order of operations being used in the calculator makes the bad situation even worse.
Since almost all science and engineering documents mandate giving higher priority to implied multiplication by juxtaposition in calculations, the only way to correct the present confusing situation is to ban the common core nonsense of not giving a higher priority to implied multiplication by juxtaposition than division because there is no way we can change tons and tons of science and engineering documents existed before the common core nonsense to conform to the stupid and wrong PEMDAS ONLY operation that ignores the rule of giving a higher priority to implied multiplication by juxtaposition than division.
The leftist politicians who hate the USA know that a unified country is a strong country and a divisive country is a weak country. That's why they tried to force the the American children to use PEMDAS without teaching them about giving a higher priority to implied multiplication by juxtaposition than division, which is still being used in engineering and science, in order to create confusion and conflicts among the different generations and different groups of people for the purpose of weakening our strength and unity.
PEMDAS without juxtaposition should be history just as Obama's other Common Core nonsense should be history in the USA.
Earlier I found out that Google Chrome has partially hidden my comment on Oliver Knill's thread but it is not censored on Firefox.The Common Core cohorts promoting the PEMDAS ONLY nonsense who have also hijacked my connection to Google are scared of the truth being exposed.
@@THaWoM Here is a note from Wolfram alpha, you should read it, it's about the division symbol when used in 'in-line math' :
mathworld.wolfram.com/Solidus.html
"Special care is needed when interpreting the meaning of a solidus in in-line math because of the notational ambiguity in expressions such as a/bc. Whereas in many textbooks, "a/bc" is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages such as the Wolfram Language, it means (a/b)×c. For clarity, parentheses should therefore always be used when delineating compound denominators."
So even Wolfram alpha say that YOU should check if parenthesis are needed for the expression to be calculated properly.
It also say that they know that for such expression the result they give may be wrong !!
You may read too that document from a Berkeley professor
math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html
It is a document about the order of operation and the expression 48/2(9+3)
On a french site, I found out a reference to this document (discussion in french writen by a french scientific researcher, from the CNRS [ french acronym for National Center of Scientific Research] about the need about a new convention of calcul or 'Does the mathematician still have a say in the evolution of the calculation conventions of the secondary school?')
images.math.cnrs.fr/Une-nouvelle-convention-de-calcul.html
In that page, the author talk about his exchange with Professor Bergman, who said that prof. Alan Schoenfeld, « who is the intersection of the Math Department and the School of Education here, also agrees that such expressions are simply ambiguous ».
And Alan Schoenfeld is a math professor at Berkeley
gse.berkeley.edu/alan-h-schoenfeld
Concerning the calculators, you may find some online casio calculator manual. In them you may find their inner 'order of operation'.
You'll find that for some implied multiplication is to be evaluated at step 7 and general multiplication/division at step 10
but in other, they are both evaluated at the same step...
I know that when I was at school (in France), one of the check to have an 'school approuved calculator' was if 1/2Pi was interpreted as 1/(2*Pi) (Good) or (1/2)*Pi (Wrong)
But mainly a special course is present in the math book (still now) to check how 'YOUR' calculator respond to several cases, just to be sure HOW to enter YOUR math expression in it later to have the result that YOU want. (and not re-interpreted differenly by your calculator)
I must have actually been taught this as my mind has been yelling "implied multiplication first" every time I've seen these "problem'" on the net. I've known the "order of operations" since the 1960s, but don't remember ever having been taught the PEMDAS [or equivalent] acronym nor any mnemonic phrase.
Me too. Only came across the pemdas/bodmas mnemonics and their fanatics as an adult when I stumbled on these questions. I was horribly shocked at how terrible math education is. And how they seem to think this childish oversimplification is the top "law" of math 🤦♀️
The way i see it is that 2 is tied to the parentheses and you gotta distribute that first or you’re breaking the order.
You did pemdas Wrong ….
@@kurtshaw229
Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
Convention doesn't trump LAW and the Distributive Property is a LAW.
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2 = 3 Monomial Factor
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
@@TheJocelynrae
BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations
6÷2(1+2)=
6÷2(3)=
3(3)=
9
2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside ....
There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more.
Commutative Property
6÷2(1+2)=
6(1+2)÷2=
6(3)÷2=
18÷2=
9
Distributive Property
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction.
6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2
Operational inverse of division by the reciprocal
6÷2(1+2)
6(1/2)(1+2)=
6(1/2)(3)=?
Multiply in any order you want you still get 9
Proper use of grouping symbols
6
-----(1+2) = 6÷2(1+2)=9
2
6
-------- = 6÷(2(1+2))=1
2(1+2)
A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator...
Another argument people tend to use incorrectly is factoring....
6 = 2+4 No parentheses required BUT
6÷(2+4) parentheses required
2+4= 2(1+2) only one set of parentheses required.
6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set...
The 2(1+2) must be placed within the first set of parentheses containing the (2+4)
6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2)
Let y = 0.5
6y(1+2)=?
6y*1+6y*2= ?
6/y⁻¹*1+6/y⁻¹*2= ?
If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9
The rules of math have to remain logical and consistent across the board...
THESE ARE THE FACTS....
Thank You for showing the INTERTERNATIONAL correct way of solving a math problem....
I always understood it as variables.
6 is one variable in the equation. It is separated out from other variables by the division symbol.
2(3) is the other variable in the equation.
The lack of a multiplication symbol implies the 2 and 3 are part of the same variable. However to solve for that variable the parentheses implies the 2 and 3 are to be multiplied.
So, 6 / 2(3) = 1
2(3) implies that it is a factorisation
such that you can factor an ammount of 2 from an unknown ammount, which you can find out to be 6 if you multiply the factor in again.
same thing if you factor for instance a from a^2-a which you would get a(a-1) the original ammount was a^2-a which you should take care of before anything else since it is in brackets only then the division
That's how I understood it. You could replace the (2+1) by x, giving 6/2X, which would give 1.
I was taught that 2(3) is the same as 2x3, you don't have to have a multiply sign
@@danielzhang1916 if it's easier for you to solve 2(3) by writing 2 x 3 then you have to write it correctly. Which is
(2 x 3).
I'm glad I had a great math teacher in highschool. He told us PEMDAS was wrong years ago.
Pemdas is not wrong. This is entirely an ambiguities in grouping problem, not order of operations problem.
@@linsqopiring6816 Yes. So it is. Shown in another video here at RUclips by a maths teacher.
@@linlasj name?
To be more precise, it's woefully inadequate and simplified such that it breaks with algebra and most more advanced math classes. x(stuff inside here) is one such exception that it does not cover correctly. So people and teachers.. well, maybe the teacher certification tests need to be harder. lol.
There are no ambiguities in math. If you ever find an ambiguity it's because you did something wrong.
1 will always equal 1, it will never sometimes equal 2.
This is great! The comment about "mathematicians being lazy" is good. There are actually implicit parenthesis all over calculations that PEMDAS does not account for. The viral solution of "9" falls apart when the terms are variables. If PEMDAS was used by engineers we would probably have more bridge disasters.
No. answer 9. even with variables.
That’s why there arent variables
@@adamwalker8777 it's 1 and 9
@@random22453 9
PEMDAS is a mnemonic, it is no more of a "Lie" as is E=MC^2 because obviously, PEMDAS doesn't include the exceptions and neither does E=MC^2 include resting mass. So, of course there are a lot of exceptions depending on how a person decides to handle implicit multiplication and implied parenthesis for fractions vs pure left to right... A*B / C*D is handled differently when it appears as a fraction because there is an implied parenthesis for the denominator, as opposed to A*B÷C*D where there is no implied fraction and order of precedence is simply left to right. And depending on the teacher, some include that in their lessons and some don't. I don't really see this as a disaster for PEMDAS or any other mnemonic, it simply means that teachers and students need to be aware that there are subtleties in the rules and depending on the level of math (ie the scope) of knowledge, they might have to adjust the order of operation. The smart move is to ensure that parenthesis are in places where you want your precedence to be highest, but basic math questions leave a lot of that as part of the question to gauge the student's level of understanding for the basic concepts. I think this video points out some flaws with PEMDAS, but the bigger issue is integrity... the thumbnail claims PEMDAS is a lie... as usual in order to boost views everyone shouts "This is a lie"... no, it's not a lie, it's click bait to the rescue.
This is nonsense. I am an engineer and PEMDAS does still apply. The problem is when people convert papers with mathematical equations to single line type using / for division instead of a proper numerator above the denominator separated by a horizontal line. When converting to single line equations they should use parenthesis where needed to avoid mistakes and confusion or ambiguity. If they don't it is either the printer that does not know or it is pure laziness. The most important thing is to avoid ambiguity so costly errors are not made.
I always assumed that because the 1+2 is inside the parentheses, you would either factor the 2 into the 2(1+2) to get 2+4=6 or just prioritize the 2(3) over 6/2 because the 3 is inside the parentheses
Actually you do not assume ....you were just taught correctly. As 1 is the correct answer.
@@JennaGibbens9 is the correct answer, if they want 1 to be the correct answer add brackets.
The "2" is part of the "(1+2)" expression, so "2(1+2)" has to be completed first. THEN 6 ÷ 6 = 1.
@@JennaGibbenscorrect!!
@@shiijei26389 is INCORRECT!! Brackets are not necessary, there are akready parentheses. 6 ÷ 2(1+2) is different than 6 ÷ 2 x (1+2). In the first, the 2 is part of the parentheses expression, so 2(1+2) MUST be completed first. In the 2nd, the 2 is not part of (1+2). I'm sorry that you are wrong, and I'm sorry that you were taught wrong. And I'm sorry that you have been brainwashed. But that is your problem not mine/ours. 6/2 x (1+2) = 9 is diff than 6/2(1+2) = 1. That is same as 6/[2(1+2)]. I hope you don't work in some important engineering, medical, or scientific field, and use your kind of math. Because if you do, I'm very scared right now.
Even in college calculus and engineering classes (and computer programming) we were never *told* that PEMDAS was a limited case -- but we very quickly absorbed by example this juxtaposition precedence and learned to add parentheses if needed for clarity when linearizing an equation or formula.
It seems that if you have to 'add' parentheses to an already stated question you actually change the question. But if you actually use PEMDAS as written and not worry about the left to right issue, then in this question presented the answer still comes to 1(I seem to remember learning both ways in HS and then the traditional left to right order in college)
Look up PEMDAS rules. Ties(MD or AS) are done left to right.
The answer is 9 and other answers are not correctly done using PEMDAS.
NYT has an article from 2019 covering this.
8÷2(2+2) = 16.. not 1.
PEMDAS is a mnemonic for the order of operations. The order of operations must always be equal otherwise two people would get two different answers.
8+2*4 = 8 + 8 = 16 using the order of operations.
10 * 4 = 40 just going left to right.
Which is correct? The order of operations is not just some random process someone came up with.
Why the order of operations?
P - Parentheses - groupings. If someone emphasized a group then that should be done first. a + (b - c), b-c should come before adding a.
E - Exponents - exponents are just a group of multiplications. 3^2 = (3*3)
M - Multiplication is just a group of additions, 2*4 = 2 + 2 + 2 + 2, thus 8 + 2 * 4 = 8 + (2 + 2 + 2 + 2) = 16.
D - Division is just a group of subtractions, 12 / 4 = 12 (- 4 - 4 - 4) = 3
-Multiplication and Division are inverses of each other thus they are of the same precedence, 12/4 = 3, 3*4 = 12.
Addition and
Subtraction are just the last operations, the most basic so they're saved for last.
In this issue 6/2(1+2), the 1+2 are first, then it comes down to 6/2(3) which is only division and multiplication, the same precedence, so they're done from left to right, 6/2 = 3, 3(3) = 9.
This could also be written as 6 * 1/2 * (1+2). Multiplication can be done in any order thanks to the communitive property, this 1/2 of 3 is 1.5, 6 * 1.5 is 9.
The only answer for 6/2(1+2) is 9.
@@leighz1962 PEMDAS does not treat juxtaposition properly, so it gets the wrong answer, as the video demonstrated.
@@leighz1962 Damn bro never knew the new york times, a bunch of journalists, know more than actual mathematicians! Might as well remove mathematics as a major and replace it with more courses in journalism as they clearly know more!
The way I read it is that when you have a parenthesis that is preceded by a number, this means you are using x times the amount in the parenthesis for the rest of the calculation. So although it is a multiplication you are doing, it is seen as it’s own group that needs calculated first before it is calculated with the rest of the equation. It does look weird when the equation is just using numbers, but when you start using variables in equations, it makes complete sense.
Actually when you see a parenthesis preceded by a number; It mean this number has been removed from the parenthesis and to properly evaluate the parenthesis it must first be distributed back.
2(1+2)=(2+4)=(6)=6 Or 2(1+2)=2(3)=(2*3)=(6)=6
Nowhere in PEMDAS order of operations is outer multiplication of parentheses stated as being done with P. It is the the MD portion.
6÷2(1+2) = 6÷2×(1+2) = 9
6÷(2×(1+2)) = 1
@@leighz1962 PEMDAS does not cover juxtaposition. When there is juxtaposition you need to use PEJMDAS.
There are two ways that I would solve this equation. One way is to use the distributive property to eliminate the ambiguity of whether to follow multiplication/division from left to right when simplifying the set in parenthesis.
6÷2(1+2)=6÷(2+4)=6÷(6)=1
The other way is to simplify the set inside the parenthesis first then, you are left with
6÷2(3) and realize that the parenthesis are still there and still must be solved before multiplication/division from left to right.
6÷2(3)=6÷(6)=1
Either way, the answer is still 1.
no. the inbetween operator extends to the next operator.
Correct. Parenthesis must be cleared first.
Nein.. there are technically hidden parentheses around everything.
6÷2(1+2) is also ((6)÷(2)((1)+(2)))
If you are solving using PEMDAS, the answer is 9.
6÷(2(1+2)) would be 1.
If you're distributing you have to distribute the entire value, it's not 2(1+2), it's 6/2(1+2), thus 6/2 * 1 + 6/2 * 2 = 6/2 + 12/2 = 18/2 = 9.
@@J-kd2qc you've not handled juxtaposition in that solution. As the video explained, PEMDAS cannot be used when an equation uses juxtaposition - you need PEJMDAS.
Often grateful for my 70-80s Canadian education. This is another of those times because I learned this as you described. I'd never heard of PEMDAS until an argument about the example or a very similar equation making its rounds online.
So did you answer 1 or 9
Thank you!! I graduated from the Naval Academy. This is how we were taught in our math, engineering, and science classes as well. This issue isn't just a fun brain teaser, it's important. If you are designing a bridge or a building, or worse - a more complex dynamic system, math matters.
And if you attempt to use a computer or calculator to assist computation for such endeavor you had better be aware that that device is likely to use PEMDAS to parse expressions. Simply do not write ambiguous expressions -- use parens to coerce an exact parsing.
It is a "fun little brain teaser" though because NO ONE writes an elementary expression like that. It's completely academic. Furthermore, I hope you meant to say you were taught PEMDAS in the Naval Academy.
"If you are designing a bridge or a building, or worse - a more complex dynamic system, math matters." I agree. That's why mathematicians, scientists, and engineers don't write stupid little "trick" problems like this. Nevertheless, even as written, the expression is NOT ambiguous. It's 9.
@@garymartin9777 The expression isn't ambiguous. It's 9.
@@garymartin9777100%.
This is the actual correct answer.
That is an answer by computation alone.
The correct answer is 1. If we read the equation plainly, it is simple to see how the answer is derived.
6 is being divided by twice the sum of 1 and 2.
Very well explained and I agree with the answer 1. This is how mathematics was taught in college and how I would apply the rules.
Has grade school math changed this much?
Because this is how I was taught in grade school in the 80's....
I never even heard of PEMDAS until I was an adult.
@@antonioliles5027 as someone who went through 2000s grades school, math at an elementary level has been slowly made easier with different dumb and lengthy methods. Nowadays time tables arent even memorized because it takes memorization and teachers don't spend enough time worrying about helping that be memorized. From what I see what most grade school teachers use to teach math is some weird method where you put numbers into boxes and somehow multiply them/divide them. Math on an elementary level has basically become something entirely based on how a teacher teaches instead of actual formulas or methods.
@@geodom85yt40 You just described common core... yes it is horrible.
This I how I was taught 40 years ago and I went on to get an honours degree in maths.
The answer is 1.
I remember getting taught this ruling in primary school(UK) during the 90s.
This was when we were learning BODMAS.
It was explained at the time that you had to do the multiplication at the same time as removing the brackets as it was considered higher on the order of operations than regular multiplication.
I also remember very distinctly as a ruling because they purposely gave us questions with stuff like 2( and 2*(, just to make sure we were paying attention.
Could we call this an inconstancy in notation?
Even to a beginner like myself, it is clear that the numerator and denominator are to be treated as isolated expressions.
That means that, for the sake of clarity, if that is the intended effect, the vertical-stacking notation of a fraction is preferred.
A horizontal-linear notation immediately invokes the PEMDAS association, however.
I seem to remember learning (in the 1970s) MIDAS, which stood for Multiply Immediately, Divide, Add, Subtract. (This was probably early enough that parentheses/exponents weren't yet covered.)
That would seem to emphasize that multiplication has priority over division, though it's less clear whether order or operation take precedence when you're down to addition and subtraction.
Excellent explanation. Now I understand why elementary school teachers keep telling me "you're doing it wrong" when I knew this is how we did it at school (in a fairly math heavy program). I mean, it's all just convention (as long as both the person writing the expression and calculating the expression follow the same rules, it's all good), but still, the gap was disturbing to me.
To see something that I have been doing intuitively for all my life being formerly explained is mind-blowing!
Using the textbook examples was what made me realize that I had done it this way all my life.
Formally?
The issue with this is that no one uses the "a/b" notation. Engineers, physicists, and mathematicians use FRACTIONS. The original problem is meant to be confusing. PEMDAS is correct, you just have to write equations properly instead of writing them in ways that are intentionally misleading.
The real problem seems to be that mathematicians will automatically assume that everyone will interpret things exactly the same way they do and fail to see that other interpretations are possible. So when in doubt, use all the parentheses needed to make absolutely sure that people do the operations in the proper order.
Nonsense. Mathematicians are communicating with likewise educated (at least college grad in the sciences) people using a convenient standardized notation, and that notation has a lot more symbols and conventions than the few you learned in elementary school. Even if mathematicians tediously parenthesized everything, making it far less readable, mathematically uneducated people still wouldn't be able to make heads or tails of it.
Other interpretations are only PLAUSIBLE when rules aren't CLEAR! I don't see how the fuck this kind of shit is so hard for people to understand! But when influencers do stupid shit and convince morons that the correct way is wrong everything goes to hell!
That is not needed. As long as you understand that a(b+c) = ab+ac then you will never find any ambiguous equation ever again.
@@ihadtochoosethisuser
a*(b+c) = ab + ac
True?
If True then: a*(b+c) = a(b+c)
True?
If True then: 2x/a*(b+c) = 2x/a(b+c)
Still true? Or do we need to start adding in more brackets at some point to clear up some ambiguity?
2*x = 2x?
True?
If True then: 2x/a*(b+c) = 2*x/a(b+c) = 2x/a(b+c) = 2*x/a*(b+c)
True?
Unfortunately our 'rules' have ambiguity, especially when the person initially writing the problem is sure there is only a single way of interpreting it. Anyone who claims that there is only one true and correct way of evaluating an ambiguous statement needs to stop because those are the types of people who will end up causing ambiguity by failing to see different potential interpretations when writing down their own formulae.
@@SubjagatorYou're intentionally breaking up perfectly implied multiplication in order to introduce unnecessary ambiguity. No one who is trying to actually do arithmetic or algebra would ever do this; you're just trying to win points on the Internet. Do you need a cookie that badly?
I remember being sucked into several of those Presh Talwaker videos where he posted arithmetic expressions like this, and explained the debate about PEMDAS and the ambiguous answers to simple math problems. The way I was taught math, in 5th-6th grades in the early 1980s, was as explained in the video above: That the expression 6 / 2(1+2) would be equal to 1. However, "New Math" and the way students are taught today, (and the way Presh Talwaker explains it) would evaluate that expression as 9. My primary-school math teachers explained "Order of operations" as that any number directly adjacent to a parenthetical expression, in this case 2(1+2), would always be evaluated first as multiplying the number outside the parenthses by the value inside it. So, that answer to the above problem would be 1, but to obtain the answer 9, (according to 1970s/early 80's math) you would write it as (6/2)(1+2).
Yep; "2(1+2)" is analogous to "2a."
I was taught that the divisor symbol is a relic of the type-writer era. The fact is that we didn't have tools like LaTeX so we needed an easy way show above the line and below the line for division. Feels like this point is keeps getting skipped over by everyone explaining the division symbol on youtube. Also, the example at 5:30 is given by a teacher that shouldnt be teaching. 2x/3y should be written as 2xy/3 if the answer has any hope to be 11.
Or, to get the answer of 9, just put in an explicit multiplication: 6 / 2 * (1+2)
No need for the extra parentheses
@@JK-tq5oe You arent making things less ambiguous with the * operation. Write the damn equation better. 6(1+2)/2 or 6/(2(1+2)). Super simple.
@@RaspKNo it is not, because there are no variables, and the parentheses' purpose (to state a quantity) is fulfilled after resolving the quantity within them. Therefore, they are to disappear and no juxtaposition is applicable.
For me, when I was taught PEMDAS, I was taught that multiplication came before division, and addition came before subtraction. I suppose I could have mis-remembered, or that the teacher was mistaken, but is that the way it should really be taught instead? Or is it only correct in the specific case of multiplication by juxtaposition? Actually though, what I was taught left out Exponents: "Perhaps Math Does A Sequence."
It would certainly be wrong to do addition before subtraction - subtraction is literally just addition of a negative value. So if you had 10 - 5 + 8 then doing addition first would be 10 - 13 = -3 while treating them as equal you get 13. Substitute a = -5 you get 10 + a + 8 and you see that now by just expressing it with a variable we'd have _changed the answer_ if addition came first! ;)
In practice it's often just juxtaposition that explicitly precedes division, but to avoid ambiguity there are a number of journals that have a specific style guide specifying multiplication in general comes first.
So it sort of depends on context tbh, and usually you can know from context the intention of a equation and which interpretation was intended. In practice I don't think I've ever seen a real world instance where this was actually an issue because any time there's ambiguity most peeps in maths and related fields will simply write it down in a way that avoids that ambiguity instead - a good example from the video is the case of x/2.
If I saw written 1 ÷ 2 × c or 1/2 ⋅ c then the inclusion of the explicit multiplication symbol would make me think it was intended as a separate operation - ie, equivalent to ½c or c/2. The spacing too can make things more or less ambiguous, so if the 1/2 is tight then the intention is _clear_ that it actually means ½c as opposed to 1/2c which would _always_ be interpreted as 1 / (2c).
Maths is a language, and like any other language, people take shortcuts, and usually those shortcuts are clear and obvious to anyone that speaks that language and don't really need to be clarified because context and intention usually eliminate ambiguity for those that know the language well. So people violating rules for brevity when intention is clear, or conventions that are not explicitly spoken but become ubiquitous through their usage are just things that happen.
@@Phidaissi Thank you for your reply, that made a lot of sense and I feel like I understand it now. And I suppose if Math is a language, PEMDAS could be described as similar to someone coming up with the English "i before e" rule that people learn, but it turns out there are far more exceptions in actual practice.
@@WooperSlim Glad that made sense!
@@Phidaissi With 10 - 5 + 8 doing addition first is perfectly fine.
It would be 10 + 3 = 13 or
18 - 5 = 13 though.
What you did was 10 - (5 + 8) which isn't the same question.
You just added incorrectly.
A and S have equal priority so the order doesn't matter.
@@WooperSlim Since M and D have equal priority the order doesn't matter. Same with A and S.
You can always do M before D and A before S and always get the same correct answer as someone who does D before M and S before A.
Like with 10 - 4 + 7
S first: 6 + 7 = 13
A first: 10 + 3 = 13 or 17 - 4 = 13
and with 16/8×4/2 (this notation isn't great though. You shouldn't have multiplication or division directly after division on one line without brackets to remove ambiguity)
M first: 16/2/2 (again, bad notation) = 8/2 = 4
D first: 2×4/2 = 8/2 = 4
See? Same answer regardless of the order.
The ambiguity here though is with the notation, not the rule.
There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. I.e. does a(b) = (a×b) or a×b?
Both are widely used but lead to different answers when used directly after division.
if you have 6÷2(3) one interpretation of the notation is
6÷(2×3) using PEMDAS now gives 1
The other interpretation is
6÷2×3 using the same PEMDAS now gives 9.
I do think the juxtaposition interpretation of implied brackets is the more common academic interpretation though.
PEMDAS applies to TERMS first and OPERANDS second.
2(1+2) is a single term and must be resolved completely before the 6/ is applied.
I came up with 1 but I used distribution and that process eliminating the parentheses isn't finished until you complete the whole of 2(1+2) then you divide the resulting 6 by 6
when she said i should get 9 i was like, wait WHAT am i retarded?
Since the majority of people believe the correct answer is 1, let me prove to you that's wrong.
We all agree we get to this point 6÷2(3). If the answer is 1 then the equation will be 6÷2(3)=1. Now replace 3 with "a". 6÷2(a)=1. Solve for "a". I guarantee you "a" will not be 3. But "a" will be 3 if the equation =9 instead of 1.
The correct answer is 9.
@@Kingdom-zu6tm incorrect. If it's 6 over 2 and a... that means it's 1/a = 1/3 .... 3. It's entirely based on how you group 2 AND a together.
@@ri3m4nn 1÷3 or 1/3 does not equal 3. 3÷1 or 3/1 does equal 3. Why are you grouping anything? The math problem is straight division and multiplication. In that case you solve from left to right.
@@Kingdom-zu6tm 6/2a=1, 6=2a, 3=a
I've always used the idea that non notated multiplication was always considered tight and always done first (it would be included in the parentheses step). Division would be notated with a slash at this level and would be secondary and controled by a large horizontal line to be clear if needed. Using actual division and mulitplication symbols, then yeah, you do them in order, but that is very elementary mathematics and is a slightly different system. Maybe I'm wrong I don't know.
perhaps by error, but many use PEMDAS literally meaning Multiplication before Division and not treating them as equals subject to the left-right rule
This is the actual problem most people are facing. I was taught in school that M comes before D, just like in PEMDAS.
The whole problem is that one even has such a term People should not use or need any memory rules for these kind of things. They should just learn them.
@@okaro6595 point taken, thanks for your posts!
I think that the notation for writing multiplication and division are taught but not clarified well enough to instill how division/fractions relate and interact. Pemdas isn't wrong but the concepts aren't clarified enough to ensure pemdas can be used properly as a tool. How we write and interpret multiplication and distributive property is one example. If a number should be in the numerator or denominator and if it should be written as a seperate number instead is another. Many who discuss this likely understand but those that don't may not be able to guess where their issue with understanding is making it difficult if not impossible to fix the gap in understanding without aid. An inconsistent foundation makes further building an unstable and hazardous endeavor.
Thank you! It's been 50 years since I finished engineering school and have never worked in my field, so when I saw this problem with the "new" solution I thought maybe I had lost my mind. It turns out that just maybe it is not me who is crazy -- just everybody else.
No, it's you and the lady in this video...
The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y.
This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2
All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. There are no coefficients in this expression...
6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b is 3 not 2
Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
Convention doesn't trump LAW and the Distributive Property is a LAW.
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property, when FULLY applied, is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property, when FULLY applied, REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2 = 3 Monomial factor
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2 no parentheses required
3×1+3×2=
3+6=
9
You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression.....
When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses. TERMS are separated by addition and subtraction not multiplication or division. 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
6÷2(1+2)= 3(1+2) no rules have been broken
2×2×4(a+b) partial Distribution 2×2(4a+4b)
However the TERM outside the parentheses when simplified equals 16 and 16(a+b)
2×2(4a+4b)= 2(8a+8b)= 16a+16b which is the same as 16(a+b)
2×2×4(a+b) when fully Distributed is 2×2×4×a+2×2×4×b and the LIKE TERMS can be factored out of the expanded expression. The LIKE TERMS being 2×2×4 So... 2×2×4(a+b)
6÷2×1+6÷2×2+6÷2×3-6÷2×4= 6÷2(1+2+3-4) as the LIKE TERM 6÷2 was factored out of the expanded expression...
Let y = 0.5
6y(1+2)=?
6y*1+6y*2= ?
6/y⁻¹*1+6/y⁻¹*2= ?
If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9
The rules of math have to remain logical and consistent across the board...
@@RS-fg5mf Brilliant! Now, why do the rules change if logic is always consistent?
@@forgottenpeopleplacesandol4258
Convention when it comes to directly prefixed coefficients and variables... 2y there are no parentheses by which you can Distribute... I can explain it to you but I can't understand it for you...
@@RS-fg5mf Somehow I managed to get through an engineering program and also managed those skills to achieve substantial retirement benefits while never drawing a paycheck from anyone and I am enjoying my accomplishments just fine despite my lack of understanding. I don't know how I made it without your abilities. But I did.
@@forgottenpeopleplacesandol4258
🤣🤣🤣🤣🤣🤣 if you say so... LMAO
You never worked in your field and never recieved a paycheck?? Hmmm. And this supports your wrong answer, how exactly??
And FYI.. Any career field that relies heavily on math would NOT use the obelus by rather the vinculum so you're argument is lacking...
The problem is the equation can be written more than one way, and there should really be a second set of parenthesis. A way to test it is to write it as 6 OVER 2(1+2). Then it becomes clear.
"Nah, I am lazy and identify as right. You are wrong and need to change."
Adding parentheses changes the equation.
6/2(1+2) =/= 6/(2(1+2))
Just swap the divided by 2 with a multiplied by 1/2. U r welcome.
@@om-qz7kpIt is so simple. We are just following the rules. Thanhs
@@harrymatabal8448 that is what I said. Have u understood my comment?
Problem with teaching PEMDAS is that they have to unlearn it later. It is much better to teach the most used rule from the start.
To a degree yes you do need PEMDAS but you also have got to remember you need law of mathematics supersedes PEMDAS first so in like this equation is the distributive law will be applied first and then PEMDAS then the answer is still one. Nobody ever remembers that..
@@nathanlarson1033 I have never learned PEMDAS at all. I have managed just fine.
I teach math at the middle school level.
There’s nothing wrong with PEMDAS as long as you teach its caveats.
Addition and subtraction are inverse operations, so all subtraction can be rewritten as addition of negatives.
Multiplication and division are inverse operations, so division can be rewritten as the multiplication of its inverse-or vice versa. That’s why sometimes we see one-half times something or that something over two.
Regarding multiplication, these are implied as a group, even without written brackets:
2(3)
2x
xy
A traditional division symbol is in-line and resolved from left to right.
A slash / is the same as a fraction bar, so the denominator that follows is implied to be grouped with its numerator. It’s resolved before other multiplications preceding it. If the denominator includes addition or subtraction, those must be grouped in brackets.
Grouping symbols aren’t necessary in these cases, but can be helpful.
I WAS GOING INSANE! I have a physics degree and I am an acoustician and I got in an argument with my entire family and extended family because I was helping a cousin with a question like this and kept telling him he is doing the math wrong, and I had everyone telling me IM DOING THE MATH WRONG. I literally use advance mathematics every day and EVERYONE is telling me the wrong answers to all of these 6th grade math problems, and im sitting here pulling my hair out and solving them in moments time and verifying my work and getting the right answers over and over and over again. I'm glad to see I'm not insane. My understanding of math is accurate and the rest of my family is just not at a level of understanding this stuff that I should even care or let it get to me.
Did you get nine as the answer? As a physicist you should be able to solve this very easy, if you got one you made a math error and I can show you what went wrong.
@@DoseofScienceDoS The whole point is whether you acknowledge the "hidden" parenthesis or not. Did you watch the video? Elementary school PEMDAS says it should be 6 / 2 * (1+2), which is 9, but physics generally uses "hidden" parenthesis when using division, so the question becomes 6/(2*(1+2)), which is 1. Programming also typically uses the PEMDAS way, and I constantly have to put in an abundance of parenthesis just to calculate the correct function I actually need. This video discusses this, as it is not possible to nicely write a divider in a single line, single line equations are fairly ambiguous. One of those being a/bc, would you regard that as a/b * c, the same as (ac)/b? For most engineering, physcis, maths post elementary school, a/bc means a/(b*c)
@@DoseofScienceDoS No Dosed he got 1 He is a physicist.
If you got the answer one I can show you how you did the math wrong
@@Of_UnCommon_Sense
Let’s do a simple rearranging
6/2(1+2)=
6(1+2)/2=
(1+2)/2*6=
Rearranging is a way to check you got the right answer as outlined by Charles Hutton.
Answer all three with the same answer and it will be correct.
Now you know 9 is correct.
i'm 72. i ALWAYS get these internet math quizzes wrong. one of the things modern people gloss over, is that the BAR separating numerator from denominator was at one time seen almost as a parenthetical. also, using the distributive rule to simplify what the expression means takes precedence over an 'earlier' operation. thanks, i was beginning to suspect early dementia.
That's pretty funny. I studied physics and game design in college and when I went back to get my teaching certifications I had to take an algebra exam, and even though I had been doing calculus with analytical geometry for years in college I pretty much failed the exam because my understanding of PEMDAS needed to be "corrected" 😅
Same here. I have a BS in math. We never learned this. We just understood what to do.
Ambiguous or missed bracketing causes the errors. Nest what needs to be nested and work out from the middle.
I’m totally agree. All this “so intelligent” youtuber trying to explain in the youtube are wrong because they never go to higher level in mathematic. Im glad you say it
Willful ignorance and the blind leading the blind... SMDH
@@RS-fg5mf I’ve read a lot of your comments and agree with you. I was surprised by how many people were ignorant of the topic (It seems like it’s the entire comment section). Thank you for taking the time to correct people (lord knows someone has to say it).
@@verypanda1801 Thank you for the kind words and support... Much appreciated.
I'm not sure I'd even consider this "higher level" math. I would've gotten 1 as an answer by grade 7, at least the way i learned basic mathenatics and order of operations back in the 70s (born in '64). I don't know where she went to school or how they teach it these days, but it scares me that anyone could get 9 out of that, much less that - as someone mentioned - an app like Excel wouldn't accept it without making 2(3) into 2*(3), adding what i think is a confusing level to it. I was taught that implied operations - 2(3) or bc in her examples - always took precedence.
@@captainz9 The key here is that math doesn't define a rule that says implicit operations take precedence. That's only a convention, and it's not universal. So without any context this operation is ambiguous, as you can't tell if you have to apply that convention (resulting 1) or not (resulting 9).
The convention exists pretty much to reduce the number of required parenthesis in common formulas. And as a side effect, to confuse people.
I've been pointing this out for years. The real issue, in my opinion, is that PEMDAS ignores distribution completely, when distribution is the only reason that the parentheses can be used as multiplication in the first place. We have a perfectly good multiplication symbol that we can use to represent 2 x 3. We don't need 2(3) to exist at all. However, in distribution, we get: ab + ac = a(b + c). Now, the parentheses function with multiplication, but strictly speaking you're multiplying through the entire term. In other words, using the 2(1 + 2) part of the equation, because of distribution we should get the same answer whether we distribute the 2 into (1 + 2) and then add, or if we add and then multiply. And we can see 2(1 + 2) = (2 + 4) = 6, just the same as 2(1 + 2) = 2(3) = 6. That's the whole point of distribution, and PEMDAS destroys this concept, which is critical when you're looking at 2(a + b), since you can't add a + b, but you can still consider (2a + 2b) if you need to.
Also, PEMDAS says nothing about when to do things like factorials, trigonometric functions, etc. so it's not even useful after elementary mathematics anyway. But that's a different issue.
If you do it properly with distribution, 6 div 2 x (1 + 2) = 3 x (1 + 2) = 3 + 6 = 9, matching PEMDAS. It's the ambiguity of 6/2(1+3) when written inline, is it (6/2)(1+3) or 6/(2(1+3)).
Thank you for the interesting explanation. I would have answered 1, and if asked why not 6, I would say BODMAS, but I’ve always instinctively seen multiplication by juxtaposition as somehow “more tightly binding” than operations using a sign.
I’ve done plenty of physics/engineering labs where the measurements don’t match the calculations unless you really mind your brackets. 100% agree no one outside of k12 math does it the Pemdas method.
PEMDAS not a "method". It's a reminder intended to help beginners remember some of the basic concepts of order of operations. And everyone at every level uses order of operations to help evaluate mathematical expressions.
However, "PEMDAS" represents a beginners' version of order of operations, intended for use with elementary arithmetic. As such it's not wrong.
Blindly using only an elementary set of rules in a more advanced context, and then calling them wrong because they're not adequate to the task, misrepresents "PEMDAS".
Isn't there also an element of (sorry for a none technical term here) "do everything under the line first"? For example, a square root sign only applies to the first number after it, but in may equations there's often a long line above a bunch of numbers / letters which I believe is meant to be the same as a bracket. So, as an example, the equation to determine the roots of a quadratic equation is [-b +/- SQRT b^2-2bc /2a ]. The rule I thought also applies to to division when written by "free hand": yc/rb would mean rb has a big line above it and that should be interpreted as having to be done first. That is, it's y times c divided by (r times b) and not y times c divided by r then the result times by b. An interest thought then is that it is NOT the same to write yc/rb like yc rb as one would have to add the implied brackets as well. And, of course, anyone who has tried to use a typewritter or computer line text knows it's really hard to draw out lines so the conventions you mention just make sense when writing a paper too!
The line is called a vinculum, and yes, it acts the same as brackets. I've read a lot of comments saying that whether the obelus (÷) or solidus (/) is used should affect the precedence rules, but I don't think there's much official support or consistency to this. Many authors use yc÷rb to mean yc÷(rb) and the same with the solidus. See my latest video ruclips.net/video/4x-BcYCiKCk/видео.html for examples. And some calculators treat 1/2x as (1/2)x (wolfram alpha for instance) so the solidus is no guarantee of grouping.
@@THaWoM
The biggest problem we have here is the confusion and misconception that a convention given to variables in an algebraic expression/equation applies to parenthetical implicit multiplication.....
6÷2a the 2 is the coefficient of a by algebraic convention NO parentheses (grouping symbols) necessary. It forms a coefficient/variable compound quantity... 2(3) is not a coefficient/variable compound quantity.
Real numbers (constants) can be coefficients but real numbers do not have coefficients.... Grouping symbols only give priority to (OPERATIONS INSIDE) the symbol not outside.... Do you people lack the intelligence to know the difference between inside and outside?? 2(3) is not a parenthetical priority and is not a coefficient/variable compound quantity... It is called implicit multiplication not implicit grouping... Implicit as in the physical operator is implied as if it were there when a constant, variable or TERM is juxstaposed next to the parentheses without a physical operator.... TERMS are seperated by addition and subtraction.
When you have an algebraic expression like 6÷2a when you replace a with a constant or value such as (1+2)=3 we know the value is 3 so we would then write 6÷(2×3) NOT 6÷2(3) OR we could write 6÷(2(1+2))... When you replace a variable with a constant value proper grouping symbols are required to maintain the grouping that WAS the coefficient/variable compound quantity...
To give priority to parenthetical implicit multiplication over division is to break the Order of Operations and the various properties of math. NOT to be confused with the simplistic use of PEMDAS.
BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations
6÷2(1+2)=
6÷2(3)=
3(3)=
9
2(3) is not a parenthetical priority and is exactly the same as 2×3 M not P or B in PEMDAS/BODMAS. Brackets/Parentheses only give priority to operations (INSIDE) the symbol not outside ....
There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more.
Commutative Property
6÷2(1+2)=
6(1+2)÷2=
6(3)÷2=
18÷2=
9
Distributive Property
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
The Distributive Property states to multiply all TERMS inside the parentheses with the TERM outside the parentheses not just the factor next to it.... 6÷2 is one TERM juxstaposed to the parentheses...
Operational inverse of division by the reciprocal
6÷2(1+2)
6(1/2)(1+2)=
6(1/2)(3)=?
Multiply in any order you want you still get 9
Proper use of grouping symbols
6
-----(1+2) = 6÷2(1+2)=9
2
6
-------- = 6÷(2(1+2))=1
2(1+2)
A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator...
Parenthetical implicit multiplication is nothing more than multiplication without the need for a physical multiplication sign and the only correct answer is 9 not 1
Algebraic proof against this BS....
The reciprocal of a is a^-1
6a(1+2) = 6÷a^-1(1+2)
Operational inverse by the reciprocal.
6a(1+2)=9
6a(3)=9
6a=9÷3
6a=3
3÷6=a
a=1/2 or 0.5
0.5^-1= 2
6÷0.5^-1(1+2)=
6÷0.5^-1(3)=
6÷2(3)=
3(3)=
9
6÷(2(1+2))=
6÷(2×1+2×2)=
6÷(2+4)=
6÷6=
Distribution is a method of eliminating the need for parentheses by bringing the terms INSIDE the parentheses to the outside.... You can NOT take
6÷2(1+2) and write 6÷2×1+2×2
You can write
6
---------
2×1+2×2 because the vinculum is a grouping symbol and is the same as writing 6÷(2×1+2×2)
I'm saddened by the fact that so many people share the same misconceptions BUT I'm even more sad and concerned with those who want to blame Common Core or politics....
Common Core has absolutely nothing to do with this debate. It is a teaching method and does NOT change the rules or the answer....
Those who think that this debate has anything to do with Common Core or a political conspiracy are just flat out CLUELESS and STUPID
@@THaWoM : First off there should be absolutely no reason for a grade school kid to ever be introduced to PEDMAS. Why are they be given ambiguous equations in the first place? This just means whoever is supplying the equations is lazy or sloppy and don't know what they are doing. An equation is essentially a sentence and as with all sentences you should employ correct mathematical grammar and punctuation. They should be taught the exhaustive use of parentheses and brackets. Their understanding of what they want an equation to say increases immeasurably. There should be no ambiguity.
Secondly this cavalier use of the obelus wreaks havok with trying to perform algebra. I was taught that everything to the left of the obelus is the numerator and everything to the right is the denominator.
2x/3y-1 SAYS Two ex DIVIDED BY Three why minus 1.
That is (2x)/(3y-1)
And that is how it should be carried out. Any variations should be indicated by parentheses.
The part of the quadratic formula under the square root symbol has a name. It is called the discriminant.
And sorry to say, you have it written incorrectly.
It is b squared minus four ac.
Not 2 bc.
It is for finding the roots, or factors, of an expression which has the form y = ax^2 + bx + c
I can still remember exactly how it was always written and said out loud, and I learned it over 43 years ago.
Minus b plus or minus the square root of b squared minus 4 ac, all over two a.
The plus or minus means that there are two solutions, also called roots, and the sign of the discriminant tells whether the roots are real or not real.
Another way to say it is it says how many zeroes there are, zeroes being the places on a Cartesian coordinate where the parabola crosses the x-axis.
If the discriminant is zero, there is only one solution, or root, or zero, or factor, (all mean the same thing, basically IIRC) meaning the two factors are the same, and the parabola just touches the x-axis in exactly one place. IOW Something like (x +2)(x +2).
If the discriminant is positive, there are two real roots. Physically this means the parabola has it's vertex below the x-axis and there are two y intercepts, or zeroes, or factors.
If negative, there are no real roots, as negative numbers do not have a real square root, only an imaginary one, some multiple of i, which is the square root of negative 1, which does not exist.
These are called complex numbers.
c is where the parabola intercepts the y-axis.
All of this has meaning that applies to the physical world.
When you point out that the expression yc/rb is taken to mean the y and c are multiplied together, as are the r and the b, and then the division is done, is exactly correct in real world usage, at least in the world of science.
Because if the order of operations rule that is at issue here applied, this would be the same as saying yc/r multiplied by b. But that could be very clearly written as ycb/r.
And no one ever writes ycb/r as yc/rb.
It simply makes no sense. There is no logical reason to write it out that way if what is meant is to have the b in the numerator.
@@williejohnson5172 well to begin with
2x implies that 2 has been multiplied to x
Equivalent to
10 implies 5 had been been multiplied to 2
Next
2x/3y-1 is an ambiguous if presented without context. Based on the problem you solve 3y-1 may or may not make sense and with a some spacing it will make sense in both Interpretation. Here's another
1/5x²+6/5x-6y
Equation is ambitious and solely depends on context of what you are solving. With a bit of brackets, it will make sense in bodmas for anyone without context.
These abstract, ambiguous math problems need to go away. There's no application for these numbers so you can have multiple correct answers, depending on where you decide to drop a parenthetical expression. The math problems we did in school weren't like these. Because the textbook writers and teachers know that ambiguously notated expressions can have multiple solutions. These kinds of problems exist only to aggravate people on Facebook.
You are so correct. When I was a child we were never taught PEMDAS nonsense so I've never had a problem with scientific notation all my life.
We were thaught bodmas
PEMDAS or not, the correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended is 9 not 1.
The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y.
This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2
All variables have a coefficient. Constants can be coefficients but constants do not have coefficients. There are no coefficients in this expression...
6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b is 3 not 2
Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
Convention doesn't trump LAW and the Distributive Property is a LAW.
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property, when FULLY applied, is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property, when FULLY applied, REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2 = 3 Monomial factor
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2 no parentheses required
3×1+3×2=
3+6=
9
You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression.....
When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses. TERMS are separated by addition and subtraction not multiplication or division. 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
6÷2(1+2)= 3(1+2) no rules have been broken
2×2×4(a+b) partial Distribution 2×2(4a+4b)
However the TERM outside the parentheses when simplified equals 16 and 16(a+b)
2×2(4a+4b)= 2(8a+8b)= 16a+16b which is the same as 16(a+b)
2×2×4(a+b) when fully Distributed is 2×2×4×a+2×2×4×b and the LIKE TERMS can be factored out of the expanded expression. The LIKE TERMS being 2×2×4 So... 2×2×4(a+b)
6÷2×1+6÷2×2+6÷2×3-6÷2×4= 6÷2(1+2+3-4) as the LIKE TERM 6÷2 was factored out of the expanded expression...
Let y = 0.5
6y(1+2)=?
6y*1+6y*2= ?
6/y⁻¹*1+6/y⁻¹*2= ?
If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9
The rules of math have to remain logical and consistent across the board...
Many years ago in grade school, multiply superseded division and not equal - as an engineer, I still have my reference! For equations written in line, I always provide appropriate parentheses, brackets and braces to force a particular mathematical procedure!
You are absolutely right, I was taught to do multiplication and division operations from left to right, but then, at college, I was kind of implicitly taught that when the operator does not appear, then it gets precedence, so, A/BxC is (A/B)xC, but A/BC is A/(BxC)
Then you were taught wrong.
@@grantofat6438: Why? I bet you were taught at school that "i before e except after c" and then you should remember the "anCIEnt soCIEty of sCIEnce" and that you should "NEIther be FEIsty nor foREIgn", so you were also taught wrong.
As with everything, like language, math is a construct, a set of rules, and if everyone understands each other, it finds a purpose. It is VERY true that at school we all are vehemently taught that we must operate from left to right when the signs have no precedence (division and multiplication, sum and subtraction, etc.), but when you go to college, it is never said but implicitly used time and time again that a non explicit multiplication has precedence. Just try it, give a problem of the type Y=A/BC to college graduates (engineers, physicists, chemists, etc. -maybe not mathematicians-) and they will most commonly operate the multiplication first, BUT, give them the same problem written as Y=A/BxC and most will divide first.
Short conclusion: ALWAYS USE PARENTHESIS so there is no doubt at all of what needs to be operated first.
I know from my many calculations for structural engineering, that Pemdas is wrong.
Of course, I will never see 90 again, and in my time we used paper and pencil, mechanical adding machines and the most important tool of all, a slide rule.
There were no computers or internet.
PENDAS isn’t wrong, but there are two different things happening here. The first is the multiplication before the bracket, from memory it was called implicit multiplication, it comes first because a number outside a bracket means it had been initially factored out elsewhere. The second is the way we try typing something into a computer whereas in the day of writing it down by hand it was easy to put in the correct numerators and denominators but by PC you get a long string and as per other comments here, you should add extra parentheses to clarify.
Thank you! This perplexes me today when I see these "what is the answer" on basic algebra and math questions. I'm almost 63 now. I graduated from a USA high school in 1977. I think that I was taught to solve inside parenthesis first and multiply (or divide if that's the case) by what's attached next to it, which gives me 6 divided by 6 equals 1. On social media and other math videos they try to convince you the answer is 9 using a twisted order of operations which differs from the way I was taught like some form of alien math. This has me so bugged out that I'm searching for what changed over the decades, and what in life is really true or false, especially when it comes to math which should be absolute!
I graduated in 1996 and I was taught the same and came up with the answer as 1.
The TERM outside the parentheses is juxstaposed to the parentheses as a whole not just the numeral next to it. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2.
6÷2(1+2)= 3(1+2) NOT 6÷(2×3)
I agree with what you've said but it's been consistently reconfirmed that math is by no means absolute. Perhaps in it's form as a commonly used construct we all agree on certain things but it can be surprising how many things are decided on from necessity to decide rather than on a naturally grounded explanation.
@@RS-fg5mf that is wrong , since you divided before removing the parenthesis .
the "problem" , has do to with the way Americans interpret parenthesis .
They "think" it means, solving what is on the inside. That is incorrect.
to rationalize an equation with parenthesis , you need to remove them completely before moving forward with any other function.
b÷a(x+y) = b÷(ax+ay) | when b=6,a=2,x=1,y=2 then b÷a(x+y) = 1
@@avibhagan you're wrong. That's exactly what grouping symbols do. Give priority to operations INSIDE the symbol not outside the symbol. There is no rule in math that says you have to open, clear, remove, take off, eliminate or get rid of parentheses.
The RULE is to evaluate operations WITHIN the symbol of INCLUSION as a priority and nothing more... 6÷2(1+2)= 3(1+2) No rules have been broken because it does not literally mean you have to evaluate the parentheses first and foremost...
Even if you decide to get rid of the parentheses... 6÷2(1+2)= 6÷2×1+6÷2×2 by the Distributive Property. Parentheses REMOVED...
Yep. I have exactly the same issue with PEMDAS. It's the juxtaposition, the implied multiplication, the algegraic grouping that has been neglected by PEMDAS. (But you don't have to make Multiplication always higher order than Division if you respect algebraic grouping.) I wouldn't say PEMDAS is wrong, I'd say it is incomplete. Add grouping like PEJMDAS it would fix it.
Sounds like an issue in how PEMDAS is explained (though, I learned it as BIDMAS). When I was told P/B, I was told to complete or solve them - as you put, as algebraic grouping.
The first 2 in 2(1+2) is very much part of the P. 2(1+2) = (2×1+2×2) = (2+4) = 6. Then, it becomes 6÷6=1.
But, if all you get told or all you remember is to simply calculate the internals of the P and then leave it at that, you get this ridiculous scenario.
Whilst I can agree that change in name might help with that, I'd rather see the maths of it taught and explained properly.
I would agree the letters of the term PEMDAS might be misleading/incomplete, this video's interpretation of it was never how I was actually taught though in the 70s, i was always taught the implicit multiplication of "bc" is interpreted first/differently than "b*c" - though taken standalone like that they may be the same, as part of an equation with other terms to evaluate they may not be. As you say, adding the J in might be appropriate for people with a blindly simplified understanding, but it never would have flown in even Albegra1 class.
This perhaps show us the flaw in blindly following acronyms rather than engaging in critical thinking and actually learning a subject rather than just blindly following along and misremembering things years later based on a stupid acronym.
@@temisu_namisu
The problem with having implied multiplication and explicit multiplication as separate things is that is just adds a different type of ambiguity. Suddenly 2(1+2) is different to 2*(1+2). That difference means either a) you can never contract a formulae in the form x*(y+z) to the form x(y+z) because those are now different things, or b) you are allowed to change x*(y+z) to x(y+z) to clean up but then you risk adding ambiguity if someone sees the simplified version and doesn't know where it came from originally.
All you have done with the implicit > explicit multiplication is moved the ambiguity to a different place.
People know we have this ambiguity problem with rules, parentheses is the solution to make sure people evaluate it correctly.
I learned the multiplication by juxtaposition rule when I was in seventh or eighth grade during the mid-fifties. For the past few years I have been a volunteer in our local grade school and have argued that the workbooks the students used were wrong but to no avail. I have, however, showed the better students (those who might advance past primary arithmetic) how the rule will be applied in the future.
Absolutely. Implied multiplication is different from explicit multiplication. Juxtaposition makes it a single term. You must treat it as a single term in all cases. You cannot just arbitrarily change terms.
For example, the number 12 can be rewritten as 3 × 4. So if you had a problem like, what is 24 ÷ 12, you can't then just rewrite 12 as 3 × 4 and come up with:
24 ÷ 12 =
24 ÷ 3 × 4 =
8 × 4 =
32.
You have to resolve all the terms. Terms are separated by explicit operators. Implicit multiplication is a part of the single term.
As a practicing physicist with degrees in physics, mathematics, and engineering, here's my opinion:
1. I never learned PEMDAS or BODMAS or any other such mnemonic devices. I was simply taught "multiplication and division is done before addition and subtraction".
2. Parentheses come first, then exponents, then multiplications and divisions IN ANY ORDER, then additions and subtractions IN ANY ORDER. You certain CAN evaluate them left to right, but it isn't necessary. You'll get the same result using ANY order.
3. Nobody uses the obelus (÷) symbol after elementary school, except for a few specialized uses such as calculator keyboards and the APL programming language.
4. As for a/bc -- I'd say that would usually be interpreted as a/(bc), because otherwise it would be written ac/b. But I would consider it technically wrong and should be rigorously written as a/(bc). Writing a/bc is informal usage, and generally used only when you're constrained to write on one line. In real life, one would write a horizontal bar with "a" on top and "bc" below, to make clear what is a dividend and what is a divisor.
5. In real life, nobody would write an expression like 6 ÷ 2(1+2), precisely because it may be interpreted differently by different people. Most everyone would write the division using a horizontal bar to make it clear what part of the expression is a dividend and what part is a divisor.
The order does matter in subtraction and division. 10-20=(-210) and 2/3=⅔ but 20-10=10 and 3/2=³⁄₂ even though division has the same priority as itself and subtraction has the same priority as itself.
Order of operations does matter. In programming, you have to write your opperation in a linear fascion and the computer will execute them from left to right. So something like 8/2*(2+2) might not be out of place.
@@aryan_kumar - We're saying two different things. Of course 2/3 is not the same as 3/2, and 20-10 is not the same as 10-20. Nobody is saying otherwise. But subtractions and divisions can be done in ANY order -- left to right, right to left, or any other order. For example: 10 - 3 - 4 - 6, left to right: 10 - 3 = 7, 7 - 4 = 3, 3 - 6 = -3. Right to left: -6 - 4 = -10, -10 - 3 = -13, -13 + 10 = -3. Same answer.
Likewise with division: 2 ÷ 3 ÷ 6 ÷ 4, left to right: 2 ÷ 3 = 2/3, (2/3) ÷ 6 = 1/9, (1/9) ÷ 4 = 1/36. Right to left: (1/4)(1/6) = 1/24, (1/24)(1/3) = 1/72, 2 / 72 = 1/36. Same answer.
I really like the 4th point in your statement.
@@MrMarclaxbut you will explicitly mention 2*(2+2) and you will not write 2(2+2) there and that's where the difference in interpretation is
The ending of the video is how I learned in school (also always going from left to right). When I have a good math day, the expressions and equations I’ve evaluated always prove to be the right answer. I think this “dilemma” has more to do with lazy / forgetful teachers and the problems of transcribing inline equations.
At school we were told to do the implicit multiplication before the division. Interesting to see it tarted in the book. Very insightful video.
So was I. Implicit first.
There really is no problem with PEMDAS! It is a question of notation at different levels of math.
PEMDAS works at the foundational level of math where expressions are non-composite.
As the level of math advances, mn/qs "implies" taking mn and qs together. It can no longer just simplify as a × n/q × s !
There are many more operators that cannot follow PEMDAS (factorials, summations, even simple trigs --> sinAb is not bsinA, etc.).
In short PEMDAS for basic order of operations, like LIATE for integration by part are general and convenient ways of manipulating tasks. They do not work for every situation. As complexity of a task increases, mathematical notations take over to guide the order of operation.
It's not that PEMDAS is wrong or a lie, it's simply that it's not designed to handle expressions such as this. PEMDAS and others are used to teach the order of evaluation for the BASIC operations it refers to. Grouping values together is not one of these operations any more than factorials, moduli, trigonometry or tetration are. As such, it's pointless trying to apply it to such expressions. There is no acronym that can cover all mathematics in this way. We learn the rules of interpretation as we learn the more advanced functions. When there's any doubt we should use additional brackets for clarification.
True.
Oh, seems like we've already met on some other channel - your nickname looks pretty familiar :)
I think I'll save this comment to my Google Keep collection.
100%
You're totally right. If there are no brackets, multiplication comes first. This is how it works in Algebra.
IT does not matter. The problem is that people are using PEDMAS wrongly. Some are coming up with a 6/2*3 . There is no 3. It is 6/2(4+2) or 6/2(2+4). In 2(1+2) the 1+2 is commutative and can be swapped to 2(2+1).The 2 gets distributed to the 1 and the 2. It is 6 /2(4+2) or 6/6
@@detroittigersandotherbaseb7220 wtf are you saying there? 6/4+2 = 3.5, and 6/2+4=7, neither of which are the correct answer...
@@irrelevant_noob Oops, I left off the 2() and just had the additions. I fixed it.