Okay wow, I was not expecting this video to get this much traction. I was just kinda happy that a few people on the 3Blue1Brown discord server liked it and then just kinda left it at that, thinking this was a fun little project to flex my video making skills and not much more. But we're now at almost 3000 views (Edit: scratch that, I'm absolutely floored right now) and still growing and I'm not really sure how to think about this. I wanna say something corny like "thank you so much, it means alot to me, etc." but that feels weird while this is still somewhat surreal to me. I wasn't planning on making anything more for the channel in the near future so don't expect anymore videos soon, but I will definitely make more at some point. So in order to not dupe myself, instead I'm just gonna ask, what did you think of the video? Particularly if there were any specific parts which especially grabbed your attention or that you didn't think worked so well. These two points, especially the latter, are in my experience the best kinds of criticism you can get. So yeah... thank you... I guess? Also on an only slightly unrelated tangent. I think Henry (Minutephysics) has a problem with communicating his intent because he's made the exact same mistake again in his most recent video on The Butterfly Effect: ruclips.net/video/vLIPUdru82c/видео.html He once again tries to explain how something can lead to misinterpretation, explains the problem and then people retort with "But that's not how I interpret it". I previously thought he must've misunderstood how PEMDAS works which left his argument open to be rebutted but now I'm convinced he's just really bad at clarifying what he's even trying to argue. Commentors think he's arguing that The Butterfly Effect is bad based on his own faulty interpretation, completely missing that the fact that the butterfly effect can so easily lead to misinterpretation in the first place is the real problem with it. EXACTLY the same thing that happened with the PEMDAS video. Tangent over. Also I now hate the word "misinterpretation", it's annoying to type.
man I got worse at math since I was put in spreal needs and they gave me year 1 level work (LEGIT we all got the same I was the eldest everyone else was year 1)
@@BlackholeSteve so true. I got better at math by doing olympiad problems etc. and when put at university I felt like loosing a bit. It's very important to keep up a work that is hard
We had a C programming lab session for us this semester, and one of the questions was to evaluate a expression. It was not like a function or something to do with variables, it was just a math expression. All we had to do was copy the thing and equate it to a variable, and print it. And the answer we got (the actually mathematically correct answer which the computer and the calculator gave) was rejected by the teacher because it didn't follow the BODMAS rule... It wasn't anything ambiguous at all, she genuinely thought that division had higher precedence compared to multiplication. We ended up deliberately adding brackets to get the "correct" answer. This is so sad.
I can see how people would think division has _lower_ precedence than multiplication (all the factors to the /‘s left go in the numerator, and the ones to its right go in the denominator). But this is pretty bad. Stubborn teachers are the worst.
@@jjdoughboy2103 I could never really get irrational numbers, too. One of the areas where I have to strictly work with formulary. Never had a problem with fractions, but I still think most order of operations problems would go away by not writing formulas inline. The irrational number thing is curious, never had problems with differential equations, even partial differential equations are easier to think through than irrational numbers. Also never had problems with calculus or discretisation and numeric solution methods. Something I got even less than irrational numbers are distributions, but I'm an engineer and should never have tried to understand that. A step response is a step response, who cares what the mathematical basis is :-)
The basic problem here is that it's not an *order* of operations, it's a *hierarchy* of operations. Syntax, both in language and in math, creates structural trees out of linear input. Multiplication doesn't precede addition, rather, they're more closely "glued together" in the structural tree. I wrote an article about this for the Mathematics Teacher journal that was published in 2017, but I think it's unavailable if you're not a member of the NCTM. It doesn't address the problem of multiplication by juxtaposition, but it does argue that PEMDAS isn't a useful approach, and suggests that students be taught to seek out terms and factors first, and then the terms and factors within those, and so on. This duplicates the hierarchy, rather than suggesting that it should be a linear order.
Well, the issue is mainly the infix notation that everyone is used to and that's the one that is ambiguous or need some kind of order or brackets to make clear which order was meant. When you use pre- or post- fix notation you don't need brackets at all and it's completely unambiguous. Though for humans it's a bit difficult to read and interpret pre or post fix, though I think it may be just be a matter of training. Though infix has the advantage that the operators act automatically as seperators between operands. Because if you write 5+6 in postfix it would be 5 6 +. I used spaces to seperate the operands, however it becomes tricky for humans to write or read such a notation. Easier to read and understand may be prefix or function notation since you could do sum(5, 6) The example from the video 6/2(1+2) == 9 would be this in prefix mul(div(6,2),sum(1,2)) == 9 or this: */6 2+1 2 == 9 In post fix it would be something like this: 6 2/1 2+* == 9 With pre and post fix you explicitly have to encode the order you want in the notation. It's completely unambiguous as you would use a stack to evaluate the notation. You could also write it as 1 2+6 2 /* == 9 The only issue we have are things like negative numbers :) So we would need either a different symbol for a negative number or a different symbol for subtraction. An alternative would be to be more consistent with the spacing. So always insert a space between every operator and every operand. Then we could use the minus for both. Though it would be easy to mistake it.
@@nahyankamale9989 This is the kind of thinking that led to PEMDAS, namely, focusing on the appearance than the content of the subject, in this case, mathematics. Despite this comment being a joke, I wanted to share this observation.
My take as a programmer is it's parenthesis first and if it's still ambiguous, you haven't used enough parenthesis. Relying on order of operations which differ slightly in almost all programming languages (because a language syntax will end up having one even if you don't think about it because _something_ has to be evaluated first); is madness!
Eh... I don't like cluttering my code with extra brackets, so I'll either remember the operator precedence, or look it up. I guess my code is liable to break if the next version of the language changes the operator precedence.
@@takatamiyagawa5688 what kind of expressions/equations are you writing friend? It's not like you're going to be waking up next to your code and be like, “Good morning beautiful”
Order of (arithmetic) operations is the same everywhere as far as programming goes. The difference is what a(b) is - a term? a highest-order operation? a multiplication operation equivalent to a * b, so therefore a / b (c) is equivalent to (a/b)(c). To all programming languages, a(b) is an attempted call of the function a, so it's not a problem because this situation is not possible.
there is a bit of a difference between programming mathematical equations and writing them on paper. When I write equations on paper I use the fewest symbols and notations possible. However, when writing the same equations for a program I use plenty of brackets also. Mostly because you can't write it the same way as on paper.
I'm a programmer too. But you also need to be able to interpret written equations correctly, without relying on the author having used spurious parentheses. Once the author has written the equation according to universally-accepted rules, it's up to the reader to be able to evaluate it correctly. Too many can't and preach PEMDAS when its not universally applicable.
A big problem for learning is the oversimplification of subjects. I think "PEMDAS" is a great example of how making things "easier" can really mess you up when it gets more complicated.
Yess, exactly! I guess it's not so bad for people that will not continue to study the subject, but I can't even remember how many times I had to re-learn things during my education because someone couldn't bother saying that something is just a simplification
@@danielblumowski34 Ikr I wish they'd atleast admit it's a simplification. In chemistry we learned that the innermost electron shell always has 2 electrons and after that it's always 8 per shell and there's no more than 4 shells. That confused me because a lot of elements have more than 26 electrons and my teacher refused to explain it because it's a higher level. I had to ask my older brother to explain it and then I actually understood
@@redshift739 I think it depends on the country because I was told something different about electron shells but still it confused me a lot. Then I learned about orbitals and it brought even more confusion because the way they are usually tought is wrong too. I only "understood" the topic when I started finding information on my own. It's so frustrating. Is saying that something is not entirely true or that it lacks mathematical rigour so difficult? I guess many teachers don't even know that what they're teaching is simplified.
It's ok to simplify as long as you make it clear that is what you are doing. Sometimes, simplification and partial explanations are necessary parts of the learning process. Throwing someone in the deep-end to sink or swim is not a good teaching method.
I have serious arguments with my math teacher when he teaches blatant lies _(some examples of his stellar teaching: imaginary numbers don't exist because you actually can't take the square root of a negative, but we are going to pretend we can for this one single lesson, and go back to the REAL numbers that are actually useful, and also 0.99999... doesn't equal 1 because there's always a small gap silly, same with all those other weird infinite sums, we're just rounding.)_ and I tell him this is really going to bite everyone in the ass when we get to calculus or any sort of higher level math at all.
Mathematicians, like writers, all too often need to understand that they're responsible for communicating their ideas clearly. If a sentence is unclear you don't blame the language, or blame the reader, you rewrite it.
"If a sentence is unclear you don't blame the language" is only true if we're talking about natural languages. If however a language is engineered for a practical purpose, you absolutely can and should criticise it for its ability to fulfil that purpose.
@@interrogatix I love how this discussion is being made in the incredibly impractical language that is english lmao. And yeah I agree, math already has a perfect easy to understand language in the form of PEMDAS (or PEDS/PEMA), people slacking by taking out parenthesis where they should be and making up their own math rules among other things just makes it hard to tell what they mean.
@@interrogatix I mean, it's just a matter of practicality. Mathematical common practice is a convention based on what mathematicians the world over have settled on over the years. Any attempt to change it is going to be almost as difficult as changing a natural language. So I don't think it's outrageous to say people should probably focus on being more clear with the conventions they have, in the short term. Mathematical notation can absolutely be criticized but at certain points you have to ask yourself what good it's doing. It's the same with english; I can spend days drafting spelling reform and punctuation reform for written english, but in real life it's still going to be written the way it's always been written.
@@interrogatixI'm not sure who told you that math is an engineered language when it's not. It's just another language that evolved naturally and happens to have a governing body like French
Being clear, unambiguous is the most important. I never hesitate to add a redundant parenthesis, either in papers or in programming. Also, in a lot of cases, units force the order of operations.
@@beageler On the last page, lower left of: P. Welch, "The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms," IEEE Transactions on audio and electroacoustics, vol. 15, no. 2, pp. 70-73, 1967., you can see an example of unnecessary parenthesizes that add clarity. It is a classical paper with more than 12000 citations. And, in programmation, I was talking about parenthesis that doesn't alter the order of operation, such as (A+B)+(C+D). Sometime, they are useful, again for clarity. That was my point. Perhaps my comment lacked some additional clarity, but, there is a limit to the effort I give to a simple subject as this one.
@@minimumlikelihood6552 Life is too short to be pissed off all the time. Especially when I rethink something and see right off that I was wrong, like fractional notation not eliminating everything that could confuse people who don't get order of operations. Or when I remember that readable code is a thing. Again, thanks for taking your time for that answer and for being nice. :-)
The problem lies in the fact that the slash doesn't make clear what is being divided and what's not. As an engineering and high school tutor I despise the slash and/or obelus symbol when used in grades older than third. Use ONLY fractions for division, which will also assist with simplifications before multiplication because it makes the numbers multiplied smaller. If you must use the "kid" symbols, then make sure that everything divided is in brackets indicating the denomator and anything else exists on the numerator.
If I use /, I always make sure to put everything in brackets, as in ((a+b)/(c*d)), because otherwise it can be extremely hard to find out what is the divisor. And I have learned not to use the notation a b/c for a+b/c anymore, since it can be read as a*b/c
i love the fact that here in germany in the last years of school, and at university we basically just didnt bother. Every division gets written as a fraction and that way no ambiguity ever occurs
In all my years in school, it was only in elementary schools that did simple division such as 9÷3. Calculators express division with a forward slash 1/4 and many expressions now show proper fractions with an over and above character ( ¼ ). PEMDAS itself is a basic guideline for handling expressions, but when it comes to formulas expressions that don't have symbols it can become confusing to beginners who might not have seen something like √(25)(-2)². Even then, it's hard to express formulas on a computer because mmany of the complex functions in math can't even be typed without looking them up.
We do the same in the U.S.A but the correct way is the answer is 1 way. There's no ambiguity.. it just that people focus in the U.S.A to memorize PEMDAS and they forget in 8th and 9th grade they finished teaching PEMDAS to add in understanding of juxtaposition were 2(2+3) is the same as (2(2+3)) in groupings. So, most forget this and the companies that make calculators or anyone writing a program that's a calculator has to strictly follow PEMDAS and omit juxtaposition because they assume kids in the elementary school will use this program and this should be the default. Most have different modes where you can switch them to use juxtaposition. You either learn how to change modes or if you like me.. just do the calculations by steps manually. After 9th grade in the U.S.A we do things correctly according to Mathematics. Before this they make us strictly Follow just PEMDAS and then 8th to 9th grade when learning algebra they then add in the last rule which is juxtaposition. All Americans remember PEMDAS but they forget what they learned in algebra class about juxtaposition.
I think the true but pedagogically wrong answer is: if you're solving an actual problem and not just doing an arithmetic exercise then ambiguous notation doesn't matter so much. You'll know whether a/bc means (a/b)c or a/(bc) by knowing where the expression came from.
That's actually a pretty important point I hadn't considered. Obviously ambiguous notation isn't really a problem if you know what the expression actually represents, but that ambiguity becomes a huge problem when you're trying to do arithmetic divorced from any context to clarify it. Maybe that hints at another problem with our teaching methods.
Mathematics is a language, and in all language we should make every effort to communicate our ideas clearly. These PEMDAS problems are poorly structured expressions (usually deliberately so). Yes, a computer or a calculator needs some rule to interpret these bad structures. But rather than teaching people how to think like a computer, we would be better served teaching them how to write clearly unambiguous expressions.
Just what I was about to write. If someone wants rigid unambiguous notation, use reverse polish notation (in fact is what some early models of HP hand held calculators used). That is perfect for a computer. But we humans are not very good at doing algebra in reverse polish notation. Instead, we can be magnanimous with the use of parenthesis. The problem with school is that it does not focus in teaching students to think, but on producing straight jackets for everyone’s minds to wear. I teach first and second year math for engineering at a spanish university and can attest that my students appear to have been deprived of any ability to use their heads to think.
Mathematics uses languages and it has ways of specifying formal grammars. See LALR and Earley parsers. A mathematical notation should try to use a deterministic context-free grammar if possible; to keep things simpler. Also we're run into program languages that have much more involved order of operations, and to avoid bugs it is generally regarded as good advice to use parenthesis sometimes even when not strictly needed. An unambiguous expression in a particular grammar might still be difficult for a human to follow.
@@stephenJpollei, because there are more things in heaven and earth, Horatio, than are dreamt of in your philosophy. Just for the sake of rhethorics, computers are what that philosophy dreams, whilst natural languages live in heaven and earth ...
@@MiguelGarcia-zx1qj My philosophy is pretty big; I think there are infinite expressions and infinite grammars. Computers are just mathematics; see Alonzo Church, Alan Turing, Kurt Gödel, etc. I also like some conlangs such as Toki Pona and Lojban. BTW due to Gödel, I think there are infinite logic/math systems :) Another aside, Lojban grammar is almost but not quite parsable with LALR(1), but PEG( parsing expression grammar) was required. “Glendower: I can call the spirits from the vasty deep. Hotspur: Why, so can I, or so can any man; But will they come, when you do call for them?” ― William Shakespeare, King Henry IV, Part 1
As someone who's not a native english speaker and was not taught math in the english-speaking countries... I never knew there was an acronym for it lol. My teacher just said "First brackets, then multiplication and divison, then addition and subtraction." No silly acronyms, literally just the rule, and many example exercises to solve until it's kinda like muscle memory.
@@dexternepo I'm from Poland. Translating the acronym to our language would give only consonants, not easy to remember. And since this stuff is taught in like 4th Grade, you can't assune everyone knows English.
You don’t understand how happy I feel after the gaslighting of telling me I’ve been doing this problem wrong. It’s not that hard to understand left -> right and that M & D and A & S are equal, however I had people tell me I’m wrong and that I “didn’t follow PEMDAS”. I’m not a math major, but math has always been my strong suit, so to have so many people incorrectly correct my math was extremely confusing
Honestly, using ÷ is just bad notation in general. It makes it seem like division is a seperate operation from multiplication, when in fact it is not. "Division" is multiplication of one number with the multiplicative inverse of another number. Instead of a ÷ b, you should always write/understand it as a * b^-1 Not a divided by b, but a multiplied with the (multiplicative) inverse of b. It's fine to call it simply the inverse, since the additive inverse is usually called the negation instead of the inverse, so there shouldn't be confusion.
@@joda7697 wait so if I write 4(1+2) * 3(6+4)^-1 isn't this still considered ambiguis writing even if I wrote it using exponents instead of division sign? sorry in advance if I misinterpretted your point.
@@arenmee540 No how is this ambiguous? this is clearly 36/10 I know you intended it to be 12/30, but 36/10 is objectively how that would be read, no way around it. Otherwise, you'd have to write 4(1+2)(3(6+4))^-1
throwback to when my elementary teacher told me you can't subtract more than what you started with. I mean why not instill a deep sense of academic distrust in students when they first start doing school, nothing could possibly go wrong.
Every god damn time, first I questioned subtraction, was told I was wrong, then they were wrong. Then roots with negative numbers. Then dividing by zero. Every time they have said I couldnt do something, I would be told I could later. Just tell me that is in a later class, don't lie to me.
@@MCNarret for real. Nothing wrong with saying "we are just focusing on positive integers" instead of saying "negative numbers are impossible" at least connect it to real life, you can't have -1 of a physical object, but you can 100% have -1 written on a piece of paper.
@@MCNarret You can't divide by zero though, you can only give an approximation of equations that approach infinity relative to each other. Anything/zero doesn't equal infinity, it equals undefined, which isn't really equaling anything
@@Reece8u You say that, yet if everything points to a defined value, then it is that defined value. I am not saying all divide by zeros are equal, nor are infinities, but if you assume a limit to evaluate an equation or value, approaching infinity might as well be infinity. I see no practical difference.
Math teachers are always like this. "Only even numbers can be divided by 2" Sike, a bit later you learn about decimals/fractions "You can't add something and get a smaller number, you have to subtract for that" Sike again, negative numbers are a thing "You can't square a number and get a negative result" Sike yet again, imaginary numbers are a thing
Teach them correctly from the start... Wow who would have thought! It reminds me of back when I was in school and I had an interest in math so I was learning some more advance things at home than our teacher was covering. When I pointed out that you can have a larger number subtracted from a smaller number in math class and it would be negative the teacher just told me that we weren't covering that now as it would be too confusing to people. Over the years though I've come to think the teacher was completely wrong. Having your students learn "truths" that are false does far more harm. With any sort of learning environment the goal is to build on the lessons that came before. But when you basically throw out the lessons of what came before, or do it in a different way that ends up confusing people. It's like with algebra where some of students have trouble grasping that there are now letters in math when really it's something they have been doing already. In grade school you get a problem like 3 + __ = 7 and you are taught how to solve it. Now that same student gets to Algebra class and sees 3 + x = 7 and they are like WTF there are letters in math, this is complicated. The constantly changing rules not only adds complexity to those who may already be struggling but it also serves to highlight to all of the students you can't trust your teachers. They are just there to push what's written in the book which itself is potentially questionable. I've had classes where teachers told us not to look up answers in back of the book because they were not always correct, they claimed it was done to catch cheaters but I suspect it was done because of the order of operations being done differently. But at the end of the day it creates a sense of distrust between layman and supposed experts in their fields when the average layman's experience comes from school which had many cases of contradictions and down right falsehoods.
Mistakes in solutions are usually misprints, typos or the person writing the book just messing up some minor calculation. Most textbooks have thousands of problems, of course people writing them will make a mistake every now and then.
In my elementary school we had this program called AM (Accelerated Math) where computers would generate custom math tests for students to progress at their own pace. I got really far ahead in 3rd grade and encountered order of operations, which normally isn't taught until 4th grade. My 3rd grade teacher misremembered PEMDAS and explained it to me as a strict order, so multiplication before division and addition before subtraction. It was really annoying getting scores of 28/30, and I didn't realize what I was doing wrong until it came up in the main curriculum a year later.
Unrelated to what you said, but in my elementary school we had Accelerated Math as well. I was always behind on it, not because I was dumb but because I just hated doing it.
@@ow_su I learned pemdas prior to sixth but my 6th grade math teacher did explain how it was a flawed system and from then on I see it is a better option just have rules explained rather than acronym
PEMDAS and PEMDSA would yield the same answer in all situations. PEMDAS and PEDMAS don't however and everyone pre-generation X when no one had a pocket calculator learned multiplication before division
"Do it the way mathematicians do it" is begging the question of which mathematicians? Most papers I read are in the vicinity of type theory where juxtaposition almost always indicates function application and multiplication, if it comes up at all, is noted explicitly. While I don't think type theory should be the standard for math classes, I'm not at all convinced there's a universally accepted system. Three examples doesn't make a consensus.
I think it’s just really important to understand that conventions and definitions can differ when you’re in different fields. Stuff like Einstein notation isn’t strictly standard, but when you’re reading about something that uses tensors, you just know what it’s supposed to mean. But that doesn’t mean that there shouldn’t be a baseline understanding of what the most common interpretation is like, and at the moment, this is quite well defined, albeit a little all over the place.
@@weakspirit_ I mean math syntax as it is IS the problem to took form over function and needlessly complicated it because they were to lazy to use proper syntax of writing everything down properly.
This is almost like the FOIL situation as in school making you mindlessly memorize acronyms. In 6th grade, we were taught FOIL. It worked pretty well for me in 6th grade, i could multiply two binomials pretty quickly. But then, one day, I wondered to myself. _Why does foil work?_ So, I thought. First, Inner, Outer, Last, why did we need that order? I tried doing a different order, and sure enough, I got the same result. Then, I realized that FOIL was basically just telling you to multiply each term in one of the parentheses by all the terms in the other parentheses. I tried this out, but with more complex equations that weren't just binomials. It worked on *_EVERYTHING_*. And, then I thought some more and realized that could be put in even simpler terms: It is basically just the distributive property but with more numbers. I am very glad my 6th grade self had made that discovery, it made multiplying polynomials WAYYYY more intuitive.
FRRRRR as someone who can see stuff like this easily it pains me when people say "foil" because it's an acronym for the sake of someone pointing at a binomial and going "NO WAY IT SPELLS A WORD 😱😱😱"
man im doing advanced higher maths (uk, basically like your first year of uni in highschool in terms of the curriculum) and this simple explanation is so much better than a random ass acronym. I wish school actually taught the rules of maths instead of weird memory games.
Yeah I remember only being taught FOIL. But recently when I was tutoring a kid who was learning how to multiply binomials, I found that he had been taught to use the distributive property first, and then a couple of sections later was taught FOIL as a short cut AFTER learning why it worked. This way of teaching it so much better than just having students memorize an acronym, because, just like you said, distribution works for ANY polynomial. Oddly, my brains switches the order automatically, like I will be trying to use FOIL but my brain goes "first, inside, outside, last".... so FIOL I guess lol. Which makes sense now that I think about it because that mirrors what I would do if I used distribution.
We didn't get told about FOIL until 2 years after we'd originally learnt this, and that was only because a trainee teacher came over and taught us it. Obviously I knew how it worked, but what annoyed me more was that my go-to order was FIOL instead! It upset me being told such a static way to do it when my method was slightly different and worked perfectly fine.
100% I was taught how to expand first, then my teacher told us that FOIL thing. I believed she was making things even more complicated by that FOIL method when she actually thought she was making it easier. I just decided to ignore it and now I don't even know how does the FOIL work.
In Germany we learn "Punkt vor Strich" which literally means "dot before line" (we commonly use : for devision). After that we learn things in parenthesis go first. So at least a bit better then PEMDAS
is it though? Because here you directly learned to put * before / which while being what PEMDAS does, its intention is not doing that, however yours is
@@pedrosso0 in Germany lots of schools teach the syntaxes like this: multiplication = · | division as : | addition as + | subtraction as - || so since multiplication and division are executed before addition and subtraction the rule would be dots before lines. Although I have no idea as to why we use · and :
well, it means that division will act on fewer things, and get a little closer to being a unary operation in an expression like... "2 + 3 * 4 - 5 / 6" treating - and / as unary operations (which is to say, as "2 + 3 * 4 + (0 - 5) * (1 / 6)") and treating them as operations that occur first (which is to say, "2 + ((3 * 4) - (5 / 6))") are pretty close to achieving the same result I think GEMA is a great convention now that I have been properly introduced to the idea, after trying to preach this "ditch subtraction and division as independent operators, they're the same as inverse addition and inverse multiplication" idea
The only reason I understood PEMDAS was because of the fact that I had been exposed to the C programming language before I ever took Algebra. It gives multiplication and division the same weight in almost every scenario, and also does the same for addition and subtraction. I was also taught that division and subtraction are just simpler ways of undoing addition and multiplication without using negative numbers or fractions. More to the point, I was shown somewhere that subtraction is often really implemented as adding a negative number, and division is really implemented as multiplying a number by the multiplicative inverse in a lot of code, sometimes even at the hardware level, because it simply wasn't necessary to duplicate the logic as it works the same way regardless. Starting with that as a foundation, it's not hard to make the leap from the idea that the compiler treats addition/subtraction and multiplication/division as two sides of the same coin, to figuring out that this might reflect an actual mathematical principle rather than just being a convenient arbitrary language rule or quirk of CPU architecture.
I once taught a first grader about how to use negative numbers in about 5 to 10 minutes, and he proceeded to calculate problems using them correctly. All it took was for me to say that: Positive numbers are basically like arrows pointing from 0 to the right, and negative numbers just point to the left. Adding any 2 numbers means to put the base of one of them to the tip of the other, and see where the new tip points. The best thing about this is that it even works for euclidian vectors and is very intuitive for a child to learn.
Using geometric explanmations for maths is so much better in any case. I really have no idea why so many math teachers hate using it. It took me 2 years in college to get irrational numbers, cause it took them 2 years to show us that it's actually adding a plane to the numerical line. After that I understood what irrationals are (mostly the fact they aren't irrational, but just a logical physical extension of line into a plane) and could actually properly operate on them using my previous math knowledge.
@@kjl3080 Yeah they definitely mean the complex plane and imaginary numbers, because irrationals are something different. These are the numbers that can not be expressed as a product of one integer and one multiplicative inverse of another integer.
@@Ussurin First of all you meant the complex plane and not irrationals, but hey we all mix stuff up from time to time. Secondly, this arrow approach is also good because it already makes the additive identity (so 0) have a visceral importance, as the point where the first arrow starts. The field axioms for the additive and multiplicative numbers are maybe formulated very abstractly and with difficult language, but they _can_ be explained to a child with more simple language, and if they build their intuitions right on those, they will have a much much smoother learning experience going forward.
The real reason the Facebook equations are causing so much confusion is because they are intentionally being written in a ambiguous manner, because the point is to get people to interact with it and generate traffic for the page. People arguing over the answer is the end goal. If pemdas wasn't an issue they would find some other way to do the same thing
Well they are written so those stuck with pedmas would get a different answer then those with higher math studies. And it's always because of a number next to parenthesis. (which people think is that same as a normal * multiplication). But any algebra will tell you : 2(a+b) will always be (2a+2b) It's not confusing per say usually..
@@peterthomas5792 I'm talking about the Facebook meme and who created the equation. Were they somebody using pemdas or were they somebody doing actual math
My problem with the equation including 2(...) is that unless multiplication symbol is explicitly written 2*(...), it's a single element in my mind. Same as 2x for example. Either way, I rarely ever write division in line on paper, as I always use fractions instead. Also in programming there's no confusion, since you have to use all the operators explicitly.
@studiouskid1528 2(x) is not a function though... This all comes down to ambiguity. This question literally has no answer, it's just nonsensical. It's like asking "What color is geography?"
wow, didn't know that students in the US and UK have these mnemonic rules. for me personally it is ok to make up a mnemonic rule only when you have to memorize some stupid big formula without particular deep meaning. i think that conceptual things like this are better memorized just by understanding its core and mnemonic can be harmful in this situation.
In france we just make the order obvious by using fractions or adding parentheses, in school we just learn to begin by the parentheses and that × and ÷ comes before + and - i've never enountered an issue between × and ÷ in my life, and passed elementary school nobody uses the ÷ symbol anymore
You know, I was always a little confused about what people had against PEMDAS, until this video made me realize that PEMDAS wasn't the thing that I learned in school.
exactly lol I'm not american so I never even heard about pemdas until I finished my second year in uni. From the very start in school, operations were explained as as opposites to one another, so there wasn't ever any confusion
Honestly, I was just NOT taught PEMDAS at school, only the order of operations and when I heard americans talking about it... I got instantly confused. And when I saw people saying multiplication should go FIRST, I got EVEN MORE confused. And I guess... Well, I guess this explains a lot lmao
PEMDAS is just an acronym for the order of operations. If you learned the order of operations, you learned pemdas, you just didn't learn the acronym. Nothing wrong with not learning the acronym lol
I think learning that MD and AS are actually the same operation (division is multiplying by a fraction, subtraction is adding a negative number) would simplify it. Then it would be a proper hierarchy of P-E-M-A.
@ I did watch the video. The order of operations is PEMDAS (or PEDS or whatever you wanna call it). Though that was a month ago, I still stand by my opinion. All of his contradictions were wrong except for the one where he said it should be a 4 letter acronym, which made sense.
It baffles me that math teachers and mathematicians can't figure out why math is commonly stated to be a student's least favorite subject. So many unclear rules, and the different methods of teaching that contradict another form of teaching is frustrating. And this is coming from someone who never minded the subject (It was the only subject I was truly confident in my abilities to pass in).
1:11 you learn polynomials (if they are a real thing i cant even remember) which are basically some number with () in front and whatever is in the brackets needs to be multiplied by the number before brackets for example 3(2+3) is 3(5) which is 3*5 which is 15 therefore 6/2(1+2) is 6/(2*3) brackets are used here to simplify the equation otherwise it would be bad math, 6/6 which is 1
Same. It drives everyone around me crazy but if you abuse parenthesis you're going to get it right every time. It ain't pretty but it's straightforward.
I feel that the issue is that people see multiplication and division ( and similarly addition and subtraction) as two different things, when they are infact one and the same. Dividing by 2 is the same as multiplying by 1/2 or 0.5. Order doesn't matter, what matters is clear notation.
@@TeraAFK The thing called math in grade school can't really be called math, it's a game of remembering. But anyway: division and subtraction are literally defined as the opposites of multiplication and addition, the same way root and logarithm are the opposites of exponents. a * x = b -> x = b / a (the definition of division); a + x = b -> x = b - a (the definition of subtraction). Multiplication and addition are commutative (that is, a (operation) b = b (operation) a), so they both have 1 opposite. a ^ x = b -> x = log(a, b) (definition of log) x ^ n = a -> x = ⁿ√a, where x ≥ 0; n ∈ ℕ and n ≥ 2 (definition of arithmetic root)
When I first learned PEMDAS in grde school literally added my own brackets PE(MD)(AS) in my notes cause it seemed the most obvious way for me to remember their equal weight. Nice to see others agree!
I think the thing that did irritate me learning math in school was finding out that in later more advanced math was that when subtraction was happening it was the addition of a negative number, which made sense to me, but was also confusing because I did not understand why they didn't just teach that to begin with instead of making things complicated.
Why is "just write clearly" a non-solution? If ambiguity is not intentional, then whoever tries to convey the message is likely to try and avoid it, no matter in natural language or in maths. Mathematical notation is obviously not immune from ambiguity and no amount of additional rules is likely to remedy this. So to me it's like: "make sure that the meaning of what you write is clear either by itself or from context"; "to get a better idea of what's clear and what's not-read more of what others (mathematicians in this case) write," and hope for the best.
Agree, the whole point of math is to understand. If 50% of the Internet (including mathematichans)fails to understand you, you could've probably written it better. I'm talking abt the viral math problem Edit: clarification
In programming languages, the meaning is indeed strict and only has one meaning. I agree the acronym is not helpful and it's easy enough to just remember the meaning. One good explanation is to show that the order is what it is because it makes it easy to write polynomials without extra parens.
In C++, some operators have left-right associativity and others have right-left associativity (the direction the operators are evaluated when they have the same precedence). Also, when multiple operators are next to each other, the compiler reads the biggest operator possible. For example: x---y evaluates as: x-- -y and not: x - --y
Curiously, I think this is only a problem with some education systems. When I was young in Spain of course I learned the order of operations which is as the system you describe. However it never had a name akin of a memo trick to remember it. And, most importantly, teachers wouldn't use tricky expressions just to fool you, they would use normal expressions. So there was no room to argue if the multiplication or the division had to go first. As an adult I have been exposed to this nonsense, of course, but I have always dismissed it as such. It is not like PEMDAS is wrong, it is useful as long as you don't try to be a smarty pant. And if you do try to be one, then the expression you come up with will not be relevant for anything more than being a smarty pants. And on a side note, if I would write 1/(2N) as such, not 1/2N. And the only reason Feinman may have done it differently is because just 3 lines above that the expression made more sense, with a big division line which makes it plentifully clear what you should do first.
Thats why no one uses ÷ sign in math. Its better to use fraction bar, Just like this 6 ------- 2(1+2) Or 6 --------- ×(1+2) 2 There, no misunderstandings
Also, advanced calculators can give you the wrong answer if you don’t input the values correctly. After using TI-84s for high school and college courses, you tend to learn that you have to add extra parentheses just so the calculator _knows_ what the equation is.
I fucking hate acronyms in education. They are the clearest indicator that the education system wants you to memorize things, not learn them. I am thankful to my primary school teacher for not telling us to memorize the multiplication table.
You had me right up until the last 10 seconds. I think everything was on point, but the landing was flubbed when you over-simplified your point--and I believe even contradicted yourself--when at the end you said "Instead of wasting time teaching language in math class, maybe just teach math." The thing is--and you all but acknowledged this earlier when you said that the important thing is communicating the underlying idea, and that "different ways of saying the same thing are still all saying . . . the SAME THING"--that math IS a language. It has it's own vocabulary and syntax and grammar that need to be understood in order to communicate well with it. That is one of the most powerful paradigm shifts that come to my high school students when I'm teaching functions and they understand that equations, graphs, charts, and lists of points are all essentially just dialects that are different ways to describe the same underlying function. And with that understanding they start to grasp the abstract idea of what a function actually is. Yes, math has it's own grammar rules, and sometimes those rules are arbitrary--but sometimes, when understood properly, those rules make perfect sense, and actually couldn't be any other way. "Math is the language by which God created the universe." --Galileo
6/2(1+2) isn't math, it's god damn nonsense masquerading as math. We have the ability to adequately communicate every aspect of mathematics. We have the means to describe such a problem in no unclear terms. Pemdas only works, because we have an underlying foundation that works unambiguously, and strictly based on logically (or rather natural law). There isn't a single operation in that "problem" that 99 percent of any developed nation couldn't solve. This bullshit literally makes people worse at math. Excusing poorly written "problems" is the fucking problem. There isn't a right answer to this math problem because it isn't math. Pemdas is not math, straight up. It has no logic. It is not backed by law. It is pure abstraction. Sure, base 10 as a standard is pure abstraction. All the symbols we use to communicate math is pure abstraction. But what they resent is not abstract, and the logic they follow is not. Math does have its own grammar rules, and pemdas brakes them. Instead of teaching kids how to properly articulate in line math problems, we get bullshit like this. Any language that cannot reliably communicate its ideas is not language at all. Fortunately math does have a robust, easily understood language. It's just counting, after all.
Formally there should be brackets. We dismiss brackets when there is no ambiguity. If there is ambiguity, one should use brackets. So 6/2(1+2) needs one more pair of brackets imo. Without these Id say it's (6/2)(1+2), but if it was written by someone that didnt intend the ambiguity Id think it's 6/(2(1+2))
It’s not true in real life though. The area of a parallelogram with the parallel sides of length a and b and height h is never going to be written as h/2(a+b). That looks immediately wrong to anyone with some experience in mathematics at a higher level. Brackets are not necessary if you wanted 6/(2(1+2)) because there’s only one interpretation.
@@qsykip Ik thats what I tried to explain in my last words. I meant in practice if someone wrote this I woule think it's h/(2(a+b)) To me it would mean (h/2)(a+b) only if it was written 'formally'... I mean it feels like the rules would say it's (h/2)(a+b), even though in fact it's just ambiguous from a formal point of view I have a hard time explaining myself in english, lol
@@holomurphy22 I‘ve understood you now. I think it’s just somehow an unwritten rule. Everyone knows it, but nobody really talks about it. The point of this video was probably to bring to attention the fact that all this PEMDAS/BIDMAS stuff isn’t complete, and shouldn’t be regarded as a formal rule. It’s one of those things that American and British (and a few others) schools keep teaching that isn’t strictly correct in real life, but because it’s taught like this, people start to defend it as though it’s the law of the universe.
@@qsykip I dont know much about pemdas and such rules. In France we only hear a thing like 'multiplication before addition' but nothing about division (or maybe I just dont remember). I still feel a tiny bit uncomfortable when I read h/2(a+b) but I get the sense (moreover there is in reality a context from which you can deduce the meaning). Actually math papers are written in some sort of LaTeX and during course it's written on a board, so I almost never see things like .../... Maybe someone who studies computer science sees it more often. Even though I guess there couldnt be ambiguity either because it has to be computed by the computer, but maybe when they write things unformally... Idk
I'm a writer, not a mathematician, but something I've come to believe is that one of the biggest disservices you can do someone is teaching them to follow arbitrary instructions rather than teaching them to think about what they're doing.
I've been staring at this 6/2(1+2) thing for like ten minutes and I cannot find a single way that you could possibly get 16 even ignoring the proper order. Can someone please explain to me how people have gone THAT wrong
Ok, so, I thought about this for a bit too long but... Multiply 6 and (1+2) to get 18, then, instead of dividing 18 by 2, subtract 2 from 18. Idk why someone would change the operation but that's the only way I've found to get 16
As a former Sped kid, with reg classes too, I always just thought I was stupid PEMDAs was hard for me to understand. I always felt more confused by it, actually your explanation makes more sense, I feel like I actually had to take less formula problem solving, felt more intuitive for me. Relying on Pemdas, is like relying on really bad made crutch. People will defend it though because that's what they were taught and they don't like being wrong.
@@universenerdd i'm assuming you're genuinely confused and not a jerk, so "sped" => special education. needed some special consideration for some aspect of learning.
This video is incredibly amusing. I've long ago abandoned PEMDAS and never put any extra thought into why and blindly accepted the abstract algebra axioms for multiplication and addition that include division and subtraction respectively as the same operations on inverses. So I never really considered how MD/AS are redundant. great video.
". . . you can't teach anything if it contradicts intuitions people have already built, and if you ignore this fact, people will learn nothing or worse." Exactly. If someone has the wrong idea about something, they need to be shown why their current understanding of the thing is wrong before trying to teach them concepts that require a correct understanding of the initial concept as prerequisite. Otherwise, those ideas will not and cannot make sense to them because they are built on premises that appear false in the perspective of the misinformed person who first needs to unlearn the lies they internalized. And if you try to teach them concepts that require prior understanding of the initial concept as prerequisite, they may only develop a warped interpretation of the other concept based on their misunderstanding or misinformation of the prerequisite concept that it is built on, giving them even more misinformation to unlearn before they are able to learn about these concepts correctly.
> _"If someone has the wrong idea about something, they need to be shown why their current understanding of the thing is wrong before trying to teach them concepts that require a correct understanding of the initial concept as prerequisite."_ This does not exclusively apply to mathematics. All the way down to social and political issues, which some like to treat like pure matters of opinion sometimes, if you can show someone that you understand his point of view and demonstrate to that person's own satisfaction why it's rooted in a mistake, then and only then can you genuinely hope to change that person's mind.
7:17 in vietnamese education, we have a saying "multiplication and division before, addition and subtration after", and parentheses is the almighty one. All of that was taught in primary school. Unary, exponent, roots, logarithm are just new dudes that got introduced later and we have to learn to use them without any catch phrase nor acronym. Despite that, the "6/2(1+2)" problem was still very much viral in our country. They never said anything about juxtaposition, it was told to be the shorter way to multiply. Later I just figured out how they works on my own intuition.
The question is "Why would I care"? The good thing to know about math is that it helps solving something that can be constructed in math language. Anyone, who think "6/2(1+2)" is a real deal to solve, are wrong IMO : it doesn't help solving anything, it's just you are guessing an answer for an improperly put question, what is a waste of time. Once you are not using that division operator in your expression, you're good to go without any ambiguity. So let's just say I agree with your summarize, and would be a bit happier, if people stopped falling for these "math" tricks
If you ever... and I mean ever say a+b/c+d, teach us PEMDAS explicitly as a tool and marks us wrong because they meant (a+b)/(c=d) is trying to make us fail math.
I just wanted to take a moment to thank you for making a video explaining what I've been trying to explain to people ever since these these viral math equations first started appearing. I hope lots of people see it. I'm disappointed that relatively few people have seen the videos on this topic from "The How and Why of Mathematics" while most of the popular videos on this question (one of which has been liked over 100K times) provide the wrong answer by over-relying on an incomplete or incorrect understanding of the order of operations. When I was in school, many students had difficulty remembering how to apply the PEMDAS. Some made the mistake of thinking that multiplication always came before division, but the bigger problem was that people would forget that multiplication by juxtaposition is an exception to the rule that multiplication and division should be done left to right. And now we have at least a couple of generations of people who either misremember what they learned or never learned it correctly in the first place, and some of them have become K-12 teachers who over-rely on PEMDAS when teaching the order of operations. As "How and Why" notes, the way calculators and spreadsheet programs require and interpret linear input has probably been a major contributor to people forgetting that multiplication by juxtaposition takes priority over "regular" multiplication or division.
Yes, hand held calculators becoming allowed in the classroom is a part of this that shouldn't be understated. But beyond that, juxtaposition is a from of shorthand for math. In the generally referenced equation, 6/2(1+2)=Y, the strict PEMDAS followers believe it should be written as 6/(2(1+2)=Y, and lend no credence to the FACT that 6/2*(1+2)=Y and 6/2(1+2)=Y are NOT the same equation. Due to the shorthand, no additional parenthesis or brackets are necessary if the solution sought is Y=1. The issue for many of them is they don't fully, or more at, erroneously, know how to resolve parenthesis. They believe that even when juxtaposition is used in expressing the equation, somehow, the parenthesis magically disappear after you solve what's inside of them. Which is true, if there is no juxtaposition. Just today I have a cat telling me how dumb I am and how I can't understand what he is saying. He even C&Pd a source, but he is so tied to PEMDAS that he can't read the very source he cites which in fact explains how the parenthesis are NOT resolved until the operator is distributed. Oddly, they won't even try to deny that 2(1+2)=2(3)=6...EXCEPT...They INSIST it becomes 2*3=6 as soon as they solve inside the parenthesis. It doesn't matter how logically you explain it to them.
@@andyfletcher3561 Please expand. Juxtaposition as an implicit 1st step was only ever used to tie a number to its unit, largely in science based classes. Other than that, it was never used as implicitly done first and was just regular multiplication in anything I ever took, up through Calc 2. Part of the problem, I think, is usually you start seeing equations using fraction format which implies parenthesis above and below the divisor line. To add to that, in typing, the / symbol is used as both a divisor line and as a replacement to ÷. This means an equation using / is ambiguous without parenthesis, and really should be defined before the equation.
@@JG-df1qd ''blah blah blah blah blah''. The only reason you know any multiplication is involved without an explicit sign is because it is attached to the parenthesis. The only form of the equation that makes any difference is whether or not there is an operations symbol. ALL of you claiming this is false are also claiming that 22=(2x2)=4. You just lack the logic capacity to understand it. It's no different than knowing 2f=2xf=?
I think it's easier when you understand that when you have parentheses, it means you need to simplify that expression first. That may mean distributing, combining exponents, performing inner addition where possible, etc. In the starting example, 6/2(1+2), it should be clear to distribute the 2 inside of the parentheses. Since, 1 + 2 are like expressions, you can simplify them to 3 first, or distribute the 2 and end up with (2+4) inside the parentheses: 6/2(1+2) = 6/(2+4) = 6/6 = 1. So, it would be incorrect to confuse the parentheses as meaning to multiply, although it's the correct operation as the distributive operator in this case. Also, coincidentally, if you write division as a fraction, you tend to perform division last when the expression is simple enough to do so, or if unable to simplify, leave the result as a fraction. It's also easier to apply algebraic rules to fractions than it is with a '/', obelus, or division symbol (the one with a dot above and below a horizontal bar). I think lack of understanding of these comments is why we see a lot of the simple math shitposts online where one generation acts as though they're smarter than the other, and both sides tend to perform the math incorrectly. It's a general lack of understanding of basic math and algebra, or poor teaching practices in my honest opinion.
That's where I was confused. I was taught pemdas and got 1 as the answer, because the parenthesis don't just vanish at convenience. Even if you did it solving the parenthesis first, it would be 6/2(3). There's still work to do with those parenthesis, like you said, distribute the 2. 6/(6). Ah, now the parenthesis is "complete". 6/6 = 1.
Just write clear expressions! For the life of me, I don’t understand why this is such an issue. Notation is meant to communicate mathematical ideas, so just write clear, concise equations and it won’t be a problem.
@@ellag3265 , yes, but parsing the expression from left to right is not the standard way of parsing it. If "clear" means "unambiguous in the absence of any precedence rules", then you can _never_ omit parentheses, meaning that even expressions like 1+2×3 (which equals 7) must be written as 1+(2×3). Is that really what you're advocating for? The point of precedence rules is to make unwieldy brackets unnecessary in the most common contexts. The trouble is that people apparently disgaree on which precedence rules to use.
@@ellag3265 brackets solve this issue entirely but form over function seems to be all anyone wants I say fucket brackets everywhere just to top it of remove / and use ÷ because fuck the moron that thought using 2 lines is acceptable instead of 1.
I was never taught any acronyms and didn't know it was a thing (schooled until pre-uni). My teacher taught the juxtaposition variant and explained it simply and cleanly, gave homework followed by a test - everyone in class understood easily. (( I couldn't figure out how it became a 9 without throwing the maths I've learned out the window first, almost gave me a panic attack lol ))
There's a moment in every programmer's life where they realize that different programming languages have different operator precedence. It shatters the idea of there being a universal order of operations like we learned in grade school. As an aside: I went to a bilingual school in Canada, and took math classes in both English and French. In French, we were taught BEDMAS, while in English we were taught PEMDAS. That certainly helped me understand the idea of equal precedence lol
Luckily for me the teacher used BEDMAS but she was a great teacher which meant she was competent enough to tell is that PEMDAS exists too. I miss her, she would make a good highschool teacher. Still pissed that she didn't know what a newt is though, but I'll give her the benefit of the doubt and just say that she probably only knew of a newt as an eute.
Not really... the calculation is always the same, the rules of the system just determine how you write your calculation... It's like how if I tried to do all my calculation in base 8 instead of base 10 I would get different answers, and of course I would... it's base 8.
I know its besides the point BUT DOUBLE MATH CLASSES??!! I feel bored within 5-10 minutes and suffering that thing twice in a day, oh ugh dont wanna imagine
Nice video, but you really monologue a lot and beat around the bush. The main idea is that asking the order is meaningless, and that this would be obvious if students just learned operations from the ground up instead of internalizing that they need to memorize random details which are actually unimportant.
It's embarrassing to see people call each other "Bad at math." over not understanding this. People seeing 6 divided by 2(1 + 2) as 6/2(1 + 2) makes sense if they were taught that way. The opposite of people not seeing it that way is also fair if that's what they have worked with in the past. It's literally just different rules. There have been math Ph.D.s who "Got it wrong." because they interpreted it a different way. You wanna claim the guys who've done the hardest math possible are bad at math? Hell, I managed to come up with valid proofs for "Prove that you can factor every real polynomial into only real quadratic and real linear factors." according to my prof and did "Very well." with trying to prove the Fundamental Theorem Of Algebra without looking anything up or getting help (Independent.) as a first-year math undergrad [I had to come up with so many proofs to do it, as I didn't have the tools given to me.], but I "Got it wrong." the first time --> I am "Bad at math." because I didn't know about the other way of doing it. No, I'm not bad at math. I just looked at it from a different perspective, and from that perspective, I was right. The difference simply depends on how you perceive it --> The rules you were taught --> The order of operations/symbols.
It seems to me (from very limited experience) that division causes most of the problems. However, division is usually written, when possible, not as a normal operator, but as an unambiguous ... what is that called? Big-ass fraction. And when it *is* written as a normal operator, it's either explicitly bracketed, because what you really have in mind is the nice looking fraction you can't write, or it's being written in a programming language where you know *precisely* what the rules are. Otherwise, it's being written as a test question, where it's meant to be deceptive. In the context of math papers where something is written nicely like 1 "over" 2 root(2, N), then later given as "1 / 2root(2, N)", I would interpret that as a (relatively) unimportant error, given that they've already written precisely what they mean, and are now loosely referring to it again. If, however, the paper declared that it would use some other convention that would strictly bracket the expression as "1 / (2 * root(2, N))", then it's certainly no error, and entirely valid. I was explicitly taught in school that -4^2 is -16, which is easily justified in the same manner as (^) associating to the right is justified, and to shove that into PEMDAS, I just say that negation is treated like subtraction, so it comes after exponentiation. All of this said, I am no die-hard fan of whatever convention we may have. I've been messing around with reverse polish notation (that is, `2 3 + 5 *` => `5 5 *` => `25`) and functional languages, and how they should interact.
It's kinda interesting how maths is taught is similarly yet different in different countries. Here in India BODMAS is taught.(Bracket , Of , Division , Multiplication, Addition , Subtraction). Great video mate
We were always taught it as BODMAS rule. It goes in this order: B = Brackets O = off/ exponents( off basically means if there is a multiply straight after a bracket we have to do it first) D = Division M = multiply A = addition S = substraction If there are two same operation then we group them together and put brackets. So 2+2-6÷4÷2 will be 2+2-(6÷(4÷2)). For the question given above: 6÷2(2+1) 6÷2×3 For we apply off 6÷6 1 If you're confused why this is look at it this way 6/(2(3)) 6/(2×3) 6/6 1
My teacher did PEMDAS, but made it explicitly known that multiplication/division and addition/subtraction were equal, that if one appeared before the other that it was done before. I still don’t know why it’s called the “Order” of Operations, because the order only really happens in like, two areas. It’s much closer to a hierarchy.
I also was taught that multiplication and division are equal, but now that I think about it, you can combine + and - any way you want in expressions such as a-b+c-d-e, but for division, one should avoid something like a/b/c.
Order can mean hierarchy. For example the order of battle in a military context is the hierarchy of command ie who is the superior to who. Your being pedantic for the sake of it.
@@michaelgoldsmith9359no, he literally isn’t, because that isn’t what most people consider it to mean (except those working in the military who are specifically taught about the meaning) so in this context it is not a useful definition to use.
GEMA is actually such an amazing tool. It also teaches the fact that division is just multiplication by a fraction and how subtraction is just addition by a negative number. Great video!
Honestly, applying GEMA would require a massive education overhaul, while just bettering PEMDAS is much easier. Yes, subtraction is addition with negative numbers, but that concept is much easier to grasp when one understands that “negative” in addition means take away, which circles back to just calling it subtraction. Or division being multiplication by a fraction. Why would someone write out: 5/1 * 1/3 when they can write: 5/3? We don’t need to overhaul PEMDAS, just teach how to interpret the relationships signs create between numbers.
I never had a problem with order of operations despite never learning pemdas (I probably wasn’t paying attention when they taught it in school. I don’t think I ever paid attention with they taught math.) I also remember almost immediately grasping the idea that subtraction and addition are the same type of operation, as are multiplication and division. So I think I learned GEMA without actually knowing it. Then again, my parents grew up in another country and they were the ones who taught me basic math. They also straight up told most of my teachers in elementary school that the way they taught math was totally incomprehensible to them. The problem lies entirely with how math is taught in the USA. It needs a massive education overhaul.
@@default3623 I think the "division is multiplication by a fraction" thing helps with getting this way of thinking and reading expressions (not just always taking symbols as symbols between numbers) that might help later with polynomials, for example. But for small kids, yes, you better say that substraction is taking away (is this wrong? It is the same wording) and division is how much of one number the other number contains (which is much better than "division is when you divide" or, imo, "splitting in parts").
Except a fraction is just division again. So you are saying that dividing by X is the same as multiplying by 1/X, but that still has division in it so realy what you are saying makes no sense.
I also need to point out... Mathematicians, enginerrs and physicists rarely, if never, use / or ÷. Theh can lead to confussion and it's just better practice to write fractions
I took an equivalent to 8th grade Math in Mexico, the concept there is simply called hierarchy of operations. Later when we moved to the US and my sister needed help with 8th grade math I was almost peeling the skin off my face as I was looking at her not being able to do any problems without writing PEMDAS in her cookie cutter Handouts.
The main problem is that there are still some infurriating people out there that write "7 × 5 ÷ 11 - 8" without knowing the nightmare they've just created Use brackets ya lazy nut!!
@@pedrosso0 U right, I tried writing one that tends to end in a mess, but just noticed it works fine hahahahha "7 ÷ 3 ÷ 5 × 8 ÷ 2 × 3" Would be more problematic I guess... But yeah, it mostly comes down to people not puting brackets when writibg in a single line (so in the original, someone may have wanted to write "(7 × 5) ÷ (11 - 8)" but didn't want to, stuff like that)
This still isn’t a problem. When the symbols are written out like that, it’s very clear that you should evaluate it from left to right. If that was not the intention, then they are just wrong, but there’s no ambiguity in the actual message.
@@airiquelmeleroy 7 ÷ 3 ÷ 5 × 8 ÷ 2 × 8 = 5.6 right? If so, then the writing is a non issue for most people from what I've seen. There's no confusion due to the lack of brackets, alternate symbols, etc. So you just go left to right. Now if I'm supposed to read it as something like (7÷3)÷(5×8)÷(2×8) well then shit... 😅
the biggest issue I encounter with PEMDAS is people think that that's the order. The acronym on its own implies that's the order, multiplication then division, and if you weren't paying that much attention in elementary math, you might get that answer. But even if you did, some people outright were taught that. And in higher level math it doesn't come up for... Well a few reasons. Primarily because you aren't doing simple problems but rather solving for variables or taking derivatives and the like. But also because, simply, there's no ambiguity. everything is clearly notated with parenthesis whenever there could possibly be any confusion. This can result in comically large numbers of parenthesis, but it gets the job done, and there's no ambiguity what you mean, no hope for misinterpretation
this entire video is essentially; "PEMDAS is bad because if you write an equation to be as ambiguous as possible, it becomes ambiguous." which, btw, has nothing to do with PEMDAS, so this entire video is pointless and stupid.
Americans teach bike riding that way. They consider it laziness if you don't just get on it and figure it out, and if a child falls and doesn't want to try again, very often their parents or older siblings will tell them they're being cowardly. Actually, they teach math that way too. It has led to many children being much worse at math than they could be, into adulthood, because their teachers expect it to come naturally to them. Whereas a good educator in a sane system would actually... teach the children math. I learned mental arithmetic from the ground up as an adult, AFTER graduating college.
As a boy who aspires to be an astrophysicist one day, and with a love for math, I can’t wait to learn the actual order of operations in adulthood, maybe even now.
Went to college for an engineering degree and I was way too slow in math. I really busted my ass but I would never finish the tests. I took a break from school and my girlfriend that teaches preschoolers saw me do basic addition and she told me "wtf did you just do"? She told me I was using touch points and that really messes you up in the long run. It makes you a lot slower and hinders your ability to grasp basic concepts of numbers. I told her I never learned any other way. She showed me a more proper method teachers were supposed to teach and it's really difficult learning the basics again. Glad I'm getting videos like this in my feed now.
Wow. Just looked up "touch points". It seems like an unnecessary addition of a layer of complexity that would hinder the learning of both numbers and arithmetical operations.
Wow, no wonder I'm still crap at math, I grew heavily dependent on this method, I just listened and memorized the solving methods without questioning or comprehending why it occurs to begin with.
The reason I got 6/2(1+2) wrong is not because of PEMDAS or anything like that, it is because we've done things like 2(x+2) a lot in school, so when my brain sees a number before a bracket, it just wants to do 2x+2*2. Going back to the original math problem, it gave me that: 6/2(1+2) =6/(2*1+2*2) =6/6=1
And it’s proper answer. It’s 1. You did it right my friend. It’s 6/2(1+2) = 6/2(3) = 6/6 = 1 it’s correct and proper way of doing this since this 2 you see after „ / „ is also a part of parenthesis and need to be evaluated before dividing. If you write this equation as fraction it’s way easier to see it’s 6 / 2(2+1), and now you see that if you want to evaluate this division, you need first to evaluate everything after division sign ✌🏻
@admiralvirhz Ehh not necessarily. These kinds of problems are just created to spark debate due to the fact that it can be interpreted in multiple ways. If someone gives you division with the symbol instead of in the form of fractions, just tell them to fuck off lmao.
"6/2(1+2) =6/(2*1+2*2) =6/6=1" Sorry, but this is incorrect. If you are going to distribute, then the division is the same as a fraction, and you have to distribute the ENTIRE 6/2 (Six halves) 6/2(1+2) (6/2*1+6/2*2) (3*1+6/2*2) (3+6/2*2) (3+3*2) (3+6) 9 Think about it this way... x(1+2) Now let x = 2/3 (Two thirds) 2/3(1+3) If you only distribute the 3, you are going to get the wrong answer. So 2/3*1+2/3*3 2/3+2/3*3 2/3+6/3 8/3 Simplified 2 and 2/3
You really want to die on the hill of not understanding what a convention is, and that it's not a divine law. Conventions depends on context and evolve over time. But sure suit yourself and go on answering every comment over this artificial little problem.
Something I noticed is that people get multiplication by juxtaposition, but only for stuff taught in and after algebra. If you replace (1+2) with a variable, say 6/2x, and x=3, almost everyone I talked too gets 1. Once you start using parenthesis along with variables, say 6/2(x+2), and x=1, some people start getting 9. When you go back to the original 6/2(1+2), now a majority of people get 9. It is taught in school, but not "untaught" for math leaned before those classes.
Using the example of 6/2x with x=3, than the proper answer would be 9. The long version is 6/2(2+1). If broken down by some, to your example of 6/2x with x=3, than the equation would look like 6/2*3. The order for which requires the 6/2, now takes presidence over the 2*3. So, 6/2=3 which than becomes 3*3=9.
@@Technotranceism I think the point was that 6/2x was written using juxtaposition, so when you replace x with 3 it should be interpreted like 6/2(3) and in this case the 2(3) has higher precedence. It would be 9 if it was written like 6/2*x.
@@clomiancalcifer yep, that's always the best way to handle stuff like this imo. It's not that hard to just add extra parentheses lol. I was just going by what the video said
I wouldn’t get 6/2x=1 and neither would a computer for that matter. Unless specified by brackets, the / sign only applies to the thing immediately thereafter. Use fractions or brackets but don’t pretend it’s a good notation to say that 6/2x and 6/2*x are two different things
Well there's simply a distinction between ÷ and /. The / is essentially implies a long bar, which itself implies brackets around numerator and denominator. The ÷ on the other hand has no such implication and just operates linearly.
The funny thing is that in my country, Hungary, they never teach acronyms. They just tell you "This is how you do it" and then that's how you do it, simple as that. That's why I can't get how can people be so hopeless in mathematics. I'm not saying there are no people hopeless in math here, but it's not at the level that you see in the US and similar countries... Compared to those people, the people that are hopeless here are would be math gods in the US, that's the saddest thing...
Haha, no. Math education sucks in Hungary too, and more people hate it than any other subject. Also, teaching methodology is pretty bad. I heard teachers said stuff like "it needs to be memorized, not understood".
@@HA7DN Personally, I've always had good teacher, but even I've heard about teachers that make it miserable for everyone, plus, the only real thing I know about other countries and their education is what I see and know from the internet, so you may be right about that. All of this is really should be a debate for those who have experience in the field across countries, not a random like me
@@HA7DN Axioms fall in that category of "It's that way, because we decided it was". The real explanation is demonstrating that it's part of a set from which you can't add or remove an element without breaking it or being able to use one or more axiom to prove another one. Even in first year of university, it's not feasible. You can still explain it like the above though. The good thing is that there aren't that much axioms and most of them can be gently brushed over. As an example "1 is the successor of 0", well you don't have to really explain it mathematically. Students know counting and that's enough. But (yes, always a "but"), at some point it becomes required to teach the internals of our positional system and what is a basis etc. As an example, most adults can't add in other radii than 10. That can be show with the following problem: A tribe is using a positional system akin to our base 10 system. Instead of 10 symbols, they only use 3 symbols. Let's denote them 0,1 and 2. Complete the following addition in their system : 12 + 22. Answer is : 12 + 22 = 12 + (21 + 1) = (12 + 1) + 21 = 20 + 21 = 100 + 11 = 111 Note how only predecessor and successor got used in the calculations. It can also be done by using the addition table where 2+2=11. So, we get: 12 + 22 = 10+20+ 2 +2 = 10 + 20 + 10 + 1 = 20 + 20 + 1 = 110 + 1 = 111 (All steps are done using a lookup[ table without even giving any meaning to the symbols) That's why math are powerful. You start with something very concrete, abstract it, manipulate the abstraction with well defined rules, get a result and interpret it. People think they start algebra after primary school. Nope, they learn the foundations in primary school and pupils are even asked to write "let A stand for Alice". They also complete arithmetic expressions with empty boxes like: 2 × □ = 10. It's kept very informal (and even wordy), because most pupils do not have yet a developed brain. Then in secondary school, we can start being more abstract. In upper secondary school, abstraction becomes the norm. (manipulation of summations, including integrals and some basic group theory) So, yes, we can't always explain everything and we shouldn't. At each stage of development it's part of mathematics.
Literally pemdas but okay. All syntax is lazy use proper notation or use brackets everywhere the only solution and stop trying to resell pemdas but better without changing a damn thing.
@@snintendog its not pemdas though? With GEMA youre saying that Multiplication has the same priority as Division and Addition has the same priority as Subtraction.
I was taught BODMAS at school, but they only really kept emphasis on it in the 4th and 5th grades, and they weren't serious about it either. The only thing they made us remember is that brackets come first, and after that, there wasn't really much on it. Now, the acronym isn't even uttered in high school, many people just... forget about it, because we remember what to do first like the back of our hand, which is to do brackets first, rest can come later. I only started to remember about BODMAS again due to people arguing over these problems, and that was the first time I've even though of it for over 5 years
I think the main reason why people start to forget BODMAS (I was also taught that) is because maths in higher classes has more emphasis on algebra and geometry, which don't really use BODMAS in their purest form.
in the 6/2(1+2) case, i always thought that the 2(...) was part of the parenthetic block, ie. distribute 2 to(1+2) before dividing 6 by the result of that section.
@@pedrosso0 Under which convention? Under the juxtaposition convention he proposed -- where implied multiplication has higher precedence than explicit multiplication -- yes. But under the PEMDAS convention -- where implicit multiplication is no different from explicit multiplication, and multiplication and division are left-associative and have the same precedence -- no. Without context, there is *no* correct answer; it is simply a malformed question. Edit: Apparently RUclips's markdown uses "--" as strike-through. Bruh.
Yeah when I saw it I figured the 2 and 1+2 are part of the same "unit" so to speak because of the space between the 6, the /, and the 2, if that isn't the case then it needs to be made more clear, like by ditching the spaces or making the 2 and the parenthesis part completely separate (so it becomes 6/2(1 + 2) or 6/2 * (1+2) instead of 6 / 2(1 + 2) since that makes it at least a little more clear that the 6 and 2 are part of their own fraction instead of the 2 being part of the parentheses instead)
Distribution doesn't take precedence over any particular operator, though. One could equally argue that the term to be distributed is 6/2, since that is what is outside the bracket.
@@chrishall9107 yes that's why the question needs to be written in a less ambiguous way so you can better deduce which operation is supposed to happen where
I always assumed the confusion was because people mistook the number connected to (multiplying with) the brackets as part of the bracket part of order of operations. my belief was that their first goal was to solve and remove the brackets completely. that's why i never liked writing 2(1+3) and would instead write 2*(1+3). because once the brackets is solved the equation is 2*4 instead of 2(4)
Thats because it is. The number connected to the brackets is part of the brackets, otherwise you couldnt factor stuff out of the brackets, and a lot of algebra breaks.
Scientists & engineers consider 2(1+3) as a single 'term' so resolving it completely [the P in PEMDAS] has priority in the order of operations. 2(3) = 6 Et voila! the brackets are gone, and now one can go about 'ordinary' multiplication, division, addition and subtraction. The problem is that not even the teachers understand the nuances of it, yet they're the ones historically who developed PEDMAS as a tool. And the programers of some (if not most) calculators have changed software to meet educator requirements, so now different calculators yield different results! The system is broken.
The problem isn't really the order of operations, but rather bad notation used to show the operations to be done. If it is possible to be confused about how to simplify an expression, then the expression should be re-written.
In Singapore, I didn't recall learning about PEMDAS in school. I've always been taught to calculate equations using juxtapositions ever since we were introduced to the idea of brackets in maths.
The last line I fully agree with. I don't really see a problem with PEMDAS if it's taught on a rule of thumb basis. Math is language, the reason most of us are "bad" at it is because we don't get exposed to its notations later than our spoken and written languages. When we do pre-algebra and higher levels of math, we are parsing out how terms are related. If PEMDAS is taught properly, some intuition has already been built, and it's presented as an incomplete guide. That said, PEMDAS usually works until it is weaponized. The memes where people argue happen because somebody has intentionally written an expression with ambiguous relationships as a gotcha game. Your average person with a high school math education, and maybe a couple of semesters of university math isn't getting tripped up as they go about life. When they need mathematical reasoning, they are usually relating concrete things that they've developed some intuition about separate from abstract math. So I don't strongly endorse PEMDAS, I just feel that the hate it gets is a bit manufactured.
Thank you for pointing out why a^b^c always means a^(b^c) and why (a^b)^c is nonsensical as we might as well write a^(bc). I frequently point out this to students. Also I don't even understand why they claim parenthesis is an operation. I mean it's just a language syntax to inform a change of order of operations. It's not an operation in itself. When do you actually perform the parenthesis operation? Never :) Regarding typing a formula outside of LaTeX or similar environment or whiteboards: I really strive to avoid writing so you have to assume juxtaposition priority above a previous division as although you can assume the reader will realize, it is still somewhat sloppy. It's easy to write stuff like 1/npq and assume the reader understand we mean 1/(npq) but the parenthesis really doesn't hurt and sends out a message that the writer genuinely care that the reader don't misinterpret things. In LaTeX I typically use the $\frac{1}{npq}$ even mid-text as it will not be too small as long as the fraction is not so complicated. If it's complicated you should write it on a separate or refer to an equation IMO. I think one of the problems is that the teaching is mostly focused on how we ideally should write mathematics. But students and many teachers misinterpret it as "How you should always READ mathematics no matter what". Another massive issue with PEMDAS is that it's beyond stupid to expect people to scan through an expression four times! Students will soon develop a intuitive feeling that PEMDAS is remarkably inefficient and get incentives to try and find faster ways to understand an advanced expression. But what can I do and what can't I do? Am I somewhat of a criminal if I find my own way of doing it? It almost seems like it in some educational system. It seems to mostly be a problem in USA to be honest. In Sweden we don't have this problem nearly to the same extent. I typically teach students how to evaluate an expression by just doing a quick scan to get a sense what it's about, then start wherever they feel like. Left to right might often be a good idea if it feels right. I mean consider a+b+c*d+Narnia (we all know it's a bit unclear what's inside Narnia ;). Of course you can begin by adding a+b=e if you like and multiplying c*d=f and even do e+f=g if you feel like it and then wander into Narnia and see what interesting adventure lies ahead. No matter what awaits us in Narnia all the operations we just did will still be useable. PEMDAS will teach student to immediately wander into Narnia in an epic quest for finding parenthesis. Then leave the wardrobe, go for another adventure in search of the holy exponents (which they will interpret wrong in case of p^q^r). Then a third quest where they do some multiplication and division. Then after levelling up to a level 12 Paladin they finally perform that a+b they could have done the first time they saw it, not the fourth! Also if Narnia now turned out to be 7*3+(10+5) they are literally taught to do 10+5 as the first operation even though it can be executed as the last operation of all! Unbelievable! What's next? VANO for reading a novel? Reading it four time evaluating Verbs then Adjectives then Nouns then Others?
Physics and maths lecturer here. 100% with you on the 'write your equations more clearly', and yes, the obelus must die. A mathematician or physicist would likely never use it in hand-written notes.
It’s interesting how we never even touched the general rules. The teacher sometimes just said that a certain operation comes before another and that was sometimes explained in a contradictory way. Now that I’m in high school, the order of operations I was taught is constantly failing me.
I was completely in agreement with you up until the end of the video, when your final arguments seemed to be completely unrelated to the premise. The statement that there is no order of operations is correct in that you can form a mathematical expression in which operations are executed in any selected order. But an order of operations does not prescribe the order in which all operations must be executed, rather an order of operations is only meaningful relative to a given notation. Without a notation, it makes no sense to talk about any rules governing the order of operations. Not all notations require an order of operations. For example, prefix and post-fix notations contain no ambiguity in their representations, but they are very hard for a human to read. Our standard "infix" notation has potential ambiguities which must be resolved. The only way to resolve this is by either forcing everyone to explicitly write brackets to remove all ambiguity, or to define a commonly understood method of resolving ambiguous situations. As you mentioned, simply hoping everyone will be good and write things clearly is not reasonable, so the only reasonable solution left without changing notation entirely is to define an expected order of operations. By simply saying that we should teach the relationships between operations and that we should simply teach math does not resolve the ambiguity issue of our notation. As long as our notation has the potential to contain ambiguities, some method of resolving those ambiguities must exist, and therefore must be taught. If I am misunderstanding your final point, please help me to understand it better. This has been a great video and a fantastic explanation of the problem with order of operations, so I would love to hear your thoughts on this final misunderstanding I'm having.
Thank you for the well thought out critique, you're right I didn't properly get across my final point. I agree that "simply teaching the operations" is not enough to resolve ambiquities in the notation, it is important to consider. What I was trying to get across was that the order of operations is fundamentally just about notation and isn't mathematical in an of itself. One of the biggest problems with PEMDAS, in my opinion, is the way it's tought to students, especially those who aren't mathematicelly inclined in the first place, gives the impression that systems like PEMDAS are in some form "correct" and not just a convention we've decided upon. This can severly hinder their ability to do math since they rely on arbitrary rules, rather than an understanding of the logic behind the operations themselves. That's what I was trying to say by "just teach the order of operations", that students would be much better equipped to do math if they learned the operations seperatly from the symbols and notation we use to represent them. I could probably have gotten this point across more eliquintly. Also I did actually plan to talk about prefix and postfix notation by highlighting Reverse Polish Notation in the "solutions" section, but I ended up cutting it for time constraints.
@@interrogatix I see, so you're talking about making a more obvious separation between actual mathematical operations and concepts versus just the act of notating such things? That is clearer, thanks! I believe that is actually done in some education systems such as Montessori schools. They teach the operations and concepts first with physical tools and then only later show the notation as things get more complicated. This builds more intuition about the operations first. Sounds like maybe what you're advocating for?
Great video! I've been waiting for someone to make a video like this. I've been wanting to say the same. Order of operations does not exist in maths itself, in the sense that it's just a notation. When you follow the common notation, the idea of evaluating multiplication and division left-to-right is ridiculous, multiplication and division are the same thing! I really like how you explained this using GEMA at 7:36
As a maths teacher we have banned the word “bidmas” in our maths department, teaching the correct order and equality of operations. However, I understand that students and most people need a method to help remember certain rules, as there is so much we require them to remember (from a syllabus we are given).
banning the word "bidmas" is kind of ridiculous. Why would it be banned? It is just an acronym to help remember the order of operations. It isn't confusing if taught properly.
@@tchevrierwhy would you bother with a dumb acronym when you can just explain the rules? It's not like order of operations is particularly complicated
Okay wow, I was not expecting this video to get this much traction. I was just kinda happy that a few people on the 3Blue1Brown discord server liked it and then just kinda left it at that, thinking this was a fun little project to flex my video making skills and not much more. But we're now at almost 3000 views (Edit: scratch that, I'm absolutely floored right now) and still growing and I'm not really sure how to think about this. I wanna say something corny like "thank you so much, it means alot to me, etc." but that feels weird while this is still somewhat surreal to me. I wasn't planning on making anything more for the channel in the near future so don't expect anymore videos soon, but I will definitely make more at some point.
So in order to not dupe myself, instead I'm just gonna ask, what did you think of the video? Particularly if there were any specific parts which especially grabbed your attention or that you didn't think worked so well. These two points, especially the latter, are in my experience the best kinds of criticism you can get.
So yeah... thank you... I guess?
Also on an only slightly unrelated tangent. I think Henry (Minutephysics) has a problem with communicating his intent because he's made the exact same mistake again in his most recent video on The Butterfly Effect:
ruclips.net/video/vLIPUdru82c/видео.html
He once again tries to explain how something can lead to misinterpretation, explains the problem and then people retort with "But that's not how I interpret it".
I previously thought he must've misunderstood how PEMDAS works which left his argument open to be rebutted but now I'm convinced he's just really bad at clarifying what he's even trying to argue. Commentors think he's arguing that The Butterfly Effect is bad based on his own faulty interpretation, completely missing that the fact that the butterfly effect can so easily lead to misinterpretation in the first place is the real problem with it. EXACTLY the same thing that happened with the PEMDAS video.
Tangent over. Also I now hate the word "misinterpretation", it's annoying to type.
man I got worse at math since I was put in spreal needs and they gave me year 1 level work (LEGIT we all got the same I was the eldest everyone else was year 1)
Video was awesome bro👌🏼👍😀
Bro thanks for good informational video
now get a gooder mic
@@BlackholeSteve so true. I got better at math by doing olympiad problems etc. and when put at university I felt like loosing a bit. It's very important to keep up a work that is hard
We had a C programming lab session for us this semester, and one of the questions was to evaluate a expression. It was not like a function or something to do with variables, it was just a math expression. All we had to do was copy the thing and equate it to a variable, and print it. And the answer we got (the actually mathematically correct answer which the computer and the calculator gave) was rejected by the teacher because it didn't follow the BODMAS rule... It wasn't anything ambiguous at all, she genuinely thought that division had higher precedence compared to multiplication. We ended up deliberately adding brackets to get the "correct" answer. This is so sad.
Damn I feel you as a computer student
I can see how people would think division has _lower_ precedence than multiplication (all the factors to the /‘s left go in the numerator, and the ones to its right go in the denominator). But this is pretty bad. Stubborn teachers are the worst.
if I was you i’d make sure she knows it’s correct I’d want nuclear revenge
Clearly this teacher hasn’t discovered the order of precedence in C... then how are they a teacher?
@@nathanoher4865 Hate to break it to you, but most programming enthusiasts actually know more than the majority of teachers out there...
I didn't realize how lucky I was to have my teacher just explain the rules to me outright instead of bothering with an acronym
i never knew the acronym exist until now.. 🤣🤣
the acronym just memorize the order, you just have to know the rule about how the md and as is the same
same
same lol
i learned this in third grade and i think we learned the acronym after
i wish i was told the same thing
I always overuse parenthesis just to make sure there’s no possible ambiguity
Just write every fraction as a fraction. I feel never using the division symbol eliminates just about all those problems.
@@beagelerironically I always had a hard time reading an equation with fraction as apposed to single line
But fractions were never my strong nor were irrational numbers
@@beageler 100%
@@jjdoughboy2103 I could never really get irrational numbers, too. One of the areas where I have to strictly work with formulary. Never had a problem with fractions, but I still think most order of operations problems would go away by not writing formulas inline.
The irrational number thing is curious, never had problems with differential equations, even partial differential equations are easier to think through than irrational numbers. Also never had problems with calculus or discretisation and numeric solution methods.
Something I got even less than irrational numbers are distributions, but I'm an engineer and should never have tried to understand that. A step response is a step response, who cares what the mathematical basis is :-)
The basic problem here is that it's not an *order* of operations, it's a *hierarchy* of operations. Syntax, both in language and in math, creates structural trees out of linear input. Multiplication doesn't precede addition, rather, they're more closely "glued together" in the structural tree.
I wrote an article about this for the Mathematics Teacher journal that was published in 2017, but I think it's unavailable if you're not a member of the NCTM. It doesn't address the problem of multiplication by juxtaposition, but it does argue that PEMDAS isn't a useful approach, and suggests that students be taught to seek out terms and factors first, and then the terms and factors within those, and so on. This duplicates the hierarchy, rather than suggesting that it should be a linear order.
Well, the issue is mainly the infix notation that everyone is used to and that's the one that is ambiguous or need some kind of order or brackets to make clear which order was meant. When you use pre- or post- fix notation you don't need brackets at all and it's completely unambiguous. Though for humans it's a bit difficult to read and interpret pre or post fix, though I think it may be just be a matter of training. Though infix has the advantage that the operators act automatically as seperators between operands. Because if you write 5+6 in postfix it would be 5 6 +. I used spaces to seperate the operands, however it becomes tricky for humans to write or read such a notation. Easier to read and understand may be prefix or function notation since you could do sum(5, 6)
The example from the video 6/2(1+2) == 9 would be this in prefix
mul(div(6,2),sum(1,2)) == 9
or this:
*/6 2+1 2 == 9
In post fix it would be something like this:
6 2/1 2+* == 9
With pre and post fix you explicitly have to encode the order you want in the notation. It's completely unambiguous as you would use a stack to evaluate the notation. You could also write it as
1 2+6 2 /* == 9
The only issue we have are things like negative numbers :) So we would need either a different symbol for a negative number or a different symbol for subtraction. An alternative would be to be more consistent with the spacing. So always insert a space between every operator and every operand. Then we could use the minus for both. Though it would be easy to mistake it.
I trust a man with a pie for his pfp
@@nahyankamale9989 This is the kind of thinking that led to PEMDAS, namely, focusing on the appearance than the content of the subject, in this case, mathematics. Despite this comment being a joke, I wanted to share this observation.
It's basically the same in logic, with the different operators
@@Bunny99s Just represent negative numbers with a subexpression that multiplies by i squared. 😆
My take as a programmer is it's parenthesis first and if it's still ambiguous, you haven't used enough parenthesis. Relying on order of operations which differ slightly in almost all programming languages (because a language syntax will end up having one even if you don't think about it because _something_ has to be evaluated first); is madness!
Eh... I don't like cluttering my code with extra brackets, so I'll either remember the operator precedence, or look it up. I guess my code is liable to break if the next version of the language changes the operator precedence.
Same
@@takatamiyagawa5688 what kind of expressions/equations are you writing friend? It's not like you're going to be waking up next to your code and be like, “Good morning beautiful”
Order of (arithmetic) operations is the same everywhere as far as programming goes. The difference is what a(b) is - a term? a highest-order operation? a multiplication operation equivalent to a * b, so therefore a / b (c) is equivalent to (a/b)(c).
To all programming languages, a(b) is an attempted call of the function a, so it's not a problem because this situation is not possible.
@@takatamiyagawa5688 use a stack to convert your cluttered expressions to postfix/prefix
This is why I use the heck out of parenthesis. I’m also a programmer, so using brackets becomes useful, or everything turns into a mess.
Oh so you're a LISP programmer.
there is a bit of a difference between programming mathematical equations and writing them on paper. When I write equations on paper I use the fewest symbols and notations possible. However, when writing the same equations for a program I use plenty of brackets also. Mostly because you can't write it the same way as on paper.
I'm a programmer too. But you also need to be able to interpret written equations correctly, without relying on the author having used spurious parentheses.
Once the author has written the equation according to universally-accepted rules, it's up to the reader to be able to evaluate it correctly.
Too many can't and preach PEMDAS when its not universally applicable.
At the same time, other people don't. So, it is important to have a system when reading other people's stuff.
@@yan-amar I came here to tell people how LISP can fix the issues listed in the video (:
A big problem for learning is the oversimplification of subjects. I think "PEMDAS" is a great example of how making things "easier" can really mess you up when it gets more complicated.
Yess, exactly! I guess it's not so bad for people that will not continue to study the subject, but I can't even remember how many times I had to re-learn things during my education because someone couldn't bother saying that something is just a simplification
@@danielblumowski34 Ikr I wish they'd atleast admit it's a simplification. In chemistry we learned that the innermost electron shell always has 2 electrons and after that it's always 8 per shell and there's no more than 4 shells. That confused me because a lot of elements have more than 26 electrons and my teacher refused to explain it because it's a higher level. I had to ask my older brother to explain it and then I actually understood
@@redshift739 I think it depends on the country because I was told something different about electron shells but still it confused me a lot. Then I learned about orbitals and it brought even more confusion because the way they are usually tought is wrong too. I only "understood" the topic when I started finding information on my own. It's so frustrating. Is saying that something is not entirely true or that it lacks mathematical rigour so difficult? I guess many teachers don't even know that what they're teaching is simplified.
It's ok to simplify as long as you make it clear that is what you are doing. Sometimes, simplification and partial explanations are necessary parts of the learning process. Throwing someone in the deep-end to sink or swim is not a good teaching method.
I have serious arguments with my math teacher when he teaches blatant lies _(some examples of his stellar teaching: imaginary numbers don't exist because you actually can't take the square root of a negative, but we are going to pretend we can for this one single lesson, and go back to the REAL numbers that are actually useful, and also 0.99999... doesn't equal 1 because there's always a small gap silly, same with all those other weird infinite sums, we're just rounding.)_ and I tell him this is really going to bite everyone in the ass when we get to calculus or any sort of higher level math at all.
Mathematicians, like writers, all too often need to understand that they're responsible for communicating their ideas clearly. If a sentence is unclear you don't blame the language, or blame the reader, you rewrite it.
"If a sentence is unclear you don't blame the language" is only true if we're talking about natural languages. If however a language is engineered for a practical purpose, you absolutely can and should criticise it for its ability to fulfil that purpose.
@@interrogatix I love how this discussion is being made in the incredibly impractical language that is english lmao. And yeah I agree, math already has a perfect easy to understand language in the form of PEMDAS (or PEDS/PEMA), people slacking by taking out parenthesis where they should be and making up their own math rules among other things just makes it hard to tell what they mean.
@@interrogatix I mean, it's just a matter of practicality. Mathematical common practice is a convention based on what mathematicians the world over have settled on over the years. Any attempt to change it is going to be almost as difficult as changing a natural language.
So I don't think it's outrageous to say people should probably focus on being more clear with the conventions they have, in the short term.
Mathematical notation can absolutely be criticized but at certain points you have to ask yourself what good it's doing. It's the same with english; I can spend days drafting spelling reform and punctuation reform for written english, but in real life it's still going to be written the way it's always been written.
@@interrogatixI'm not sure who told you that math is an engineered language when it's not. It's just another language that evolved naturally and happens to have a governing body like French
@@ssgoko88 mathematicians literally made up the notations over time and usually they stick with the one that the discoverers use in their papers
Being clear, unambiguous is the most important. I never hesitate to add a redundant parenthesis, either in papers or in programming. Also, in a lot of cases, units force the order of operations.
If you don't use fractions in papers, you're doing it wrong. And needed brackets in programming are by definition not redundant.
@@beageler On the last page, lower left of: P. Welch, "The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms," IEEE Transactions on audio and electroacoustics, vol. 15, no. 2, pp. 70-73, 1967., you can see an example of unnecessary parenthesizes that add clarity.
It is a classical paper with more than 12000 citations.
And, in programmation, I was talking about parenthesis that doesn't alter the order of operation, such as (A+B)+(C+D). Sometime, they are useful, again for clarity. That was my point. Perhaps my comment lacked some additional clarity, but, there is a limit to the effort I give to a simple subject as this one.
@@minimumlikelihood6552 Fair enough, I didn't think much about my comment. Kudos for the effort in your answer :-)
WOW! Someone who changes his mind! Very rare on youtube! You deserve much respect!
@@minimumlikelihood6552 Life is too short to be pissed off all the time. Especially when I rethink something and see right off that I was wrong, like fractional notation not eliminating everything that could confuse people who don't get order of operations. Or when I remember that readable code is a thing.
Again, thanks for taking your time for that answer and for being nice. :-)
The problem lies in the fact that the slash doesn't make clear what is being divided and what's not. As an engineering and high school tutor I despise the slash and/or obelus symbol when used in grades older than third. Use ONLY fractions for division, which will also assist with simplifications before multiplication because it makes the numbers multiplied smaller. If you must use the "kid" symbols, then make sure that everything divided is in brackets indicating the denomator and anything else exists on the numerator.
If I use /, I always make sure to put everything in brackets, as in ((a+b)/(c*d)), because otherwise it can be extremely hard to find out what is the divisor. And I have learned not to use the notation a b/c for a+b/c anymore, since it can be read as a*b/c
@@JonathanMandrake I generally tend to prefer writing d/c where d = a*c+b instead of a b/c for a+b/c, especially while still in a calculation.
agreed, the only cases where i accept using "/" is when the numerator an denominator are just numbers (eg 5/7, 891/31)
Or be a chad and use a * b^-1 instead of a/b
@@joda7697 Sigma male move
i love the fact that here in germany in the last years of school, and at university we basically just didnt bother. Every division gets written as a fraction and that way no ambiguity ever occurs
In all my years in school, it was only in elementary schools that did simple division such as 9÷3. Calculators express division with a forward slash 1/4 and many expressions now show proper fractions with an over and above character ( ¼ ). PEMDAS itself is a basic guideline for handling expressions, but when it comes to formulas expressions that don't have symbols it can become confusing to beginners who might not have seen something like √(25)(-2)². Even then, it's hard to express formulas on a computer because mmany of the complex functions in math can't even be typed without looking them up.
yeah, in england as soon as i got to GCSE maths it got switched to fractions for all divisions cos it made things so much easier
We do the same in the U.S.A but the correct way is the answer is 1 way. There's no ambiguity.. it just that people focus in the U.S.A to memorize PEMDAS and they forget in 8th and 9th grade they finished teaching PEMDAS to add in understanding of juxtaposition were 2(2+3) is the same as (2(2+3)) in groupings. So, most forget this and the companies that make calculators or anyone writing a program that's a calculator has to strictly follow PEMDAS and omit juxtaposition because they assume kids in the elementary school will use this program and this should be the default. Most have different modes where you can switch them to use juxtaposition. You either learn how to change modes or if you like me.. just do the calculations by steps manually. After 9th grade in the U.S.A we do things correctly according to Mathematics. Before this they make us strictly Follow just PEMDAS and then 8th to 9th grade when learning algebra they then add in the last rule which is juxtaposition. All Americans remember PEMDAS but they forget what they learned in algebra class about juxtaposition.
I think the true but pedagogically wrong answer is: if you're solving an actual problem and not just doing an arithmetic exercise then ambiguous notation doesn't matter so much. You'll know whether a/bc means (a/b)c or a/(bc) by knowing where the expression came from.
That said, anyone who writes a+b/c+d and means (a+b)/(c+d) is a menace to society and must be stopped.
That's actually a pretty important point I hadn't considered. Obviously ambiguous notation isn't really a problem if you know what the expression actually represents, but that ambiguity becomes a huge problem when you're trying to do arithmetic divorced from any context to clarify it.
Maybe that hints at another problem with our teaching methods.
@@martinepstein9826 I hope this is a joke
@@fritzzz1372 how is that a joke. it's facts
@@DNXTMaster true, then we should stop everyone of the many professional marhematicians who do it this way lol
Mathematics is a language, and in all language we should make every effort to communicate our ideas clearly. These PEMDAS problems are poorly structured expressions (usually deliberately so). Yes, a computer or a calculator needs some rule to interpret these bad structures. But rather than teaching people how to think like a computer, we would be better served teaching them how to write clearly unambiguous expressions.
Just what I was about to write. If someone wants rigid unambiguous notation, use reverse polish notation (in fact is what some early models of HP hand held calculators used). That is perfect for a computer. But we humans are not very good at doing algebra in reverse polish notation. Instead, we can be magnanimous with the use of parenthesis.
The problem with school is that it does not focus in teaching students to think, but on producing straight jackets for everyone’s minds to wear. I teach first and second year math for engineering at a spanish university and can attest that my students appear to have been deprived of any ability to use their heads to think.
Mathematics uses languages and it has ways of specifying formal grammars. See LALR and Earley parsers. A mathematical notation should try to use a deterministic context-free grammar if possible; to keep things simpler. Also we're run into program languages that have much more involved order of operations, and to avoid bugs it is generally regarded as good advice to use parenthesis sometimes even when not strictly needed. An unambiguous expression in a particular grammar might still be difficult for a human to follow.
@@stephenJpollei, because there are more things in heaven and earth, Horatio,
than are dreamt of in your philosophy. Just for the sake of rhethorics, computers are what that philosophy dreams, whilst natural languages live in heaven and earth ...
@@MiguelGarcia-zx1qj
My philosophy is pretty big; I think there are infinite expressions and infinite grammars. Computers are just mathematics; see Alonzo Church, Alan Turing, Kurt Gödel, etc. I also like some conlangs such as Toki Pona and Lojban. BTW due to Gödel, I think there are infinite logic/math systems :)
Another aside, Lojban grammar is almost but not quite parsable with LALR(1), but PEG( parsing expression grammar) was required.
“Glendower: I can call the spirits from the vasty deep.
Hotspur: Why, so can I, or so can any man;
But will they come, when you do call for them?” ― William Shakespeare, King Henry IV, Part 1
@@stephenJpollei “I could be bounded in a nutshell and count myself king of infinite space…” :)
As someone who's not a native english speaker and was not taught math in the english-speaking countries...
I never knew there was an acronym for it lol. My teacher just said "First brackets, then multiplication and divison, then addition and subtraction." No silly acronyms, literally just the rule, and many example exercises to solve until it's kinda like muscle memory.
Americans like acronyms a lot. They unironically use TLA (a three letter acronym that stands for Three Letter Acronym).
I've never seen a single person use that acronym, though with the amount of government agencies it might be useful
That's interesting. Which country are you from? I am from India and we were taught PEMDAS.
@@dexternepo I'm from Poland. Translating the acronym to our language would give only consonants, not easy to remember. And since this stuff is taught in like 4th Grade, you can't assune everyone knows English.
Hey, nice profile pic d:
You don’t understand how happy I feel after the gaslighting of telling me I’ve been doing this problem wrong. It’s not that hard to understand left -> right and that M & D and A & S are equal, however I had people tell me I’m wrong and that I “didn’t follow PEMDAS”. I’m not a math major, but math has always been my strong suit, so to have so many people incorrectly correct my math was extremely confusing
Honestly, using ÷ is just bad notation in general. It makes it seem like division is a seperate operation from multiplication, when in fact it is not. "Division" is multiplication of one number with the multiplicative inverse of another number.
Instead of a ÷ b, you should always write/understand it as a * b^-1
Not a divided by b, but a multiplied with the (multiplicative) inverse of b.
It's fine to call it simply the inverse, since the additive inverse is usually called the negation instead of the inverse, so there shouldn't be confusion.
@@joda7697 wait so if I write
4(1+2) * 3(6+4)^-1
isn't this still considered ambiguis writing even if I wrote it using exponents instead of division sign?
sorry in advance if I misinterpretted your point.
@@arenmee540 No how is this ambiguous? this is clearly 36/10
I know you intended it to be 12/30, but 36/10 is objectively how that would be read, no way around it.
Otherwise, you'd have to write
4(1+2)(3(6+4))^-1
that’s what i thought as well, my teacher always told me to do whichever comes first (for MD and AS)
@@joda7697 yeah, juxtaposition doesn't have priority over exponents
throwback to when my elementary teacher told me you can't subtract more than what you started with. I mean why not instill a deep sense of academic distrust in students when they first start doing school, nothing could possibly go wrong.
Every god damn time, first I questioned subtraction, was told I was wrong, then they were wrong. Then roots with negative numbers. Then dividing by zero. Every time they have said I couldnt do something, I would be told I could later. Just tell me that is in a later class, don't lie to me.
@@MCNarret for real. Nothing wrong with saying "we are just focusing on positive integers" instead of saying "negative numbers are impossible" at least connect it to real life, you can't have -1 of a physical object, but you can 100% have -1 written on a piece of paper.
@@MCNarret You can't divide by zero though, you can only give an approximation of equations that approach infinity relative to each other. Anything/zero doesn't equal infinity, it equals undefined, which isn't really equaling anything
@@Reece8u You say that, yet if everything points to a defined value, then it is that defined value. I am not saying all divide by zeros are equal, nor are infinities, but if you assume a limit to evaluate an equation or value, approaching infinity might as well be infinity. I see no practical difference.
Math teachers are always like this.
"Only even numbers can be divided by 2"
Sike, a bit later you learn about decimals/fractions
"You can't add something and get a smaller number, you have to subtract for that"
Sike again, negative numbers are a thing
"You can't square a number and get a negative result"
Sike yet again, imaginary numbers are a thing
Teach them correctly from the start... Wow who would have thought!
It reminds me of back when I was in school and I had an interest in math so I was learning some more advance things at home than our teacher was covering. When I pointed out that you can have a larger number subtracted from a smaller number in math class and it would be negative the teacher just told me that we weren't covering that now as it would be too confusing to people.
Over the years though I've come to think the teacher was completely wrong. Having your students learn "truths" that are false does far more harm. With any sort of learning environment the goal is to build on the lessons that came before. But when you basically throw out the lessons of what came before, or do it in a different way that ends up confusing people. It's like with algebra where some of students have trouble grasping that there are now letters in math when really it's something they have been doing already. In grade school you get a problem like 3 + __ = 7 and you are taught how to solve it. Now that same student gets to Algebra class and sees 3 + x = 7 and they are like WTF there are letters in math, this is complicated.
The constantly changing rules not only adds complexity to those who may already be struggling but it also serves to highlight to all of the students you can't trust your teachers. They are just there to push what's written in the book which itself is potentially questionable. I've had classes where teachers told us not to look up answers in back of the book because they were not always correct, they claimed it was done to catch cheaters but I suspect it was done because of the order of operations being done differently. But at the end of the day it creates a sense of distrust between layman and supposed experts in their fields when the average layman's experience comes from school which had many cases of contradictions and down right falsehoods.
Mistakes in solutions are usually misprints, typos or the person writing the book just messing up some minor calculation. Most textbooks have thousands of problems, of course people writing them will make a mistake every now and then.
In my elementary school we had this program called AM (Accelerated Math) where computers would generate custom math tests for students to progress at their own pace. I got really far ahead in 3rd grade and encountered order of operations, which normally isn't taught until 4th grade. My 3rd grade teacher misremembered PEMDAS and explained it to me as a strict order, so multiplication before division and addition before subtraction. It was really annoying getting scores of 28/30, and I didn't realize what I was doing wrong until it came up in the main curriculum a year later.
@im_nonexistent your school did you dirty
Unrelated to what you said, but in my elementary school we had Accelerated Math as well. I was always behind on it, not because I was dumb but because I just hated doing it.
@@ow_su I learned pemdas prior to sixth but my 6th grade math teacher did explain how it was a flawed system and from then on I see it is a better option just have rules explained rather than acronym
@im_nonexistent whatever helps yall sleep at night “not bc I was dumb”
PEMDAS and PEMDSA would yield the same answer in all situations. PEMDAS and PEDMAS don't however and everyone pre-generation X when no one had a pocket calculator learned multiplication before division
"Do it the way mathematicians do it" is begging the question of which mathematicians? Most papers I read are in the vicinity of type theory where juxtaposition almost always indicates function application and multiplication, if it comes up at all, is noted explicitly. While I don't think type theory should be the standard for math classes, I'm not at all convinced there's a universally accepted system. Three examples doesn't make a consensus.
I think it’s just really important to understand that conventions and definitions can differ when you’re in different fields. Stuff like Einstein notation isn’t strictly standard, but when you’re reading about something that uses tensors, you just know what it’s supposed to mean.
But that doesn’t mean that there shouldn’t be a baseline understanding of what the most common interpretation is like, and at the moment, this is quite well defined, albeit a little all over the place.
@@weakspirit_ I mean math syntax as it is IS the problem to took form over function and needlessly complicated it because they were to lazy to use proper syntax of writing everything down properly.
@@weakspirit_ will do once you use real syntax not this abbreviated garbage.
This is almost like the FOIL situation as in school making you mindlessly memorize acronyms. In 6th grade, we were taught FOIL. It worked pretty well for me in 6th grade, i could multiply two binomials pretty quickly. But then, one day, I wondered to myself. _Why does foil work?_ So, I thought. First, Inner, Outer, Last, why did we need that order? I tried doing a different order, and sure enough, I got the same result. Then, I realized that FOIL was basically just telling you to multiply each term in one of the parentheses by all the terms in the other parentheses. I tried this out, but with more complex equations that weren't just binomials. It worked on *_EVERYTHING_*. And, then I thought some more and realized that could be put in even simpler terms: It is basically just the distributive property but with more numbers. I am very glad my 6th grade self had made that discovery, it made multiplying polynomials WAYYYY more intuitive.
FRRRRR as someone who can see stuff like this easily it pains me when people say "foil" because it's an acronym for the sake of someone pointing at a binomial and going "NO WAY IT SPELLS A WORD 😱😱😱"
man im doing advanced higher maths (uk, basically like your first year of uni in highschool in terms of the curriculum) and this simple explanation is so much better than a random ass acronym. I wish school actually taught the rules of maths instead of weird memory games.
Yeah I remember only being taught FOIL. But recently when I was tutoring a kid who was learning how to multiply binomials, I found that he had been taught to use the distributive property first, and then a couple of sections later was taught FOIL as a short cut AFTER learning why it worked. This way of teaching it so much better than just having students memorize an acronym, because, just like you said, distribution works for ANY polynomial.
Oddly, my brains switches the order automatically, like I will be trying to use FOIL but my brain goes "first, inside, outside, last".... so FIOL I guess lol. Which makes sense now that I think about it because that mirrors what I would do if I used distribution.
We didn't get told about FOIL until 2 years after we'd originally learnt this, and that was only because a trainee teacher came over and taught us it. Obviously I knew how it worked, but what annoyed me more was that my go-to order was FIOL instead! It upset me being told such a static way to do it when my method was slightly different and worked perfectly fine.
100%
I was taught how to expand first, then my teacher told us that FOIL thing. I believed she was making things even more complicated by that FOIL method when she actually thought she was making it easier. I just decided to ignore it and now I don't even know how does the FOIL work.
In Germany we learn "Punkt vor Strich" which literally means "dot before line" (we commonly use : for devision). After that we learn things in parenthesis go first. So at least a bit better then PEMDAS
is it though? Because here you directly learned to put * before / which while being what PEMDAS does, its intention is not doing that, however yours is
@@pedrosso0 Strich means - and + Punkt means : and ·
@@tiborgrun6963 they said . before line though... but I'm sure that that could mean multiplication before additions/subtraction
@@pedrosso0 in Germany lots of schools teach the syntaxes like this: multiplication = · | division as : | addition as + | subtraction as - || so since multiplication and division are executed before addition and subtraction the rule would be dots before lines. Although I have no idea as to why we use · and :
@@hefesan I guess someone just forgot the line between the dots xD ÷ :
What's all this "multiplication comes before division" malarky? The version I was taught had the D before the M!
Which just makes it even worse, huh.
well, it means that division will act on fewer things, and get a little closer to being a unary operation
in an expression like... "2 + 3 * 4 - 5 / 6" treating - and / as unary operations (which is to say, as "2 + 3 * 4 + (0 - 5) * (1 / 6)") and treating them as operations that occur first (which is to say, "2 + ((3 * 4) - (5 / 6))") are pretty close to achieving the same result
I think GEMA is a great convention now that I have been properly introduced to the idea, after trying to preach this "ditch subtraction and division as independent operators, they're the same as inverse addition and inverse multiplication" idea
Yeah, in the UK it is/was BODMAS and I definitely recall being taught that the MD and AS had the same priority.
The only reason I understood PEMDAS was because of the fact that I had been exposed to the C programming language before I ever took Algebra. It gives multiplication and division the same weight in almost every scenario, and also does the same for addition and subtraction. I was also taught that division and subtraction are just simpler ways of undoing addition and multiplication without using negative numbers or fractions. More to the point, I was shown somewhere that subtraction is often really implemented as adding a negative number, and division is really implemented as multiplying a number by the multiplicative inverse in a lot of code, sometimes even at the hardware level, because it simply wasn't necessary to duplicate the logic as it works the same way regardless. Starting with that as a foundation, it's not hard to make the leap from the idea that the compiler treats addition/subtraction and multiplication/division as two sides of the same coin, to figuring out that this might reflect an actual mathematical principle rather than just being a convenient arbitrary language rule or quirk of CPU architecture.
I think it is not the c language itself but the compiler and how much internal memory one got(mcu) I had a lot of problems with stm8 and sdcc.
I once taught a first grader about how to use negative numbers in about 5 to 10 minutes,
and he proceeded to calculate problems using them correctly. All it took was for me to say that:
Positive numbers are basically like arrows pointing from 0 to the right, and negative numbers just point to the left.
Adding any 2 numbers means to put the base of one of them to the tip of the other, and see where the new tip points.
The best thing about this is that it even works for euclidian vectors and is very intuitive for a child to learn.
Using geometric explanmations for maths is so much better in any case. I really have no idea why so many math teachers hate using it. It took me 2 years in college to get irrational numbers, cause it took them 2 years to show us that it's actually adding a plane to the numerical line. After that I understood what irrationals are (mostly the fact they aren't irrational, but just a logical physical extension of line into a plane) and could actually properly operate on them using my previous math knowledge.
@@Ussurin you mean imaginary numbers?
@@kjl3080 Yeah they definitely mean the complex plane and imaginary numbers, because irrationals are something different. These are the numbers that can not be expressed as a product of one integer and one multiplicative inverse of another integer.
@@Ussurin First of all you meant the complex plane and not irrationals, but hey we all mix stuff up from time to time. Secondly, this arrow approach is also good because it already makes the additive identity (so 0) have a visceral importance, as the point where the first arrow starts.
The field axioms for the additive and multiplicative numbers are maybe formulated very abstractly and with difficult language, but they _can_ be explained to a child with more simple language, and if they build their intuitions right on those, they will have a much much smoother learning experience going forward.
Yeah, sorry, meant complex numbers.
Mistake came from my language calling all that stuff in no similar way. Sorry for that.
The real reason the Facebook equations are causing so much confusion is because they are intentionally being written in a ambiguous manner, because the point is to get people to interact with it and generate traffic for the page. People arguing over the answer is the end goal. If pemdas wasn't an issue they would find some other way to do the same thing
Well they are written so those stuck with pedmas would get a different answer then those with higher math studies.
And it's always because of a number next to parenthesis. (which people think is that same as a normal * multiplication). But any algebra will tell you : 2(a+b) will always be (2a+2b)
It's not confusing per say usually..
@@jawstrock2215now we come to the question of did the original creator of said Facebook equation understand that principle
I disagree - they're not ambiguous at all for those that understand to universally-accepted rules in the scientific & maths communities.
@@peterthomas5792 I'm talking about the Facebook meme and who created the equation. Were they somebody using pemdas or were they somebody doing actual math
@@peterthomas5792 just use a fucking calculator to verify
My problem with the equation including 2(...) is that unless multiplication symbol is explicitly written 2*(...), it's a single element in my mind. Same as 2x for example.
Either way, I rarely ever write division in line on paper, as I always use fractions instead. Also in programming there's no confusion, since you have to use all the operators explicitly.
I was taught that 2(3) is the same thing as 2*(3) and it's just shorthand.
@studiouskid1528 2(x) is not a function though... This all comes down to ambiguity. This question literally has no answer, it's just nonsensical. It's like asking "What color is geography?"
@studiouskid1528 this has nothing to do with functions
@@Andrew-it7fb No it explicitly links them together as a unit. Calling it shorthand is just a lazy way of looking at it.
@@calebfuller4713 It's not lazy. That's littereraly what it is. What's inside parentheses is prioritized. The 2 is outside of parentheses.
wow, didn't know that students in the US and UK have these mnemonic rules. for me personally it is ok to make up a mnemonic rule only when you have to memorize some stupid big formula without particular deep meaning. i think that conceptual things like this are better memorized just by understanding its core and mnemonic can be harmful in this situation.
I'm russian, same here, we never had such mnemonics or problems with precedence, we just know right order and that's all.
In france we just make the order obvious by using fractions or adding parentheses, in school we just learn to begin by the parentheses and that × and ÷ comes before + and - i've never enountered an issue between × and ÷ in my life, and passed elementary school nobody uses the ÷ symbol anymore
This is a good way of putting it
@@charlz-darvin Иван родил девченку, велел тащить пелёнку
@@stephanevautrin7317 I'm british and have exactly the same experience as you, it might just be an american thing??
You know, I was always a little confused about what people had against PEMDAS, until this video made me realize that PEMDAS wasn't the thing that I learned in school.
exactly lol
I'm not american so I never even heard about pemdas until I finished my second year in uni. From the very start in school, operations were explained as as opposites to one another, so there wasn't ever any confusion
Pemdas at my school was taught 6/2(1+2)=9 because my teacher said that they have different precedence.
My school I think teached us about pemdas and gemdas I'm still confused till this day
bruh lol
That is correct.@@V1TheWarMachine
9:07 Wait how did someone get 16 as the answer i seriously can't figure out
it was probably a simmilar equation with different numbers
they used the question 8/2x(2+2) not the other one shown (pretty much the same problem with different numbers)
Honestly, I was just NOT taught PEMDAS at school, only the order of operations and when I heard americans talking about it... I got instantly confused. And when I saw people saying multiplication should go FIRST, I got EVEN MORE confused.
And I guess...
Well, I guess this explains a lot lmao
PEMDAS is just an acronym for the order of operations. If you learned the order of operations, you learned pemdas, you just didn't learn the acronym. Nothing wrong with not learning the acronym lol
@@Reece8u Did you even watch the video
I think learning that MD and AS are actually the same operation (division is multiplying by a fraction, subtraction is adding a negative number) would simplify it. Then it would be a proper hierarchy of P-E-M-A.
@ I did watch the video. The order of operations is PEMDAS (or PEDS or whatever you wanna call it). Though that was a month ago, I still stand by my opinion. All of his contradictions were wrong except for the one where he said it should be a 4 letter acronym, which made sense.
@@eve6262_ That is how its taught, but people get confused because of the extra letters, so making it a 4 letter acronym makes sense
It baffles me that math teachers and mathematicians can't figure out why math is commonly stated to be a student's least favorite subject. So many unclear rules, and the different methods of teaching that contradict another form of teaching is frustrating. And this is coming from someone who never minded the subject (It was the only subject I was truly confident in my abilities to pass in).
1:11 you learn polynomials (if they are a real thing i cant even remember) which are basically some number with () in front and whatever is in the brackets needs to be multiplied by the number before brackets for example 3(2+3) is 3(5) which is 3*5 which is 15 therefore 6/2(1+2) is 6/(2*3) brackets are used here to simplify the equation otherwise it would be bad math, 6/6 which is 1
as a software engineer, i just use parentheses for every operation to remove ambiguity
Same. It drives everyone around me crazy but if you abuse parenthesis you're going to get it right every time. It ain't pretty but it's straightforward.
Packing up those equations like amazon packs up my packages
Me writing ((((((((((1+1)))))))))
@@scrungozeclown836 more like ((((((((((1))))+((((1))))))))))
I feel that the issue is that people see multiplication and division ( and similarly addition and subtraction) as two different things, when they are infact one and the same. Dividing by 2 is the same as multiplying by 1/2 or 0.5. Order doesn't matter, what matters is clear notation.
it took me a while to get this but it was mind blowing when i got it
multiplication and division are completely different things; in fact, they are the opposites of each other
@@F_A_F123 go back to grade school math and come back
@@TeraAFK The thing called math in grade school can't really be called math, it's a game of remembering. But anyway: division and subtraction are literally defined as the opposites of multiplication and addition, the same way root and logarithm are the opposites of exponents.
a * x = b -> x = b / a (the definition of division);
a + x = b -> x = b - a (the definition of subtraction).
Multiplication and addition are commutative (that is, a (operation) b = b (operation) a), so they both have 1 opposite.
a ^ x = b -> x = log(a, b) (definition of log)
x ^ n = a -> x = ⁿ√a, where x ≥ 0; n ∈ ℕ and n ≥ 2 (definition of arithmetic root)
@@F_A_F123 root and exponents are the same thing, not opposites. sqrt(2) = 2^0.5
When I first learned PEMDAS in grde school literally added my own brackets PE(MD)(AS) in my notes cause it seemed the most obvious way for me to remember their equal weight. Nice to see others agree!
*and then, everybody clapped*
@@Loki-qo2kb I don't quite get what you were going for here.
For me it was BEDMAS
@@RoseInTheWeeds He's basically calling you a liar, and I agree with him
@@Felipe-wg4ir Why would he lie about something that small
I think the thing that did irritate me learning math in school was finding out that in later more advanced math was that when subtraction was happening it was the addition of a negative number, which made sense to me, but was also confusing because I did not understand why they didn't just teach that to begin with instead of making things complicated.
They do teach it.
@@viragkaroly2949not everywhere
Teaching that in the beginning requires blowing kids' minds with the existence of negative numbers.
@@starshade7826 sounds great
@@starshade7826 at least metric system exists
Math was the bane of my existance in high school, i was told the excercise was wrong and never explained where i failed.
Why is "just write clearly" a non-solution? If ambiguity is not intentional, then whoever tries to convey the message is likely to try and avoid it, no matter in natural language or in maths. Mathematical notation is obviously not immune from ambiguity and no amount of additional rules is likely to remedy this. So to me it's like: "make sure that the meaning of what you write is clear either by itself or from context"; "to get a better idea of what's clear and what's not-read more of what others (mathematicians in this case) write," and hope for the best.
Agree, the whole point of math is to understand. If 50% of the Internet (including mathematichans)fails to understand you, you could've probably written it better. I'm talking abt the viral math problem
Edit: clarification
In programming languages, the meaning is indeed strict and only has one meaning.
I agree the acronym is not helpful and it's easy enough to just remember the meaning. One good explanation is to show that the order is what it is because it makes it easy to write polynomials without extra parens.
In C++, some operators have left-right associativity and others have right-left associativity (the direction the operators are evaluated when they have the same precedence).
Also, when multiple operators are next to each other, the compiler reads the biggest operator possible. For example:
x---y
evaluates as:
x-- -y
and not:
x - --y
@@waldolemmer So simple, yet one of the top questions on Stack Overflow is asking what x-->0 means!
@@JohnDlugosz I love the responses to that question. Here's my favourite:
x can go to zero even faster in the opposite direction:
int x = 10;
while( 0
This highly depends on the language though.
wait until this guy finds out about i before e except after c
and sometimes w and...
Curiously, I think this is only a problem with some education systems. When I was young in Spain of course I learned the order of operations which is as the system you describe. However it never had a name akin of a memo trick to remember it.
And, most importantly, teachers wouldn't use tricky expressions just to fool you, they would use normal expressions. So there was no room to argue if the multiplication or the division had to go first.
As an adult I have been exposed to this nonsense, of course, but I have always dismissed it as such. It is not like PEMDAS is wrong, it is useful as long as you don't try to be a smarty pant. And if you do try to be one, then the expression you come up with will not be relevant for anything more than being a smarty pants.
And on a side note, if I would write 1/(2N) as such, not 1/2N. And the only reason Feinman may have done it differently is because just 3 lines above that the expression made more sense, with a big division line which makes it plentifully clear what you should do first.
Same, I live in South America and they expect you to remember the order yourself without any words
Thats why no one uses ÷ sign in math.
Its better to use fraction bar,
Just like this
6
-------
2(1+2)
Or
6
--------- ×(1+2)
2
There, no misunderstandings
E X A C T L Y
Also, advanced calculators can give you the wrong answer if you don’t input the values correctly. After using TI-84s for high school and college courses, you tend to learn that you have to add extra parentheses just so the calculator _knows_ what the equation is.
It's harder to write, the fact that you had to use 3 lines and a lot of dashes to write it...
I fucking hate acronyms in education. They are the clearest indicator that the education system wants you to memorize things, not learn them. I am thankful to my primary school teacher for not telling us to memorize the multiplication table.
You had me right up until the last 10 seconds. I think everything was on point, but the landing was flubbed when you over-simplified your point--and I believe even contradicted yourself--when at the end you said "Instead of wasting time teaching language in math class, maybe just teach math." The thing is--and you all but acknowledged this earlier when you said that the important thing is communicating the underlying idea, and that "different ways of saying the same thing are still all saying . . . the SAME THING"--that math IS a language. It has it's own vocabulary and syntax and grammar that need to be understood in order to communicate well with it.
That is one of the most powerful paradigm shifts that come to my high school students when I'm teaching functions and they understand that equations, graphs, charts, and lists of points are all essentially just dialects that are different ways to describe the same underlying function. And with that understanding they start to grasp the abstract idea of what a function actually is.
Yes, math has it's own grammar rules, and sometimes those rules are arbitrary--but sometimes, when understood properly, those rules make perfect sense, and actually couldn't be any other way.
"Math is the language by which God created the universe." --Galileo
6/2(1+2) isn't math, it's god damn nonsense masquerading as math. We have the ability to adequately communicate every aspect of mathematics. We have the means to describe such a problem in no unclear terms. Pemdas only works, because we have an underlying foundation that works unambiguously, and strictly based on logically (or rather natural law). There isn't a single operation in that "problem" that 99 percent of any developed nation couldn't solve. This bullshit literally makes people worse at math. Excusing poorly written "problems" is the fucking problem. There isn't a right answer to this math problem because it isn't math. Pemdas is not math, straight up. It has no logic. It is not backed by law. It is pure abstraction. Sure, base 10 as a standard is pure abstraction. All the symbols we use to communicate math is pure abstraction. But what they resent is not abstract, and the logic they follow is not.
Math does have its own grammar rules, and pemdas brakes them. Instead of teaching kids how to properly articulate in line math problems, we get bullshit like this. Any language that cannot reliably communicate its ideas is not language at all. Fortunately math does have a robust, easily understood language. It's just counting, after all.
Formally there should be brackets. We dismiss brackets when there is no ambiguity. If there is ambiguity, one should use brackets. So 6/2(1+2) needs one more pair of brackets imo. Without these Id say it's (6/2)(1+2), but if it was written by someone that didnt intend the ambiguity Id think it's 6/(2(1+2))
Wait what?
It’s not true in real life though. The area of a parallelogram with the parallel sides of length a and b and height h is never going to be written as h/2(a+b). That looks immediately wrong to anyone with some experience in mathematics at a higher level.
Brackets are not necessary if you wanted 6/(2(1+2)) because there’s only one interpretation.
@@qsykip Ik thats what I tried to explain in my last words. I meant in practice if someone wrote this I woule think it's h/(2(a+b))
To me it would mean (h/2)(a+b) only if it was written 'formally'... I mean it feels like the rules would say it's (h/2)(a+b), even though in fact it's just ambiguous from a formal point of view
I have a hard time explaining myself in english, lol
@@holomurphy22 I‘ve understood you now. I think it’s just somehow an unwritten rule. Everyone knows it, but nobody really talks about it.
The point of this video was probably to bring to attention the fact that all this PEMDAS/BIDMAS stuff isn’t complete, and shouldn’t be regarded as a formal rule. It’s one of those things that American and British (and a few others) schools keep teaching that isn’t strictly correct in real life, but because it’s taught like this, people start to defend it as though it’s the law of the universe.
@@qsykip I dont know much about pemdas and such rules. In France we only hear a thing like 'multiplication before addition' but nothing about division (or maybe I just dont remember). I still feel a tiny bit uncomfortable when I read h/2(a+b) but I get the sense (moreover there is in reality a context from which you can deduce the meaning). Actually math papers are written in some sort of LaTeX and during course it's written on a board, so I almost never see things like .../...
Maybe someone who studies computer science sees it more often. Even though I guess there couldnt be ambiguity either because it has to be computed by the computer, but maybe when they write things unformally... Idk
I'm a writer, not a mathematician, but something I've come to believe is that one of the biggest disservices you can do someone is teaching them to follow arbitrary instructions rather than teaching them to think about what they're doing.
We learned GEMS:
Grouping symbols,
Exponents,
Multiplication and division,
Subtraction and addition.
That method sounds like a gem
@@GillOfTheNorth lmao
I've been staring at this 6/2(1+2) thing for like ten minutes and I cannot find a single way that you could possibly get 16 even ignoring the proper order.
Can someone please explain to me how people have gone THAT wrong
Ok, so, I thought about this for a bit too long but...
Multiply 6 and (1+2) to get 18, then, instead of dividing 18 by 2, subtract 2 from 18. Idk why someone would change the operation but that's the only way I've found to get 16
I think it is an another problem. It was something like 8/2(2+2)=?
@@tav7630 That would make more sense
@@tav7630 Makes more sense, so probably
I took that as a joke actually since the comment said she took calc 3 lessons
As a former Sped kid, with reg classes too, I always just thought I was stupid PEMDAs was hard for me to understand. I always felt more confused by it, actually your explanation makes more sense, I feel like I actually had to take less formula problem solving, felt more intuitive for me.
Relying on Pemdas, is like relying on really bad made crutch. People will defend it though because that's what they were taught and they don't like being wrong.
"As a former sped kid"
@@universenerdd i'm assuming you're genuinely confused and not a jerk, so "sped" => special education. needed some special consideration for some aspect of learning.
This video is incredibly amusing. I've long ago abandoned PEMDAS and never put any extra thought into why and blindly accepted the abstract algebra axioms for multiplication and addition that include division and subtraction respectively as the same operations on inverses. So I never really considered how MD/AS are redundant. great video.
". . . you can't teach anything if it contradicts intuitions people have already built, and if you ignore this fact, people will learn nothing or worse."
Exactly.
If someone has the wrong idea about something, they need to be shown why their current understanding of the thing is wrong before trying to teach them concepts that require a correct understanding of the initial concept as prerequisite.
Otherwise, those ideas will not and cannot make sense to them because they are built on premises that appear false in the perspective of the misinformed person who first needs to unlearn the lies they internalized.
And if you try to teach them concepts that require prior understanding of the initial concept as prerequisite, they may only develop a warped interpretation of the other concept based on their misunderstanding or misinformation of the prerequisite concept that it is built on, giving them even more misinformation to unlearn before they are able to learn about these concepts correctly.
Oh hi you just described my entire experience with mathematics. Currently unlearning and relearning it.
> _"If someone has the wrong idea about something, they need to be shown why their current understanding of the thing is wrong before trying to teach them concepts that require a correct understanding of the initial concept as prerequisite."_
This does not exclusively apply to mathematics. All the way down to social and political issues, which some like to treat like pure matters of opinion sometimes, if you can show someone that you understand his point of view and demonstrate to that person's own satisfaction why it's rooted in a mistake, then and only then can you genuinely hope to change that person's mind.
7:17 in vietnamese education, we have a saying "multiplication and division before, addition and subtration after", and parentheses is the almighty one. All of that was taught in primary school.
Unary, exponent, roots, logarithm are just new dudes that got introduced later and we have to learn to use them without any catch phrase nor acronym.
Despite that, the "6/2(1+2)" problem was still very much viral in our country. They never said anything about juxtaposition, it was told to be the shorter way to multiply. Later I just figured out how they works on my own intuition.
Huh? I learned normal pemdas in vietnam
@@skellq4385 really? I only got the "Nhân chia trước, cộng trừ sau".
Pretty sure school made me better at Maths. Before school, I didn't even know what a number is.
Lamo
@@idek6585 Laughing Ass My Off?
@@ClamMimic Its like a variant of it
The question is "Why would I care"? The good thing to know about math is that it helps solving something that can be constructed in math language. Anyone, who think "6/2(1+2)" is a real deal to solve, are wrong IMO : it doesn't help solving anything, it's just you are guessing an answer for an improperly put question, what is a waste of time. Once you are not using that division operator in your expression, you're good to go without any ambiguity.
So let's just say I agree with your summarize, and would be a bit happier, if people stopped falling for these "math" tricks
If you ever... and I mean ever say a+b/c+d, teach us PEMDAS explicitly as a tool and marks us wrong because they meant (a+b)/(c=d) is trying to make us fail math.
I just wanted to take a moment to thank you for making a video explaining what I've been trying to explain to people ever since these these viral math equations first started appearing. I hope lots of people see it. I'm disappointed that relatively few people have seen the videos on this topic from "The How and Why of Mathematics" while most of the popular videos on this question (one of which has been liked over 100K times) provide the wrong answer by over-relying on an incomplete or incorrect understanding of the order of operations. When I was in school, many students had difficulty remembering how to apply the PEMDAS. Some made the mistake of thinking that multiplication always came before division, but the bigger problem was that people would forget that multiplication by juxtaposition is an exception to the rule that multiplication and division should be done left to right. And now we have at least a couple of generations of people who either misremember what they learned or never learned it correctly in the first place, and some of them have become K-12 teachers who over-rely on PEMDAS when teaching the order of operations. As "How and Why" notes, the way calculators and spreadsheet programs require and interpret linear input has probably been a major contributor to people forgetting that multiplication by juxtaposition takes priority over "regular" multiplication or division.
Yes, hand held calculators becoming allowed in the classroom is a part of this that shouldn't be understated. But beyond that, juxtaposition is a from of shorthand for math. In the generally referenced equation, 6/2(1+2)=Y, the strict PEMDAS followers believe it should be written as 6/(2(1+2)=Y, and lend no credence to the FACT that 6/2*(1+2)=Y and 6/2(1+2)=Y are NOT the same equation. Due to the shorthand, no additional parenthesis or brackets are necessary if the solution sought is Y=1. The issue for many of them is they don't fully, or more at, erroneously, know how to resolve parenthesis. They believe that even when juxtaposition is used in expressing the equation, somehow, the parenthesis magically disappear after you solve what's inside of them. Which is true, if there is no juxtaposition. Just today I have a cat telling me how dumb I am and how I can't understand what he is saying. He even C&Pd a source, but he is so tied to PEMDAS that he can't read the very source he cites which in fact explains how the parenthesis are NOT resolved until the operator is distributed. Oddly, they won't even try to deny that 2(1+2)=2(3)=6...EXCEPT...They INSIST it becomes 2*3=6 as soon as they solve inside the parenthesis. It doesn't matter how logically you explain it to them.
@@andyfletcher3561
2(1+2) is shorthand for 2 * (1+2), so in this example, 6/2(1+2) is shorthand for 6/2*(1+2).
@@JG-df1qd wrong.
@@andyfletcher3561
Please expand. Juxtaposition as an implicit 1st step was only ever used to tie a number to its unit, largely in science based classes. Other than that, it was never used as implicitly done first and was just regular multiplication in anything I ever took, up through Calc 2.
Part of the problem, I think, is usually you start seeing equations using fraction format which implies parenthesis above and below the divisor line.
To add to that, in typing, the / symbol is used as both a divisor line and as a replacement to ÷. This means an equation using / is ambiguous without parenthesis, and really should be defined before the equation.
@@JG-df1qd ''blah blah blah blah blah''. The only reason you know any multiplication is involved without an explicit sign is because it is attached to the parenthesis. The only form of the equation that makes any difference is whether or not there is an operations symbol. ALL of you claiming this is false are also claiming that 22=(2x2)=4. You just lack the logic capacity to understand it. It's no different than knowing 2f=2xf=?
I think it's easier when you understand that when you have parentheses, it means you need to simplify that expression first. That may mean distributing, combining exponents, performing inner addition where possible, etc. In the starting example, 6/2(1+2), it should be clear to distribute the 2 inside of the parentheses. Since, 1 + 2 are like expressions, you can simplify them to 3 first, or distribute the 2 and end up with (2+4) inside the parentheses: 6/2(1+2) = 6/(2+4) = 6/6 = 1. So, it would be incorrect to confuse the parentheses as meaning to multiply, although it's the correct operation as the distributive operator in this case.
Also, coincidentally, if you write division as a fraction, you tend to perform division last when the expression is simple enough to do so, or if unable to simplify, leave the result as a fraction. It's also easier to apply algebraic rules to fractions than it is with a '/', obelus, or division symbol (the one with a dot above and below a horizontal bar). I think lack of understanding of these comments is why we see a lot of the simple math shitposts online where one generation acts as though they're smarter than the other, and both sides tend to perform the math incorrectly. It's a general lack of understanding of basic math and algebra, or poor teaching practices in my honest opinion.
That's where I was confused. I was taught pemdas and got 1 as the answer, because the parenthesis don't just vanish at convenience. Even if you did it solving the parenthesis first, it would be 6/2(3). There's still work to do with those parenthesis, like you said, distribute the 2. 6/(6). Ah, now the parenthesis is "complete". 6/6 = 1.
Just write clear expressions! For the life of me, I don’t understand why this is such an issue. Notation is meant to communicate mathematical ideas, so just write clear, concise equations and it won’t be a problem.
The trouble is that what is clear differs from person to person. To my mind, 6÷2(1+2) perfectly clearly equals 1.
@@JivanPal I agree!
@@ellag3265 , yes, but parsing the expression from left to right is not the standard way of parsing it. If "clear" means "unambiguous in the absence of any precedence rules", then you can _never_ omit parentheses, meaning that even expressions like 1+2×3 (which equals 7) must be written as 1+(2×3). Is that really what you're advocating for?
The point of precedence rules is to make unwieldy brackets unnecessary in the most common contexts. The trouble is that people apparently disgaree on which precedence rules to use.
Yeah there is no need to leave any ambiguity in purely logical expressions.
@@ellag3265 brackets solve this issue entirely but form over function seems to be all anyone wants I say fucket brackets everywhere just to top it of remove / and use ÷ because fuck the moron that thought using 2 lines is acceptable instead of 1.
I was never taught any acronyms and didn't know it was a thing (schooled until pre-uni).
My teacher taught the juxtaposition variant and explained it simply and cleanly, gave homework followed by a test - everyone in class understood easily.
(( I couldn't figure out how it became a 9 without throwing the maths I've learned out the window first, almost gave me a panic attack lol ))
There's a moment in every programmer's life where they realize that different programming languages have different operator precedence. It shatters the idea of there being a universal order of operations like we learned in grade school.
As an aside: I went to a bilingual school in Canada, and took math classes in both English and French. In French, we were taught BEDMAS, while in English we were taught PEMDAS. That certainly helped me understand the idea of equal precedence lol
In Southern Ontario English schools we always referred to it as BEDMAS. Interesting to see the small differences throughout the country.
@@4R8YnTH3CH33F in my school in the uk it's always been "BODMAS" for us
Luckily for me the teacher used BEDMAS but she was a great teacher which meant she was competent enough to tell is that PEMDAS exists too. I miss her, she would make a good highschool teacher. Still pissed that she didn't know what a newt is though, but I'll give her the benefit of the doubt and just say that she probably only knew of a newt as an eute.
Not really... the calculation is always the same, the rules of the system just determine how you write your calculation... It's like how if I tried to do all my calculation in base 8 instead of base 10 I would get different answers, and of course I would... it's base 8.
I know its besides the point BUT DOUBLE MATH CLASSES??!! I feel bored within 5-10 minutes and suffering that thing twice in a day, oh ugh dont wanna imagine
Nice video, but you really monologue a lot and beat around the bush. The main idea is that asking the order is meaningless, and that this would be obvious if students just learned operations from the ground up instead of internalizing that they need to memorize random details which are actually unimportant.
Superbly succinct! :)
It's embarrassing to see people call each other "Bad at math." over not understanding this.
People seeing 6 divided by 2(1 + 2) as 6/2(1 + 2) makes sense if they were taught that way. The opposite of people not seeing it that way is also fair if that's what they have worked with in the past. It's literally just different rules.
There have been math Ph.D.s who "Got it wrong." because they interpreted it a different way. You wanna claim the guys who've done the hardest math possible are bad at math?
Hell, I managed to come up with valid proofs for "Prove that you can factor every real polynomial into only real quadratic and real linear factors." according to my prof and did "Very well." with trying to prove the Fundamental Theorem Of Algebra without looking anything up or getting help (Independent.) as a first-year math undergrad [I had to come up with so many proofs to do it, as I didn't have the tools given to me.], but I "Got it wrong." the first time --> I am "Bad at math." because I didn't know about the other way of doing it.
No, I'm not bad at math.
I just looked at it from a different perspective, and from that perspective, I was right.
The difference simply depends on how you perceive it --> The rules you were taught --> The order of operations/symbols.
It seems to me (from very limited experience) that division causes most of the problems. However, division is usually written, when possible, not as a normal operator, but as an unambiguous ... what is that called? Big-ass fraction. And when it *is* written as a normal operator, it's either explicitly bracketed, because what you really have in mind is the nice looking fraction you can't write, or it's being written in a programming language where you know *precisely* what the rules are. Otherwise, it's being written as a test question, where it's meant to be deceptive.
In the context of math papers where something is written nicely like 1 "over" 2 root(2, N), then later given as "1 / 2root(2, N)", I would interpret that as a (relatively) unimportant error, given that they've already written precisely what they mean, and are now loosely referring to it again. If, however, the paper declared that it would use some other convention that would strictly bracket the expression as "1 / (2 * root(2, N))", then it's certainly no error, and entirely valid.
I was explicitly taught in school that -4^2 is -16, which is easily justified in the same manner as (^) associating to the right is justified, and to shove that into PEMDAS, I just say that negation is treated like subtraction, so it comes after exponentiation.
All of this said, I am no die-hard fan of whatever convention we may have. I've been messing around with reverse polish notation (that is, `2 3 + 5 *` => `5 5 *` => `25`) and functional languages, and how they should interact.
*LISP programming flashbacks*
It's kinda interesting how maths is taught is similarly yet different in different countries.
Here in India BODMAS is taught.(Bracket , Of , Division , Multiplication, Addition , Subtraction).
Great video mate
And in post-Soviet countries like Qazaqstan and Russia, none of those acronyms are taught. :) :) :)
We were always taught it as BODMAS rule. It goes in this order:
B = Brackets
O = off/ exponents( off basically means if there is a multiply straight after a bracket we have to do it first)
D = Division
M = multiply
A = addition
S = substraction
If there are two same operation then we group them together and put brackets. So 2+2-6÷4÷2 will be 2+2-(6÷(4÷2)).
For the question given above:
6÷2(2+1)
6÷2×3
For we apply off
6÷6
1
If you're confused why this is look at it this way
6/(2(3))
6/(2×3)
6/6
1
My teacher did PEMDAS, but made it explicitly known that multiplication/division and addition/subtraction were equal, that if one appeared before the other that it was done before.
I still don’t know why it’s called the “Order” of Operations, because the order only really happens in like, two areas. It’s much closer to a hierarchy.
By order I think they mean the other definition. As in like a God demanding order.
I also was taught that multiplication and division are equal, but now that I think about it, you can combine + and - any way you want in expressions such as a-b+c-d-e, but for division, one should avoid something like a/b/c.
Order can mean hierarchy. For example the order of battle in a military context is the hierarchy of command ie who is the superior to who. Your being pedantic for the sake of it.
Alliteration
@@michaelgoldsmith9359no, he literally isn’t, because that isn’t what most people consider it to mean (except those working in the military who are specifically taught about the meaning) so in this context it is not a useful definition to use.
GEMA is actually such an amazing tool. It also teaches the fact that division is just multiplication by a fraction and how subtraction is just addition by a negative number. Great video!
Honestly, applying GEMA would require a massive education overhaul, while just bettering PEMDAS is much easier. Yes, subtraction is addition with negative numbers, but that concept is much easier to grasp when one understands that “negative” in addition means take away, which circles back to just calling it subtraction. Or division being multiplication by a fraction. Why would someone write out: 5/1 * 1/3 when they can write: 5/3? We don’t need to overhaul PEMDAS, just teach how to interpret the relationships signs create between numbers.
I never had a problem with order of operations despite never learning pemdas (I probably wasn’t paying attention when they taught it in school. I don’t think I ever paid attention with they taught math.)
I also remember almost immediately grasping the idea that subtraction and addition are the same type of operation, as are multiplication and division. So I think I learned GEMA without actually knowing it.
Then again, my parents grew up in another country and they were the ones who taught me basic math. They also straight up told most of my teachers in elementary school that the way they taught math was totally incomprehensible to them.
The problem lies entirely with how math is taught in the USA. It needs a massive education overhaul.
@@default3623 I think the "division is multiplication by a fraction" thing helps with getting this way of thinking and reading expressions (not just always taking symbols as symbols between numbers) that might help later with polynomials, for example.
But for small kids, yes, you better say that substraction is taking away (is this wrong? It is the same wording) and division is how much of one number the other number contains (which is much better than "division is when you divide" or, imo, "splitting in parts").
Except a fraction is just division again. So you are saying that dividing by X is the same as multiplying by 1/X, but that still has division in it so realy what you are saying makes no sense.
@@default3623we should just write it verticaly like this
P
E
MD
AS
I also need to point out...
Mathematicians, enginerrs and physicists rarely, if never, use / or ÷. Theh can lead to confussion and it's just better practice to write fractions
I took an equivalent to 8th grade Math in Mexico, the concept there is simply called hierarchy of operations. Later when we moved to the US and my sister needed help with 8th grade math I was almost peeling the skin off my face as I was looking at her not being able to do any problems without writing PEMDAS in her cookie cutter Handouts.
The main problem is that there are still some infurriating people out there that write "7 × 5 ÷ 11 - 8" without knowing the nightmare they've just created
Use brackets ya lazy nut!!
what do you mean nightmare? it's clearly 35/11-8 as you wrote. Since there's no brackets
@@pedrosso0 U right, I tried writing one that tends to end in a mess, but just noticed it works fine hahahahha
"7 ÷ 3 ÷ 5 × 8 ÷ 2 × 3"
Would be more problematic I guess...
But yeah, it mostly comes down to people not puting brackets when writibg in a single line (so in the original, someone may have wanted to write "(7 × 5) ÷ (11 - 8)" but didn't want to, stuff like that)
This still isn’t a problem. When the symbols are written out like that, it’s very clear that you should evaluate it from left to right. If that was not the intention, then they are just wrong, but there’s no ambiguity in the actual message.
@@airiquelmeleroy 7 ÷ 3 ÷ 5 × 8 ÷ 2 × 8 = 5.6 right?
If so, then the writing is a non issue for most people from what I've seen. There's no confusion due to the lack of brackets, alternate symbols, etc. So you just go left to right.
Now if I'm supposed to read it as something like (7÷3)÷(5×8)÷(2×8) well then shit... 😅
the biggest issue I encounter with PEMDAS is people think that that's the order. The acronym on its own implies that's the order, multiplication then division, and if you weren't paying that much attention in elementary math, you might get that answer. But even if you did, some people outright were taught that. And in higher level math it doesn't come up for... Well a few reasons. Primarily because you aren't doing simple problems but rather solving for variables or taking derivatives and the like. But also because, simply, there's no ambiguity. everything is clearly notated with parenthesis whenever there could possibly be any confusion. This can result in comically large numbers of parenthesis, but it gets the job done, and there's no ambiguity what you mean, no hope for misinterpretation
It can get reeeeaaally fun when you are dealing with non-associative operations...
Fraction doesnt exist in computer code, its all linear all the time. Brackets save code from ambiguities too
Fractions do exist though, but they are represented in an extremely strange format called floating point.
this entire video is essentially; "PEMDAS is bad because if you write an equation to be as ambiguous as possible, it becomes ambiguous." which, btw, has nothing to do with PEMDAS, so this entire video is pointless and stupid.
Conclusion: Teaching PEMDAS is basically teaching someone to ride a bike with too specific instructions
Americans teach bike riding that way. They consider it laziness if you don't just get on it and figure it out, and if a child falls and doesn't want to try again, very often their parents or older siblings will tell them they're being cowardly.
Actually, they teach math that way too. It has led to many children being much worse at math than they could be, into adulthood, because their teachers expect it to come naturally to them. Whereas a good educator in a sane system would actually... teach the children math. I learned mental arithmetic from the ground up as an adult, AFTER graduating college.
@@georgeparkins777 Anywhere else you'd be taught using instincts instincts
_echoes_
As a boy who aspires to be an astrophysicist one day, and with a love for math, I can’t wait to learn the actual order of operations in adulthood, maybe even now.
Went to college for an engineering degree and I was way too slow in math. I really busted my ass but I would never finish the tests. I took a break from school and my girlfriend that teaches preschoolers saw me do basic addition and she told me "wtf did you just do"? She told me I was using touch points and that really messes you up in the long run. It makes you a lot slower and hinders your ability to grasp basic concepts of numbers. I told her I never learned any other way. She showed me a more proper method teachers were supposed to teach and it's really difficult learning the basics again. Glad I'm getting videos like this in my feed now.
Wow. Just looked up "touch points". It seems like an unnecessary addition of a layer of complexity that would hinder the learning of both numbers and arithmetical operations.
Wow, no wonder I'm still crap at math, I grew heavily dependent on this method, I just listened and memorized the solving methods without questioning or comprehending why it occurs to begin with.
the method is NOT wrong. But the acronym is not the method. It is a simple phrase to help remember the method.
The reason I got 6/2(1+2) wrong is not because of PEMDAS or anything like that, it is because we've done things like 2(x+2) a lot in school, so when my brain sees a number before a bracket, it just wants to do 2x+2*2. Going back to the original math problem, it gave me that:
6/2(1+2)
=6/(2*1+2*2)
=6/6=1
And it’s proper answer. It’s 1. You did it right my friend. It’s 6/2(1+2) = 6/2(3) = 6/6 = 1 it’s correct and proper way of doing this since this 2 you see after „ / „ is also a part of parenthesis and need to be evaluated before dividing. If you write this equation as fraction it’s way easier to see it’s 6 / 2(2+1), and now you see that if you want to evaluate this division, you need first to evaluate everything after division sign ✌🏻
@admiralvirhz Ehh not necessarily. These kinds of problems are just created to spark debate due to the fact that it can be interpreted in multiple ways. If someone gives you division with the symbol instead of in the form of fractions, just tell them to fuck off lmao.
@@admiralvirhz "And it’s proper answer. It’s 1"
No the proper answer is 9
"6/2(1+2)
=6/(2*1+2*2)
=6/6=1"
Sorry, but this is incorrect.
If you are going to distribute, then the division is the same as a fraction, and you have to distribute the ENTIRE 6/2 (Six halves)
6/2(1+2)
(6/2*1+6/2*2)
(3*1+6/2*2)
(3+6/2*2)
(3+3*2)
(3+6)
9
Think about it this way...
x(1+2)
Now let x = 2/3 (Two thirds)
2/3(1+3)
If you only distribute the 3, you are going to get the wrong answer.
So
2/3*1+2/3*3
2/3+2/3*3
2/3+6/3
8/3
Simplified
2 and 2/3
You really want to die on the hill of not understanding what a convention is, and that it's not a divine law. Conventions depends on context and evolve over time. But sure suit yourself and go on answering every comment over this artificial little problem.
Please excuse my dear aunt Sally for being confusing.
Something I noticed is that people get multiplication by juxtaposition, but only for stuff taught in and after algebra. If you replace (1+2) with a variable, say 6/2x, and x=3, almost everyone I talked too gets 1. Once you start using parenthesis along with variables, say 6/2(x+2), and x=1, some people start getting 9. When you go back to the original 6/2(1+2), now a majority of people get 9. It is taught in school, but not "untaught" for math leaned before those classes.
Using the example of 6/2x with x=3, than the proper answer would be 9. The long version is 6/2(2+1). If broken down by some, to your example of 6/2x with x=3, than the equation would look like 6/2*3. The order for which requires the 6/2, now takes presidence over the 2*3. So, 6/2=3 which than becomes 3*3=9.
@@Technotranceism I think the point was that 6/2x was written using juxtaposition, so when you replace x with 3 it should be interpreted like 6/2(3) and in this case the 2(3) has higher precedence. It would be 9 if it was written like 6/2*x.
@@beginneratstuff Or...you could just write it unambiguously as 6/[2(2+1)] which equals...1.
@@clomiancalcifer yep, that's always the best way to handle stuff like this imo. It's not that hard to just add extra parentheses lol. I was just going by what the video said
I wouldn’t get 6/2x=1 and neither would a computer for that matter. Unless specified by brackets, the / sign only applies to the thing immediately thereafter. Use fractions or brackets but don’t pretend it’s a good notation to say that 6/2x and 6/2*x are two different things
Well there's simply a distinction between ÷ and /. The / is essentially implies a long bar, which itself implies brackets around numerator and denominator. The ÷ on the other hand has no such implication and just operates linearly.
The funny thing is that in my country, Hungary, they never teach acronyms. They just tell you "This is how you do it" and then that's how you do it, simple as that. That's why I can't get how can people be so hopeless in mathematics. I'm not saying there are no people hopeless in math here, but it's not at the level that you see in the US and similar countries... Compared to those people, the people that are hopeless here are would be math gods in the US, that's the saddest thing...
Haha, no. Math education sucks in Hungary too, and more people hate it than any other subject.
Also, teaching methodology is pretty bad. I heard teachers said stuff like "it needs to be memorized, not understood".
@@HA7DN Personally, I've always had good teacher, but even I've heard about teachers that make it miserable for everyone, plus, the only real thing I know about other countries and their education is what I see and know from the internet, so you may be right about that. All of this is really should be a debate for those who have experience in the field across countries, not a random like me
@@HA7DN Axioms fall in that category of "It's that way, because we decided it was". The real explanation is demonstrating that it's part of a set from which you can't add or remove an element without breaking it or being able to use one or more axiom to prove another one. Even in first year of university, it's not feasible.
You can still explain it like the above though.
The good thing is that there aren't that much axioms and most of them can be gently brushed over. As an example "1 is the successor of 0", well you don't have to really explain it mathematically. Students know counting and that's enough.
But (yes, always a "but"), at some point it becomes required to teach the internals of our positional system and what is a basis etc. As an example, most adults can't add in other radii than 10. That can be show with the following problem:
A tribe is using a positional system akin to our base 10 system. Instead of 10 symbols, they only use 3 symbols. Let's denote them 0,1 and 2.
Complete the following addition in their system : 12 + 22.
Answer is :
12 + 22
= 12 + (21 + 1)
= (12 + 1) + 21
= 20 + 21
= 100 + 11
= 111
Note how only predecessor and successor got used in the calculations.
It can also be done by using the addition table where 2+2=11.
So, we get: 12 + 22 = 10+20+ 2 +2 = 10 + 20 + 10 + 1 = 20 + 20 + 1 = 110 + 1 = 111
(All steps are done using a lookup[ table without even giving any meaning to the symbols)
That's why math are powerful. You start with something very concrete, abstract it, manipulate the abstraction with well defined rules, get a result and interpret it.
People think they start algebra after primary school. Nope, they learn the foundations in primary school and pupils are even asked to write "let A stand for Alice". They also complete arithmetic expressions with empty boxes like: 2 × □ = 10. It's kept very informal (and even wordy), because most pupils do not have yet a developed brain.
Then in secondary school, we can start being more abstract. In upper secondary school, abstraction becomes the norm. (manipulation of summations, including integrals and some basic group theory)
So, yes, we can't always explain everything and we shouldn't. At each stage of development it's part of mathematics.
My teacher taught it as GEMA (Grouping, Exponents, Multiplication/Division, Addition/Subtraction) it’s litteraly helped so much
Literally pemdas but okay. All syntax is lazy use proper notation or use brackets everywhere the only solution and stop trying to resell pemdas but better without changing a damn thing.
@@snintendog its not pemdas though? With GEMA youre saying that Multiplication has the same priority as Division and Addition has the same priority as Subtraction.
I was taught BODMAS at school, but they only really kept emphasis on it in the 4th and 5th grades, and they weren't serious about it either. The only thing they made us remember is that brackets come first, and after that, there wasn't really much on it.
Now, the acronym isn't even uttered in high school, many people just... forget about it, because we remember what to do first like the back of our hand, which is to do brackets first, rest can come later.
I only started to remember about BODMAS again due to people arguing over these problems, and that was the first time I've even though of it for over 5 years
I think the main reason why people start to forget BODMAS (I was also taught that) is because maths in higher classes has more emphasis on algebra and geometry, which don't really use BODMAS in their purest form.
real problem nobody wants to teach logic...
deadass, if people would just study formal logic, then it would help them greatly in math.
@@serai-xiv4293 DEADASS? is that a new PEMDAS?
@@JoRoBoYo no
in the 6/2(1+2) case, i always thought that the 2(...) was part of the parenthetic block, ie. distribute 2 to(1+2) before dividing 6 by the result of that section.
That is correct
@@pedrosso0 Under which convention? Under the juxtaposition convention he proposed -- where implied multiplication has higher precedence than explicit multiplication -- yes. But under the PEMDAS convention -- where implicit multiplication is no different from explicit multiplication, and multiplication and division are left-associative and have the same precedence -- no. Without context, there is *no* correct answer; it is simply a malformed question.
Edit: Apparently RUclips's markdown uses "--" as strike-through. Bruh.
Yeah when I saw it I figured the 2 and 1+2 are part of the same "unit" so to speak because of the space between the 6, the /, and the 2, if that isn't the case then it needs to be made more clear, like by ditching the spaces or making the 2 and the parenthesis part completely separate (so it becomes 6/2(1 + 2) or 6/2 * (1+2) instead of 6 / 2(1 + 2) since that makes it at least a little more clear that the 6 and 2 are part of their own fraction instead of the 2 being part of the parentheses instead)
Distribution doesn't take precedence over any particular operator, though. One could equally argue that the term to be distributed is 6/2, since that is what is outside the bracket.
@@chrishall9107 yes that's why the question needs to be written in a less ambiguous way so you can better deduce which operation is supposed to happen where
I always assumed the confusion was because people mistook the number connected to (multiplying with) the brackets as part of the bracket part of order of operations. my belief was that their first goal was to solve and remove the brackets completely. that's why i never liked writing 2(1+3) and would instead write 2*(1+3). because once the brackets is solved the equation is 2*4 instead of 2(4)
Thats because it is. The number connected to the brackets is part of the brackets, otherwise you couldnt factor stuff out of the brackets, and a lot of algebra breaks.
Scientists & engineers consider 2(1+3) as a single 'term' so resolving it completely [the P in PEMDAS] has priority in the order of operations. 2(3) = 6 Et voila! the brackets are gone, and now one can go about 'ordinary' multiplication, division, addition and subtraction. The problem is that not even the teachers understand the nuances of it, yet they're the ones historically who developed PEDMAS as a tool. And the programers of some (if not most) calculators have changed software to meet educator requirements, so now different calculators yield different results! The system is broken.
The problem isn't really the order of operations, but rather bad notation used to show the operations to be done.
If it is possible to be confused about how to simplify an expression, then the expression should be re-written.
In Singapore, I didn't recall learning about PEMDAS in school.
I've always been taught to calculate equations using juxtapositions ever since we were introduced to the idea of brackets in maths.
they taught you wrong then.
@@haroldgamarra7175 nah, you're are the wrong one
@@haroldgamarra7175 did you even watch the video?????
you can't calculate equations...
@@rampakeshbharat1938 You can calculate the value of an equation given numbers to plug into the expressions
The last line I fully agree with. I don't really see a problem with PEMDAS if it's taught on a rule of thumb basis. Math is language, the reason most of us are "bad" at it is because we don't get exposed to its notations later than our spoken and written languages. When we do pre-algebra and higher levels of math, we are parsing out how terms are related. If PEMDAS is taught properly, some intuition has already been built, and it's presented as an incomplete guide.
That said, PEMDAS usually works until it is weaponized. The memes where people argue happen because somebody has intentionally written an expression with ambiguous relationships as a gotcha game. Your average person with a high school math education, and maybe a couple of semesters of university math isn't getting tripped up as they go about life. When they need mathematical reasoning, they are usually relating concrete things that they've developed some intuition about separate from abstract math.
So I don't strongly endorse PEMDAS, I just feel that the hate it gets is a bit manufactured.
Thank you for pointing out why a^b^c always means a^(b^c) and why (a^b)^c is nonsensical as we might as well write a^(bc). I frequently point out this to students. Also I don't even understand why they claim parenthesis is an operation. I mean it's just a language syntax to inform a change of order of operations. It's not an operation in itself. When do you actually perform the parenthesis operation? Never :)
Regarding typing a formula outside of LaTeX or similar environment or whiteboards: I really strive to avoid writing so you have to assume juxtaposition priority above a previous division as although you can assume the reader will realize, it is still somewhat sloppy. It's easy to write stuff like 1/npq and assume the reader understand we mean 1/(npq) but the parenthesis really doesn't hurt and sends out a message that the writer genuinely care that the reader don't misinterpret things. In LaTeX I typically use the $\frac{1}{npq}$ even mid-text as it will not be too small as long as the fraction is not so complicated. If it's complicated you should write it on a separate or refer to an equation IMO.
I think one of the problems is that the teaching is mostly focused on how we ideally should write mathematics. But students and many teachers misinterpret it as "How you should always READ mathematics no matter what".
Another massive issue with PEMDAS is that it's beyond stupid to expect people to scan through an expression four times! Students will soon develop a intuitive feeling that PEMDAS is remarkably inefficient and get incentives to try and find faster ways to understand an advanced expression. But what can I do and what can't I do? Am I somewhat of a criminal if I find my own way of doing it? It almost seems like it in some educational system. It seems to mostly be a problem in USA to be honest. In Sweden we don't have this problem nearly to the same extent.
I typically teach students how to evaluate an expression by just doing a quick scan to get a sense what it's about, then start wherever they feel like. Left to right might often be a good idea if it feels right.
I mean consider a+b+c*d+Narnia (we all know it's a bit unclear what's inside Narnia ;). Of course you can begin by adding a+b=e if you like and multiplying c*d=f and even do e+f=g if you feel like it and then wander into Narnia and see what interesting adventure lies ahead. No matter what awaits us in Narnia all the operations we just did will still be useable.
PEMDAS will teach student to immediately wander into Narnia in an epic quest for finding parenthesis. Then leave the wardrobe, go for another adventure in search of the holy exponents (which they will interpret wrong in case of p^q^r). Then a third quest where they do some multiplication and division. Then after levelling up to a level 12 Paladin they finally perform that a+b they could have done the first time they saw it, not the fourth!
Also if Narnia now turned out to be 7*3+(10+5) they are literally taught to do 10+5 as the first operation even though it can be executed as the last operation of all! Unbelievable!
What's next? VANO for reading a novel? Reading it four time evaluating Verbs then Adjectives then Nouns then Others?
0:24 LMAO 69
This video makes me so mad but I know I’m not gonna win any arguments so I won’t say anything
Physics and maths lecturer here. 100% with you on the 'write your equations more clearly', and yes, the obelus must die. A mathematician or physicist would likely never use it in hand-written notes.
It’s interesting how we never even touched the general rules. The teacher sometimes just said that a certain operation comes before another and that was sometimes explained in a contradictory way. Now that I’m in high school, the order of operations I was taught is constantly failing me.
As an Italian, I had never seen something like this, in Italy they just teach you the order of operations in first grade with no acronyms.
Same in Poland
I was completely in agreement with you up until the end of the video, when your final arguments seemed to be completely unrelated to the premise.
The statement that there is no order of operations is correct in that you can form a mathematical expression in which operations are executed in any selected order. But an order of operations does not prescribe the order in which all operations must be executed, rather an order of operations is only meaningful relative to a given notation. Without a notation, it makes no sense to talk about any rules governing the order of operations.
Not all notations require an order of operations. For example, prefix and post-fix notations contain no ambiguity in their representations, but they are very hard for a human to read. Our standard "infix" notation has potential ambiguities which must be resolved. The only way to resolve this is by either forcing everyone to explicitly write brackets to remove all ambiguity, or to define a commonly understood method of resolving ambiguous situations. As you mentioned, simply hoping everyone will be good and write things clearly is not reasonable, so the only reasonable solution left without changing notation entirely is to define an expected order of operations.
By simply saying that we should teach the relationships between operations and that we should simply teach math does not resolve the ambiguity issue of our notation. As long as our notation has the potential to contain ambiguities, some method of resolving those ambiguities must exist, and therefore must be taught.
If I am misunderstanding your final point, please help me to understand it better. This has been a great video and a fantastic explanation of the problem with order of operations, so I would love to hear your thoughts on this final misunderstanding I'm having.
Thank you for the well thought out critique, you're right I didn't properly get across my final point.
I agree that "simply teaching the operations" is not enough to resolve ambiquities in the notation, it is important to consider. What I was trying to get across was that the order of operations is fundamentally just about notation and isn't mathematical in an of itself. One of the biggest problems with PEMDAS, in my opinion, is the way it's tought to students, especially those who aren't mathematicelly inclined in the first place, gives the impression that systems like PEMDAS are in some form "correct" and not just a convention we've decided upon. This can severly hinder their ability to do math since they rely on arbitrary rules, rather than an understanding of the logic behind the operations themselves.
That's what I was trying to say by "just teach the order of operations", that students would be much better equipped to do math if they learned the operations seperatly from the symbols and notation we use to represent them. I could probably have gotten this point across more eliquintly.
Also I did actually plan to talk about prefix and postfix notation by highlighting Reverse Polish Notation in the "solutions" section, but I ended up cutting it for time constraints.
@@interrogatix I see, so you're talking about making a more obvious separation between actual mathematical operations and concepts versus just the act of notating such things? That is clearer, thanks!
I believe that is actually done in some education systems such as Montessori schools. They teach the operations and concepts first with physical tools and then only later show the notation as things get more complicated. This builds more intuition about the operations first. Sounds like maybe what you're advocating for?
@@BlameTaw like teaching " one plus one is two, two plus one is three " and later saying "how do we write that? 1+1=2, 2+1=3"
Great video! I've been waiting for someone to make a video like this. I've been wanting to say the same. Order of operations does not exist in maths itself, in the sense that it's just a notation. When you follow the common notation, the idea of evaluating multiplication and division left-to-right is ridiculous, multiplication and division are the same thing! I really like how you explained this using GEMA at 7:36
Can someone explain to me how people manage to find 16 as result to the operation?
As a maths teacher we have banned the word “bidmas” in our maths department, teaching the correct order and equality of operations. However, I understand that students and most people need a method to help remember certain rules, as there is so much we require them to remember (from a syllabus we are given).
might I ask, How do you teach it?
banning the word "bidmas" is kind of ridiculous. Why would it be banned? It is just an acronym to help remember the order of operations. It isn't confusing if taught properly.
@@tchevrierwhy would you bother with a dumb acronym when you can just explain the rules? It's not like order of operations is particularly complicated
@@georgiykireev9678 because for some people acronyms HELP remember the rules.