Math Prof answers 6÷2(1+2) = ? once and for all ***Viral Math Problem***
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- Опубликовано: 1 окт 2024
- lol, am I really doing this? Ok, fine. There is a **viral math problem** about, uh, order of operations. You know, #BEDMAS or #PEMDAS. The most common form is 6/2(1+2) but it also shows up as 60/5(7-5) and other equivalent forms. What is the correct answer explained by a math prof? Sorry, I don't care. But I'm happy to share a few thoughts on why I think this issue repeatedly going viral says some things about societal views of mathematics.
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Ok, you ACTUALLY want my answer? I can't just clickbait you all and not tell you which I ACTUALLY prefer? OK fine, but I can see from the comments I'm going to upset a lot of you:D If I wrote this type of thing on the board, my natural inclination is to write division as a big diagonal dash instead that lumps the 2(1+2) on the bottom. That is, when I take this algebraic string of symbols and write it out - without using any brackets - the way I would write typical calculus expressions in my classes, then I would habitually write it in a way that use spatial relationships that interpret it as being 1. If I wanted it to be 9 I'd be explicit and put brackets around the (6/2), when writing on the board. Using spatial relationships (i.e. not a strict left-to-right application of BEDMAS) is extremely common in math, it's just that normally you don't have as your starting part a character string like this because, as I say in the video, the most important part is to be explicit about what you mean when there is a possibility of ambiguity!
I thought you explained it well in the video already- I'm honestly baffled that people continue to argue which answer is "correct" 🤷
@@NeoiconMintNet he most definitely explained but, maybe you didn't understand his explanation?
@@NeoiconMintNet How could he have blamed the question, that doesn't make sense. He didn't say "I don't get it and it's the question's fault!" he just said the equation is ambiguous. The answer requires more information about what is actually being solved.
@@NeoiconMintNet If you really want to know my mathematical standing, I never went beyond calc 1, and I passed it with a C only on the second attempt. What's yours
@@NeoiconMintNet Since you're bringing 5th grade math into this, that implies mathematical standing does actually matter to you.
I'd easily give this video a 6÷2(1+2) out of 10
It means 1 out of 10.
@@digambarnimbalkar8750 Nah, they gave this video a solid 9.
@@digambarnimbalkar8750 The question is ambiguous and badly written to modern standards so it is both 1 and 9 at the same time (depending on which interpretation you are using - academic or programming) which is the joke 😋.
If I wanted 1 I'd write 6÷(2(1+2)).
If I wanted 9 I'd write (6÷2)(1+2) or 6÷2×(1+2).
These would be unambiguous and the joke wouldn't work then and we wouldn't have the video either as there would be no discussion.
@@JustVezix Schrödinger's rating 🤔😋
In other words 6÷2(1+2)/10...?
If coding has taught me anything, just put parentheses around everything.
haha right? Computer programmers just don't have this issue:D
Especially with Smalltalk, which I don't think has normal procedural language precedence. I have programmed in C++ for a few decades, and I mostly know the rules, but as you say, when in doubt, write parenthesis, and people will say this in code reviews if they don't think it's intuitively clear.
The problem with all those extra parentheses is readability, especially with inline expressions.
..and that's how we got Lisp.
@@RemunJ66 exactly, that's why the Order of Operations and the various properties and axioms of math were formalized to eliminate ambiguity and to minimize the need for excessive parentheses.... Unfortunately some people have issues with following simple rules...
As another math ph.d. myself, my answer is simply, "I would NEVER write such an expression. And I don't think most mathematicians would write such an expression, either."
Indeed. Heck, I haven’t even used that symbol in at least 15 years!
@@DrTrefor BTW, thanks for all your great calculus videos! I've used them as supplementary viewing for Calc 1, 2, and 3 this summer and fall with distance learning.
Thanks for mentioning, always like hearing they are being used. Hope your students find them helpful:)
Well, it's wrong. The habit has become to write a number next to a parentheses, but between the '2' and the '(' should be an 'x'. No one uses divisors, but if you use them its... formatting that is only used to teach pemdas and in that -- very specific -- formatting, you are required to use every mathematical operator. This skips one, and thus we don't know what else it decided to skip. It's a "wrong" formula.
What about textbooks? I can pull examples from nearly any textbook (math or physics) I own that has a/bc in it, and you're supposed to interpret that as a/(bc). Yes, it's quite obvious in that context to interpret it that way, but I think that definitely casts doubt on the idea that mathematicians and physicists don't use implicit multiplication when writing symbols in-line.
This is not to say that one or the other is "correct", but just to cast doubt on your claim.
I'm 28 years old and just now learning I was taught PEMDAS wrong. For me it wasn't the parentheses that were the issue. Every math teacher I've had said you have to do the multiplication before division. I was never taught that they were on the same level, and we could just do left to right. If I did, they said the answer was wrong.
It is generally accepted that explicit multiplication and division are both on the same level nowdays. If it makes you feel better though, there was a time, back in the 18th or 19th century, when doing all the multiplication first was the more accepted convention. So you're not wrong per se, just a bit out of date... 😂
Mr Norman you are also correct so 6×3÷2=9
@@calebfuller4713fairly certain some teachers skip that part, like mine.
There is no left to right convention as the video presenter said. When on one line you have to use brackets to replicate both possible answers that the two line notation shows you. If you do left to right you can only ever get one of the two possible answers.
With 2 lines left to right is not needed of course, hence why no left to right convention.
@zakelwe I never said there was. I was saying I could go left to right. My point was that he said it doesn't matter what order the multiplication and division was. My teachers taught me the opposite (outdated way) so therefore there was only one answer with that method vs the current accepted way.
As a computer engineer, my instinct is to think of the 2(1+2) as similar to (1x+2x) which is "simplified" to x(1+2) and more clearly written as 6/(2(1+2)) = 1 - Rather use many brackets to provide clarity than leave the next engineer pondering what you meant
100%
My exact thought process thank you
As a mechanical engineer, I 100% agree.
As the son of an electrical engineer, I agree, too. 😊
It troubles me that *_so many_* people think otherwise!
You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol...
6÷(1x+2x)= 6÷(x(1+2)) NOT 6÷x(1+2)
6÷x*1+6÷x*2+6÷x*3-6÷x*4= 6÷x(1+2+3-4) as the LIKE TERM 6÷x was factored out of the expanded expression....
6÷(1x+2x+3x-4x)= 6÷(x(1+2+3-4) as x was factored out of the expression WITHIN the grouping symbol... You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol....
As a civil engineer, my instinct is to change that devision sign into a diagonal slash and get the answer 1 too. 😅
It’s the same, ÷ should not be used. But in essence ÷ = / = : Yes : is also used for division.. and it’s all the same.
@ Jose: Moreover, when you have implicit multiplication as a result of the 2 being juxtaposed right next to the (1+2) like that, anybody with any knowledge of math -- or at least, algebra and higher -- will treat that as a single, indivisible (no pun intended) expression. It's basically a ÷ bc (or a/bc), where a=6, b=2, and c=(1+2). And everybody knows -- or damn well _oughta_ know -- that a/bc is a/(bc) and *_not_* (a/b) × c.
@@Milesco no! a/bc = a/b*c!!!!!!
Shame on you how did you became civil enginner
@@trwent Because there's no need to? You're trying to treat it as if it's a hard rule when mathematical expression is more like a language. It's about how people interpret these equations because it's humans who reads them, and at higher level maths, people are either:
1. going to interpret implicit multiplication as having a higher precedence because that's just how it's pretty much always been done, and/or
2. Say the equation sucks and needs to be rewritten because it's ambiguous and nobody uses the obelus.
Very happy to see this nonsense described as a language problem and not a math problem. And I know my hard-science colleagues would throw a fit at the comparison to soft science; but when something is ambiguous in the English language the sentence is written in a different way. Thanks for the explanation that the mathematical expression should simply be written in a different way as well.
agreed, I'm currently trying to explain this to a friend and he's still refusing to believe that it's a language problem. and that onyone who views it the other way is simply wrong.
After all, math is a language, too.
There is still an objectively correct answer. It can be shown here: "6 / 2(1 + 2) = 6 / 2(3) = 6 / 6 = 1" because "6 / (1 + 2) = 6 / 1(1 + 2) ≠ (6 / 1) * (1 + 2)", therefore "6 / 2(1 + 2) ≠ (6 / 2) * (1 + 2)". There is no ambiguity because "n(m)" always implies "(n(m))" just like "m" implies "1m" or "1(m)".
Carl, I agree it is a language problem but maybe more..... For example, I just took my CASIO Scientific calculator [Model fx 82AU] and typed in the problem and it gave me the answer 9. I then took another calculator, CASIO Scientific calculator [Model fx-83GT PLUS], and it gave the answer 1.
The first calculator obviously is programmed to use PEMDAS and the second [same company different model] uses 'implied multiplication precedence over division 'Juxtaposition' (PEJMDAS)'. So, this means one person in an exam is getting the 'right' answer and the other the 'wrong' answer depending on the teacher's preferred answer/interpretation. This doesn't mean more than that, for two students of equal ability (but with different calculators) one gets a mark or two more/less in a test. A little unfair but I can cope with that. BUT....Now I have two nurses in a hospital, (with the two calculators I mentioned above) they calculate, via the formula given by the drug company, the dosage for a medicine. They both type in the exact same information, and one (even if she/he checks two or three times) calculates the dosage as 9 units, and the other that 1 unit is required. This is not trivial anymore. EVERYONE needs to be taught orders of operations in a consistent way that gives the 'right' answer. As a scientist I use PEJMDAS, but primary students are usually taught PEMDAS, and brackets are often not used if there is a chance of ambiguity. This, I feel, is the main reason why there is a problem - two (or more) ways of interpreting the same 'piece of language'. When does this first come up? In primary school So.... I feel it is very important that primary teachers are trained 'correctly', because it is here that this/these problem(s) are first encountered and can be tackled. Also, by doing this hopefully trust in our health practitioners, and calculator/computer company can be restored.
@@wrrsean_alt so this has been a very long ongoing and thoughtful discussion. What I find most interesting is that some people still believe there is an objectively right answer. With the calculator issue you've expressed there is to me an obvious time when people believed one way to be right and excepted it. Then some evolution happened and a new algorithm was accepted. What makes the version now right and the previous wrong? Also, usually I view math as an explanation for some process in the universe that the series or expression represents. And I'm not saying I disagree with anything or any ones point of view here. But objectively something seems to be changing in the foundations of math.
In my early years I was taught that the number preceding the bracket was part of the bracket - so 2(1+2) = (2*1) + (2 * 2) = 2 + 4 = 6. This was because I was taught algebraically that a(b+ c) has to have the brackets removed, so this becomes ab + ac.
Ya man . Now the tricky thing is identidying questions like this and when its (a+b)
That is correct. And a parenthesis isn't "solved" until you complete the multiplication or division of it.
All rules states parenthesis (or brackets) are to be solved first and foremost.
So, according to you, a(b+c) is the same as (a(b+c)). I was taught that only the contents within the parentheses are evaluated.
Sure, a(b+c) is the formula used to describe the distributive property but the expression of 6÷2(1+2) is composed of only one term and must be evaluated as such because terms are defined by the presence of addition and subtraction and not multiplication and division. You need to evaluate the entire context of the expression and not just part of it.
Also, the obelus (÷) does not imply grouping where what is before the sign is the numerator and what is after it is the denominator. That is the function of a vinculum or horizontal fraction bar where what is above the bar is the numerator and what is below is the denominator.
If you desire an answer of 1 for the given expression, you must add an additional set of parentheses.
6÷(2(1+2))=1.
@@Joe_Narbaiz a(b+c) is the same as as (a(b+c)) yes. The outside parenthesis is redundant since it is a regular + parenthesis and thus is solved as soon as you solve what is inside. Given there are no terms outside the parenthesis it offers no change.
Let's say you want the content to be the 6÷2×3 where 3 is a sum of 2 numbers, you will need to put in those extra parenthesis like (6÷2)x(1+2). Otherwise a multiplicative parenthesis will always take priority.
Actually use this quite often in economics, due to the fact that a lot depends on factors.
I was taught that there is no difference between 2(1+2) and 2*(1+2) and that it's just a shorthand way of writing it.
Ok which you all just sent this viral again:D
My understanding is that "multiplication by juxtaposition" is a separate step in the Order of Operations that comes before the "multiplication and division" step, and PEMDAS leaves it out for some reason; and that mathematicians, engineers, anyone who does math for a living, does the juxtaposition first and would solve the problem in question as 1. We really just need to clear this up by changing PEMDAS to PEJMDAS.
A lot would follow this rule, but it isn't actually a universally accepted rule of math. The problem here is that the mathimatical community hasn't bothered to settle this for a good reason. No matter what rules you make its always possible to poorly communicate a math problem. This is the same as saying when writing a sentence in english I can misscommunicate by using unclear verbs, sentence structure, or grammar. The point of mathimatical expressions is to clearly communicate an idea just like in any other language. Using ambiguous structures that can have multiple inturrputations is just poor math and you wouldn't find any formal math proof submitted for peer review using them. Math papers avoid the old division symbol because it had two different inturrputations over time. They also clearly communicate the term breakdown using brackets. This question and others like it failed to do that and that leads to multiple correct answers depending on inturrputation used.
PEMDAS leaves it out because PEMDAS is a simplified version of the order of operations that is taught to young kids. The real question is why the order of operations isn't revisited in the US after concepts like functions, multiplication by juxtaposition, and unary operators are understood.
The correct answer is 9
@@MrGreensweightHist The 'correct' answer is "fix your notation". 1 and 9 are both right and both wrong depending on if you respect juxtaposition. 1 ÷ 2x vs 1 ÷ 2 * x are two different operations.
I 100% agree! AND....the most important thing is bringing PEJMDAS to primary teachers/education authorities' attention. It is here that most people learn and take PEMDAS as being the correct rule without any other consideration. Even calculator companies need to be consistent. For example, using a CASIO Scientific calculator [Model fx 82AU] gives an answer 9 for this problem. While a CASIO Scientific calculator [Model fx-83GT PLUS], gives an answer 1.
The first calculator obviously is programmed to use PEMDAS and the second [same company different model] uses PEJMDAS. So, this means one person in an exam is getting the 'right' answer and the other the 'wrong' answer depending on a teacher's preferred answer/interpretation. This doesn't mean more than that for two students of equal ability (but with different calculators) one gets a mark or two more/less in the test. A little unfair, but this I can cope with. BUT....what if two nurses are in a hospital (with the two calculators I mentioned above), and each calculates (via the formula given by the drug company re the dosage) a medicine dose. They both type in the exact same information, and one (even if she/he checks two or three times) calculates the dosage as 9 units, while the other that 1 unit is required. This is not trivial anymore. Whether they learnt PEMDAS (or know of PEJMDAS) their trust in the calculator is sort of 'Russian Roulette' for their patient. We all need to become consistent. This is not a trivial misinterpretation of one way of looking at expressions compared to another, but an extremely important issue that needs attention.
The issue with these kinds of questions ls that many non math people have no idea wherre to start and where to end. To solve this question it is better to ask them what they mean.This is definitely not a math problem. It is similar to the match ptoblem where one asks about the displacement of the match to get a correct solution
Jesus Christ!... 3:30 🤦♂️just shoot me
What’s wrong?
@AndresFirte I am apparently. But you right. Just need to stop feeding into the dumbest social media feeds the brought me to this point 😆
Another PhD in math and engineering here. If any of us wrote an expression like that, we failed our education. Unless we walked into a bar and just wanted to start a bar fight...
Well bring it
What is 77 + 33
@@skiddadleskidoodle4585 "What is 77 + 33"
Easy, it's 7733.
However, we still understand 1/2x as 1/(2x), since if we meant it the PODMAS way, we would have written x/2. And if there is ambiguity, there is context to clear that up. Six letter acronyms are for children!
ruclips.net/video/lLCDca6dYpA/видео.html
Argument 2 wins for me, because of this: how you rewrite 1/f(1+2) as a fraction should not depend on whether f is a function or a number.
I'm with the 2nd argument as well. Since it makes more sense when you think about algebra.
Along with distributive property of Multiplication
@@manzanajoemerj.9849 distributive property doesn’t apply here.. 6/(2(1+2)) is distributive property which equals 1.. 6/2(1+2) isn’t distributive so the answer is 9
@@jshad1074 always do parenthesis first and open them...
Its argument 2
@@jshad1074 when you start to use / , I'd say any numbers come after that would be as one denominator
@@jshad1074
2(2+1)/6 = 1
do you know what does it mean when a result of division is 1? that the operators before and after the division sign are equal.
so 6/2(2+1) = 1, not 9.
I don't have to add any futile brackets.
I don't have to write 6/(2(2+1)) to get 1.
I didn't write (6/2)(2+1) to get your stupid 9.
could you fix your stupidity please?
Excellent. Thank you. Also, School math word problems are often written in the worst English grammar.
Sir.... You have solved a war in my house. Not in the way you think! You explained an issue with how my parents communicated with me in general! I did math differently with my step dad and how you explained the 2 differences explained to my logic prone step dad how I function and learned as a creative individual.
Thank you.
LOL, my wife is an astrophysics professor and I am an economist. She quite succinctly told me the error was with the person who wrote the original equation allowing for ambiguity to exist. Personally I think the law of distribution must be obeyed before we talk PEMDAS. There is more to math than just PEMDAS. Since there isn't an operator between the 2 and the (1+2) then you have to assume the 2 was factored out of (2+4).
She is right.
The question is badly written to modern standards.
ISO-80000-1 mentions about fractions on one line and how brackets are needed to remove the ambiguity now.
Back in the early 1900s this would not have been an ambiguous question but with modern programming it now is.
You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol...
You fail to understand the Distributive Property correctly. It amazes me how otherwise very intelligent people fail to understand and apply very basic rules and principles of math...
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is part of a single TERM...
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
@@RS-fg5mf stop spamming your stupidity
@@RS-fg5mf 6÷2 is not a single term like (1+2) is. By juxtaposition, it is the 2 which is multiplied by the bracket. "The How and Why of Mathematics" has a couple of videos on this topic where she looks at periodicals to see how professionals would approach it; they all use juxtaposition and get the answer to be 1.
@@shaunpatrick8345 you're wrong and so is she. Every example she gives is in the form of a/bc NOT a/b(c)
There is a distinct mathematical difference between 6÷2y and 6÷2(y) despite your misguided beliefs and subjective opinions...
6÷2(1+2) is a single TERM EXPRESSION with two SUB-EXPRESSIONS. 6÷2 is a single TERM sub-expression juxtaposed outside the parentheses as a whole to the two TERM sub-expression inside the parentheses 1+2
There are two types of implicit multiplication and they are not mathematically the same....
Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed (NO DELIMITER) and forms a composite quantity by Algebraic Convention... Example 2y
Type 2... Implicit Multiplication between a TERM and a Parenthetical value or across each TERM within the parenthetical sub-expression... Terms are separated by addition and subtraction not multiplication or division.... 6/2(1+2) is a single TERM expression with two sub-expressions. The single TERM sub-expression juxtaposed outside the parentheses as a whole 6÷2 and the two TERM sub-expression inside the parentheses (1+2)
In the axiom A(B+C)= AB+AC the A represents the TERM or TERM outside the parentheses not just the numeral next to it.
The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y.
This is an inaccurate comparison...
These two expressions utilize two DIFFERENT types of Implicit multiplication...
6÷2y = 6÷(2y)= 3/y by Algebraic Convention
6÷2(a+b)= (6÷2)(a+b)= 3a+3b by the Distributive Property...
All variables have a coefficient written or not. Constants can be coefficients but constants do not have coefficients. There are no coefficients in the expression 6÷2(1+2)...
6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b after simplification is 3 not 2
Correlation does not imply Causation. Just because both expressions utilize implicit multiplication doesn't inherently mean they are treated in the same manner...
The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them.
For people who argue 6÷2(1+2) and 6÷2y should be evaluated the same way, their argument is circular and is an informal fallacy that is flawed in the substance of their argument...
First of all, I agree with you. 6 ÷ 2(1+2) is poorly written. Besides the better way to write the problem that you included, there is another example of what this could have meant. That has to do with factoring. For example (2+4) If we uses variables first to put it into a format that is recognizable such as ab + ac how would you write this? You could write it as a(b+c) so to factor (2+4) to simplify it to the lowest prime numbers you could write it as 2(1+2) Using the distributive law, when you "solve" this expression you could follow PEMDAS and add the values in the parentheses together first (1+2) =(3) and then multiply the 2 outside the parentheses to get 6 or you could distribute the 2x1 + 2x2 and still get 6.
If you had an example of a factor a(b+c) and expanded the problem to include a division operation such as 6 ÷ a(b+c) what is the denominator? is it a(b+c)? If this is a factor, do you separate the variables a from the (b+c) before you obtain the value for the factor?
Is 5(7-5) actually the factor expression for (35-25)? If you had (35-25) how would you write it as a factor? in 60 ÷ 5(7-5) what is the denominator? If 5(7-5) a factor of (35-25) do you separate the 5 from the (7-5)? Why is there an implied multiplication operation between 5(7-5) if it was a factor? If you write a(b+c) can you call that a factor some of the time and not a factor other times? Would I have to read your mind to know when you consider a(b+c) a factor and when you don't consider a(b+c) to be a factor? If you didn't want a(b+c) to be considered a factor why not write it as a x (b+c) then there would be no confusion.
thata numbers right there
Im in 8th grade and that’s the exact same thing I thought but with other examples, I finally found someone that knows his stuffq
On this operation
That last sentence is exactly my argument against the answer 9. a(b+c) implicates the entirety as a factor that needs to be resolved first. Otherwise order it as a×(b+c) to separate the functions to 6/2 × 2+1.
Too much wordas for 1 math problem my friend
Yes, the real issue is to avoid ambiguity in communication. Example: The dog liked to guard the house and the postman could not make it to the door because he was barking viciously. AI may have a problem with the pronoun: does 'he' refer to the postman or the dog?.
Your excitement got me excited xD!
This is why whenever there is a viral question related to science or math, i would look for professionals answer..bcos there is too much unprofessional people answered this question and arguing as if they already finished the whole books of mathematics and start to be judgy towards other people opinions 😌
The problem is that if you ask to any professional mathematician about the problem in the video, the answer would be something like "I refuse to answer, let's talk about topological algebra instead".
I find this just to be really annoying. Why on gods green earth would someone write something this way? What is the point of writing anything down if it is not clear what you really mean.
It can be ether 1 or 9. I had one heated debate with that 1 million views person that pretty much says I never learned order of operations correctly. I keep saying that you don't seem to understand that you did 6 divided by 2(1+2) to become 6 divided by 2(3). You did what was inside the brackets, you still have to clear out the 3 first because of it being in brackets. Brackets are higher than division is on the order of operations.
I get both ways, I do see both answers but the 1 million views person just seems to be someone that is stuck with it always being 9 no matter what. Even if you can show that order of operations was still followed.
They didn't seem to understand that there are two answers to this because we don't know if 3 is being multiplied by 2 or is it being multiplied by 6/2. We simply don't know. Someone would never write it this way because of the ambiguity. It is always better to write out the fraction with one number on top of the other one.
The brackets part of order of operations is for inside brackets not outside. Once you have a bracket down to one number you effectively don't have brackets anymore. In this case, we have just multiplying 2*3 but the problem here is the 2 is dividing which you don't see unless you write the entire question.
Yes, use the horizontal line, not the forward slash.
It's funny seeing these sorts or problems appear on social media that use ambiguous notation, then watching dickheads argue about it in the comments. Calling each other uneducated, quoting whatever rule they learned in school, and using the argument 'I have a PhD, so my answer is correct'.
I forgot how much I hated math. Him explaining math to me is like the equivalent of a warm glass of milk.
Yuck.
You are not the only one. It's sad that most of the people don't have any clue about the wonderful mathematical universes that unravel, once these stupid problems disappear.
I agree. This controversy shows that society thinks of mathematics as a machine, full of operations and devoid of creativity. When in fact it is one of the most creative and beautiful fields, and requires extreme levels of ingenuity, creativity, and abstract thinking.
Exactly! I should hire you to be my script writer:D
@@DrTrefor lol 🤣🤣🤣
For most people, mathematics is a set of numerical expressions or questions, each of which (usually) has 1 right answer & many wrong answers (most of which, fortunately, are highly implausible). The goal is to find the right answer-or answers, for those comparatively rare cases in which there are 2 or more correct answers.
Math is a science how you use it as a function is an art but you can't change the scientific elements of the math. Smh!!
I indeed see Math as a complex machine with very specific rules, maybe because of my background (Software Engineer). So that makes me always see "6 ÷ 2(3)" as "6 ÷ 2 × 3", which is unequivocally 9. I can see the confusion on this being interpreted as "a ÷ bc" which, for what I understand, would be 1. HOWEVER, if you, the guys who really know this stuff, say it's ambiguos, then I believe you and I'm ok with that.
As a mathematician and an engineer, I love that this problem became viral because it shows the fundamental differences between rules and conventions.
You said it all and so few likes
but us folks for whom math has always made me feel stupid, i i need rules!
@@melissalynn5774The rule is called "#PEJMDAS": Parenthèses - Exponentiation - Juxtaposition - explicit mult/div - addition/subtraction.
Not only that, but following some rules and conventions over others breaks some of the arguments, imo at least. It's easy to confuse people with this type of notation because the results are usually integers...
But, if you apply juxtaposition before Order Of Operations then a decimal value can never be represented as its fractional equal without being inserted in brackets because the juxtaposition will enter in effect without applying it to the entire fraction, but the other part of the expression is already inserted in brackets.
Eg. 0.25(2+2)=x. You can, according to the concept of equality, replace the 0.25 for 1/4 or, since "/" is equally representative to ":" , as 1/4(2+2) or 1:4(2+2) .
However, in any of the latter two, by applying juxtaposition before OOO you will not get x=1 but x=1/16 if the fraction isn't in brackets.
But following OOO instead of juxtaposition 0.25(2+2) can be represented as 1/4(2+2) or 1:4(2+2) without any confusion.
That example can be replaced with anything similar, like 0.x(a+b)=y being replaced with 1/z(a+b)=y .
But we can't forget that 1 is also 2/2, 3/3, 4/4, 5/5, or x/x , and any (a+b) can be written as 1(a+b) or x/x(a+b) .
That is how I look at it, I don't know if my argument is valid or invalid since I'm not a mathematician though.
You are incorrect.
Basically the question is viral because it's a poorly written math problem. Nobody knows if the question means this (6 / 2) * (1 + 2) or this (6) / (2*(1+2))
100%
The correct answer is that YOU NEVER FRAME AN AMBIGUOUS EQUATION LIKE THAT. YOU HAVE PARENTHESES. USE THEM!!
THE EQUATION DOES NOT NEED TO BE AMBIGUOUS AND SHOULD NOT BE WRITTEN THIS WAY.
It is an expression, not an equation.
I think this problem is a bit more relevant than you make it out to be. For example, I can pull - from nearly any of my textbooks - an equation written in-line that looks something like a/bc. We are of course supposed to interpret that as a/(bc). Yes, it is obvious in that context what the correct interpretation is, but I don't think we can have the attitude of "I don't care" when expressions like this are written frequently in textbooks and they MUST be interpreted a certain way.
I think a better answer would be that the "correct" interpretation depends on the context, but I believe that was implied in your video anyway, so I'm probably nit picking.
Love your content! Your vector calc visualizations are amazing.
I agree that the problem is just that there is no context. a/(bc) is probably the more useful interpretation for a/bc but these textbooks kinda suck then as our textbooks were unambiguous and wrote fractions vertically when grouped together. When using standard text signs I always Parenthesis in abundance. I also agree that maybe a debate could be interesting but fundamentally the point of the video is that the equation isn't written correctly or consistently which is why there is no need to come to a conclusion when the input is the problem.
There was an explicit choice to NOT include a multiplication sign but they included the division sign in the original problem so it strongly suggests that the right side is the denominator and the answer is 1.
The problem is what if this actual problem shows us on the test. We all know test are there to be tricky
@@Jry088 I have seen problems like this written in text books although with a / instead of a ÷. In those cases it's usually to save space because they are trying to get the whole thing on one line and then in that case the right side is the denominator. I would not expect a trick. Also if I saw this in a notebook found somewhere I would guess the author left out the multiplication sign because they want the whole right side to be solved first otherwise they would have written X or * just like the wrote ÷ on the other side. Not writing X or * would be inconsistent with the style unless they meant it to be a denominator so if found in a notebook it would be very safe to assume the right side is solved first.
@@nickjunes that's why 1 seemed so obvious to me also. At least the way I learned a(b+c) is all included in the P in pemdas. Otherwise it would be easily separated.
I think the confusing part is the use of the parenthesis without the explicit * sign, so the problem is not 6÷2*(1+2) which would unambiguously be 9, given BODMAS and L to R execution. To examine further, , let us put (1+2) as x, so the expression is 6÷2x which is not the same as 6÷2*x. Although we normally think of 2x as 2*x but in the context of 6÷2x, 2x would mean 6 and the answer would be 1. I do think the expression is ambiguous and the author must rewrite it as (6÷2)(1+2) if he wants 9 to be the answer.
pemdas vs the bs
the problem is indeed the implicit * sign
No bodmas says it is 1.
B is for brackets, so in 6÷2(3) you have to calculate brackets first, aka you get 6÷6. Now all of your reversals works aswell.
@@kimf.wendel9113 The 2 is outside the bracket. If it was 6÷(2*3) no confusion would arise.
@@manzerm7805 yes, and that means the contents of the parenthesis is shortened by a factor. And to remove the parenthesis you need to multiply is expression inside.
All logic in maths says you solve the parenthesis first, that is why the first letter in those order of operations starts with a that. It doesn't matter what is inside, you solve it first until there are no parenthesis
At last... common sense. Agreed communication is key here.
So basically, both answers are correct. It's the question that's wrong. Just a sloppy set up
WRONG
Definitely wrong, there are a bunch of questions like this in my text book. Here in Australia.
@@Kage-jk4pj can you post pics of your textbook so we can see what it says...
@@filename1674 No you can't. 🙄🙄🙄
@@RS-fg5mf Argue with a maths professor on this one? You are unbelievably up your own rear end
What I find interesting is that everyone here looks at math as a intellectual exercise rather than a representative of real-life. A math problem is short cut to understanding what will be required. I.e. I have three pies and six people. 3/6=1/2 everyone gets a half of pie to take home. This can be related to this problem. I have three pies being distributed between 2 families each with 2 parents and 1 child. So how many pies does each person get? I can't think of any situation where we start with three pies and divide them up and get nine pies as an answer. If anybody has a real life situation to explain the 9 answer I'm listening.
I have 6 boxes to be distributed to 2 people. Each box contains 2 apple pies and 1 cherry pie. How many total pies does each person get? Each person gets 9.
It's not so much an intellectual exercise as it is just one of the wonders of math when writing an expression or an equation in a single line format.
A division followed by a multiplication, is according to the associative property, non-associative. It's the same with successive divisions, with a subtraction followed by addition, and successive subtractions.
Add in no universally accepted convention
and you have 10 plus years of internet arguing.
@@phoenix2634 thank you for your reply, but wouldn't that problem be written 6(2+1)÷2. Due to the fact that the 2+1 is referring to the boxes and not the people?
@@joeguadarrama3523 eh, that's one way writing it. Although, you're still at some point dividing the number of boxes between the 2 people for 6÷2 multiply that by the pies in the box (2+1). If you've learned the convention that gives multiplication and division equal priority, there's really no reason to not write it as 6÷2(2+1). If you've learned another convention (implied multiplication is given priority) than yeah, I'd probably write it as 6(2+1)÷2 (If I had to write it in a single line format).
Of course having advanced beyond elementary school math, regardless of the type of real world problem, I'd write it out as 6 over 2 if I wanted 9 or I'd write as 6 over 2(2+1) if I wanted 1. Or, if forced to write it in a single line format I'd use parentheses. No point in making it ambiguous.
Thanks for giving me a chance to think about this type of problem and how I'd approach it.
@@phoenix2634 so I tried something that seems to prove that well...yes...9 is the correct answer. I tried to solve for "b" for 6÷2(2+b)= 1 and then 9. Only the 9 gave me the answer where b=1 so it looks like 9 is the correct answer (even though I didn't like) but hey looks like I've learned something, despite my best efforts.
@@joeguadarrama3523 That's because 6÷2(b+2) is still ambiguous notation and you can't prove anything this way because it's a notation issue.
You can show b=1 both ways.
6÷2(b+2) = 1 with the Academic interpretation:
6÷(2(b+2))=1
3÷(b+2)=1
3=b+2
1=b so b=1
6÷2(b+2)=9 using the Modern interpretation:
6÷2×(b+2)=9
3×(b+2)=9
b+2=3
b=1
You can prove b=1 both ways.
It's an ambiguous question and badly written.
I use equal precedence when explicit * is used, but give implicit multiplication higher precedence.
You talk a lot, now get to it~
The best I've heard it explained is that even after reducing 2(1+2) to 2(3), you still haven't dealt with the parenthetical expression. In other words, the P of PEMDAS still isn't finished. And by restructuring the equation as (6/2) * 3, you've changed the equation entirely. Instead of distributing the 2 throughout the parentheses to satisfy the P, you've just kind of removed it. Instead of turning a(b+c) into ab + ac like you're supposed to, you've changed the equation to (1/a) * b + c.
TLDR: The multiplier of the parentheses must be distributed to satisfy the P in PEMDAS.
Except that P is for inside parentheses only.
Juxtaposition is either a separate step after Exponents, like in PEJMDAS, or it's a notation convention that needs to be interpreted and written explicitly before you start to simplify at all.
Easy to show with
3²(4)
If the P step is still present, how can you do P before E here? What's the next step?
It's bad teaching to say outside parentheses is part of the parentheses step.
@@GanonTEK this might actually be hard to understand because the answer to 3²×4 and 3²(4) are the same but the way they are calculated is different.
3²×4 = (3×3)×4 = 9×4 = 36
3²(4) = ((3²)(4)) = ((3×3)(4)) = ((9)(4)) = (9×4) = (36) = 36
This becomes more obvious if you begin with 3²(2+2) instead of 3²(4).
3²(2+2) = (((3²)(2))+((3²)(2))) =
(((3×3)(2))+((3x3)(2))) =
(((9)(2))+((9)(2))) =
((9×2)+(9×2))
((18)+(18)) =
(18+18) = (36) = 36
The thing is 3²(4) can be calculated as 3²×4 = 9×4=36 but if it were to be part of a bigger equation 3²(4) doesn't become 3²×4 but (3²×4).
wow you really didn't get the video you just watched start to finish huh
I would never write down an ambiguous expression like this and expect my readers to figure out what the hell I meant. Don't be a fool, keep your cool, use parentheses.
100%
I liked your points at the end on how society views mathematics. Would love a whole video dedicated to that!
This is actually a great idea and a BIG topic imo
ruclips.net/video/eLMccl_z9Xg/видео.html
Viral Math Equation 6÷2(1+2) = ?
Watch this video for answer - ruclips.net/video/zqXvBLXw5Tc/видео.html
How society views math doesn't solve the problem.
You're all wrong, because the answer to everything is 42.
Don’t forget your towel
Applying the math I learned, I came up with the answer 1. Which I find is more aesthetic than the answer 9. Math CAN be beautiful...
But I agree that the way it's been written does make some dissension; one can not have two answer to a math problem.
aesthetic is also related to culture and convention; so the result being 1 is at least as much the result of convention as it being 9.
Math convention dictates after parentheses, one should follow left to right; multiply and division are deemed 'the same'. I do not agree, I'm old skool... @@valdir7426
Your answer is 1 just because you find it more aesthetic? Are you friends with your brain?
I am more inclined to use my right brain hemisphere, but occasionally my left one kicks in...@@universe25.x
A zillion years ago when I actually did math, I had an RPN (reverse polish notation) calculator. I think using both helped solidify the relationships in my head. At the time I really thought RPN was superior, but had limitations. You had to think to decide which order to type things in. This thinking gelled the thought process of how the numbers related to each other.
I think many math students could benefit from learning RPN as a side project.
I would often do a problem with both, and if my answers disagreed, it let me know that I had some more thinking to do.
I really like how you described this as an English communication problem. Bravo.
People: ITS 9! ITS 1! ITS 9! ITS 1!
me: Its both
It's 1.
Ask any physycist or engineer if 9 is the coorect answer and you wiil receive am emphatic No. The expressiin, a/bx is always understood to be a times the inverse of bx.
I needed this video when I was in school 18,000 years ago, for my high school teachers. I hated math, and to this day still struggle with it. Don't get me started on comprehension questions!
Dr. Bazett, I really appreciate your comment about making sure that we write math problems in an unambiguous manner. This applies to many different aspects of business today, such as contracts, reports, articles, and much more. The biggest problems I have encountered in business have been related to this specific matter, ambiguity. Thank you for this video.
The thing is the equation is written correctly. If it wanted to be a different equation, it would be written differently. Everyone who got 9 as the answer has changed the equation itself. You don't change the equation to match your answer. You solve the equation to find your answer.
On the board or in the book, I agree. When programming, it comes out 9 - c language, python, fortran, octave, libreoffice...
My whole life has been a lie
=1
Could be.
Irl in scientific community(at least math and physics) it would be interpreted as 6÷(2(1+2)), but in science we almost always initially do the algebraic transformation till we get the answer(here it would be a÷b(c+d) ), and only then substitute numerical values. Plus, as the author said, horizontal line is almost always used instead of "÷" or ":" symbols.
I agree, this problem needs clear notation.
i am moko and think it could be written better however it is correct in it definition the answer is 1
I appreciate and agree with both:
-your very valid point--BE UNAMBIGUOUS. BE CLEAR in the math you're writing--especially when you involve division, because it changes the result so dramatically depending on what you mean!
-your pinned comment where you default, as I do, to interpreting this as taking care of 2(3) before the division. Though it is nonetheless, as originally presented in the problem, an ambiguous expression.
As in most things in life, good communication is essential.
I think the big issue is calulators need to interpret this badly written formula (because people will make bad formulations) the same abnd they don't.
TIL you need a PhD in mathematics to solve a first grade level math problem
Well in 5th grade, you can say 9. But if you want to publish with the intellectuals, The Physics Style Guide says 1
Mathematics is all about presenting ideas in a symbolic form to make abstract and complex ideas simple.but those symbols should be clear. If the symbols used are ambiguous then you are presenting it in a wrong way.
I love that you show better ways to notate this to remove the ambiguity. I think the problem is that math is usually taught as math in a vacuum. The "right" answer is what real world application you are calculating. Instead of focusing on how it's written, the real answer should be to go back to the real problem to clarify the question. Arguing over a poorly notated equation is just silly.
100%
I "liked" this video, because it cuts through much of the BS found on so many other videos discussing this "problem of operational order" to observe that the problem is ambiguous statement. The video gets an "A" for cutting through the chaff, but a C for the work to get there, so a B overall. On a bad day, that would have been an A, a D, and a C+ overall.
Why this grade?
I couldn't tell until several minutes in that the video was in fact all about the need to clearly state problems. That should have been the first thing that was said. And there was much too much mucking about with the details, rather than decorating the main point with some details.
If it were me doing it, I'd probably have started with the good math is good communication angle, then moved into a statement showing how to unambiguously state the problem, next showing how the other videos continue to deal with ambiguous statements, finishing up with a reiteration of the unambiguous statement, with a conclusion that the other method is a waste of time, because it doesn't address the real problem, much less address it. In other words, start with showing how to do it right before talking about how it's done wrong, and finish with repeating how it's done right.
This was the best explanation for this problem I've heard so far. Essence: Don't write your problem in an ambiguous form!
Except it's not ambiguous to anyone competent in maths. The answer is 1, and that's it. All other answers are wrong.
@@peterthomas5792ion see your PHD so ur wrong
With no multiplication sign, the only indication that (1 + 2) is multiplied comes from its being grouped with 2.
I’ve been caught up in this debate every time it pops up on Facebook. My argument was that a college level math teacher wouldn’t write a problem on the board like 6 / 2(1+2). Instead it would be written like 6 with a line under and then 2(1+2). We would instinctively tackle the 2(1+2) first to simplify and then end up with an answer of 1. But the angry comments yelling at us about PEMDAS strongly disagreed.
Pemdas says it is 1, P stands for Parenthesis.
To solve a a(b+c) parenthesis you end up with ab+ac. So 2(3) is not solved, it is shortened, 2x1+2x2 is the solved state which is to be reduced to a 6.
The best possible take
You guys should remember when u change division to multiplication u can take the reciprocal of dat number so d answer is 9
THIS VIDEO SOULDVE BEEN 10 SECS LONG BY SIMPLY SAYIN THE ANSWER IS 1 OR THE ANSWER IS 9 😂😂😂😂😂
As a fellow PhD, I have been presented this problem a number of times. Initially I was in the hard lined order of operations, but the more I revisited the topic, I started noticing the number of examples of when PEMDAS is overridden without confusion. (e.g., cos2x and 1/2x) Now when presented, I go into the ambiguity of the expression should have been addressed by the author and not the reader. A good analogy is the importance of being aware of removing any potential ambiguity when writing a sentence involving a list and not using the Oxford comma.
I went to a birthday party with the strippers, JFK and Stalin
PHD in WHAT, you clown?
There is a slight argument that scalar multiples may be interpreted that way but even then it's ambiguous. But when all symbols are in the same general realm (being variables or numbers, but all the same) - that argument goes away. And even then, it's just bad form to create ambiguity and virtually all math people... scratch that, people that use math to communicate e.g. +physicists etc. - would write it in an unambiguous way
@@ibarskiy Exactly. I have a Bachelor's degree in Mathematics and what I usually tell people is that if I had written a formula like that on a test paper where I was showing my work, I would have gotten points off for writing something so ambiguous
@@ibarskiy Just repeat 5th grade, and this time, stay awake.
2 more examples of strange notation:
-cm^2 instead of (cm)^2
-sin ax + b instead of sin(ax) + b and the sam with other trig functions, log,...
sin^2 x instead of (sin x)^2 but sin^(-1) x is to be interpreted as arcsin x. I also never liked is the use of superscripts notation to represent something other than powers and exponentiation.
Yep. Good examples there.
The cm one is interesting because many see cm not as c×m but as the word "centimeter". It probably is c×m though really because of the meaning of the prefix centi-.
We don't have centi-square meters really so that's probably why cm² is fine.
I like your comment.
@@GanonTEK It's not c times m. Otherwise mm would be m squared.
@@maxxiong mm means milli-meter though, the two ms don't mean the same thing so is not m².
They aren't variables.
Milli- means 10^-3
mm means then 10^-3 meters
@@GanonTEK As a matter of convention, when you typeset a symbol in upright font, the entire word is one variable. So I don't need to write (cm)^2 just as I don't have to write (score)^2.
the division sign ➗ is reserved for first graders and R&B bands. i have never seen this used in algebraic expressions for a good reason, it is ambiguous and poor form. thats like saying that 11 = 1 ( implying a multiplication sign between the digits)
The answer is: The question is POORLY written and vague. The author should write it clearly by including parentheses and not encourage nonsense.
I'm sorry but I never see an engineer resolves a/bc as a/b*c but always as a/(b*c)
lmao
galileo.phys.virginia.edu/classes/609.ral5q.fall04/LecturePDF/L18-ENERGY.pdf
Page 4
W = 1/2m(at)² = 1/2mv²
www.tcd.ie/Physics/study/current/undergraduate/lecture-notes/py1h01/Lecture_Mech_5.pdf
Page 23
1/2mv(f)²
www.citycollegiate.com/workpowerenergy_Xc.htm
K.E = 1/2 mv²
@@Araqius lmao
It's a formula. It's understood as ½mv² not horizontally. No engineer will put it that way, unless ofc if it's just a note.
@@Bakadesu-
In physics or engineering, there is [unit].
@@Araqius Then I guess I'm not an engineer then. Lol
6÷2(1+2) does NOT translate to A/BC
The correct evaluation when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly is 6÷2(1+2) translates to A(B+C) where A is equal to the TERM outside the parentheses not just the factor next to it...
A= 6÷2
B=1
C=2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
6÷2y = 6÷(2y) = 3/y by Algebraic Convention
2y is a directly prefixed coefficient and variable that forms a composite quantity BUT 6/2(y) = 3y
Stop confusing and conflating an Algebraic Convention given to coefficients and variables that are directly prefixed to parenthetical implicit multiplication... They are NOT the same thing...
ABC÷ABD = C/D by Algebraic Convention
ABC/AB(D)= CD
The TERM outside the parentheses is to be multiplied with the TERM or TERMS inside the parentheses.... TERMS are seperated by addition and subtraction not multiplication or division.
I came here for math and learnt philosophy
I never thought of this problem this way but you are right. The problem was thrown out to create a little controversy because the originator understood people could and would come up with 2 different answers and both would be correct because enough info was not given.
It's more insidious than that.
It was created not to edify, but to explicitly be ambiguous and to drive "interactions" on a given FB page or Tweet.
The idea is not to arrive at a "correct answer" [none is given, and no winners declared]. The idea is simply to create drama and dissent, which leads to more clicks, more page views, more comments, and arguably more reputation for the page, and thus possibly more monetization, etc., in some form or other.
They're not here altruistically to teach people anything, but to sow discord and make money off of it, whether driving clicks to other pages / sites / videos, or growing some subscriber base and then selling the page to some new chump willing actually pay something for it for some unknown reason, with a built-in subscriber/liker/follower base that can then be advertised to or whatever.
@@MGmirkin Exactly right, the authors of the videos saying the answer is 9 are doing it for money, despite the harm that they do to society. It is shameful.
No some people just forgot what they learned i school and got confused. As such they turned to social medias to verify they weren't the only ones to forget how math works.
Then more fot confused becuase they were in doubt aswell, and then a confusion spread.
@@kimf.wendel9113sometimes that's the case, but different people have been taught differently as well. Some people have been taught that multiplication by juxtaposition has priority over other multiplication and division and some were taught that it's bo different than any other multiplication.
@@Andrew-it7fb that doesn't mean the latter group is right. If they were taught that + was "divide by" there would not be an additional right answer, they would just be wrong.
Normal people use either fractions or division symbol followed by bracketed expression. No ambiguity in that.
Finally found a sensible answer to this kind of stupid questions 😭! Thank you!
Yes indeed.
It's just ambiguous notation.
Academically, multiplication by juxtaposition implies grouping but the programming interpretation does not.
Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity.
Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too.
Multiple professors and mathematicians have said so also like:
Dr. Trevor Bazett here, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter etc.
Even scientific calculators don't agree on one interpretation or the other.
Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why.
It's just a really poorly written expression written like that on purpose to be misleading and go viral. It's a trick. Anyone who thinks there is only one correct answer is simply wrong and doesn't understand the situation.
Society would lead you to be lazy and interpret 2/xy as 2/(xy). Any formulas you deal with in the future applies this implication too. The only reason why this ambiguity hits so hard is cause this is primary school math. That’s why you see the “old” vs “new” method. There is no old vs new. It’s just kids haven’t learned math past 5th grade yet
Nothing sensible about it... He simply neglects to point out the actual understanding and application of the Order of Operations and the various properties and axioms of math as intended...
When you actually understand and apply the basic rules and principles of math correctly as intended you get the only correct answer 9
@@jjh7611 No... You failed 5th grade and confuse and conflate an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity by this convention to Parenthetical Implicit Multiplication. They are NOT the same thing...
From another Ph.D. in mathematics: Thanks for doing this video. These types of problems are pointless. Those who have memorized orders of operation rules get an answer consistent with those rules. Those who haven’t memorized those rules get an arguably plausible answer. If you are entering an expression into a poorly designed calculator interface or writing an exceptionally complicated expression for a program, then order of operations rules must be clearly understood, but these situations should be avoided as they are error prone. Break the expression up into two or three lines. Get a calculator with a postfix user interface (6 2 1 2 + * /). I learned the order of operations rules in high school and have rarely used them since. Mathematicians have a knack for writing expressions so that they will be clearly understood without even thinking of rules for order of operations. There is beauty in a well crafted expression. Programmers will improve the clarity of a computation by expressing it in two or three lines.
It seems logical that the use/or not of symbols has implications. I know no-one can be definitive about implications
To me, a number hard up against a bracket implies connected as against the 'partition' that a symbol would imply.
This is indeed not even a problem in math. It is a problem in language, or on convention, and that has nothing to do with the math. I am a math teacher, and I just ask to the students to be careful on ambiguous expression, and to add parentheses in case we might doubt. Parentheses convey also better the functional relationship in general, and are good for the beginners. Then conventions and abbreviations are easily learned consciously by practice. Things like PEMDAS are nuisible for teaching math. That's perhaps why some ignorant philosophers believe that all mathematical truth are conventional, and that math is fiction (which I think is non-sensical).
I agree with the points in this video. This channel is so underrated though.
Thank you!!
I didn't realize this was a thing. Myself (American) and my British classmates surprisingly had different answers and I did not understand how when I learned it clearly one way. Turns out there's a different method
great take, math is a language with its own symbols
And you didn't even address the "elephant in the room" which is that these "viral math problems" are **intentionally** written ambiguously **in order to stir controversy** and drive clicks / likes / interactions (comment). No actual answer key is given, and there is typically no "right answer" for that reason. They're not actually looking for a right answer, just driving clicks and some putative monetization scheme or whatever.
And that is the even bigger problem that seems to never get addressed.
I wish that every single one of those posts on FB had a giant button for "report click-bait/interact-bait" which would get the post removed and give the page a strike. Too many strikes and the page gets suspended or deleted. And, let's be real, most of the "pages" forwarding this stuff are simply garbage pages with no real use in the real world. They exist only to exist and to keep churning out this garbage, and maybe making the owners money in some nefarious way. Either by selling the page to someone else once it has some kind of userbase of 'subscribers'/'likers'/'followers' or by driving views someplace else (youtube videos, some web site, etc.).
Frankly, they should all be banned for shady practices.
As to interpreting it as written, my inclination is to view:
6[div]2(1+2) or 6/2(1+2)
as implicitly different from
6[div]2[mult](1+2) or 6/2*(1+2)
Due to the way it's constructed.
Primarily for the reason that when one see something written in the form x(y+z), we tend to see or think of it 'distributively' as equivalent to (xy+xz), or likewise x(x+y) as (x^2+xy).
So, like if I saw 6/x(y+z) rather than, say (6/x)*(y+z), I would tend to see/interpret it as more equivalent to 6/((xy)+(xz)) than to ((6y)/x)+((6z)/x). That may be wrong. But given how we **factor polynomials** and such, that's where my brain goes to. When you have a number or letter directly outside of a parenthesis, you distribute that number or letter over the contents of the parenthesis. So, x(y+z) becomes xy+xz, and 2(1+2) becomes 2(3) becomes 6. IMO, part of dealing with "parentheses" is dealing with distributing anything "over" them that is immediately adjacent and implicitly **part** of the parenthesis (barring any other explicit parentheses grouping the adjacent character with some other symbols/logic).
As you say, if they'd put an explicit extra parenthesis to read (6/2)(1+2), that would make the order of operations more explicitly clear:
(6/2)(1+2)
=(3)(1+2)
=(3)(3)
=9
Without the extra parenthesis it's far more ambiguous, leading some folks to believe that the extra parenthesis were **intentionally** left out, not grouping the first two items together, and thus the latter parenthesis and adjacent number should be viewed distributively.
But, again, this goes back to the whole elephant in the room, which is that it's **intentionally** written **ambiguously** to foster exactly these kinds of ambiguities/arguments, not actually to **edify** anyone (nobody ever actually come on after the discussion to say "here's the correct answer, and why; congratulations to the people who got it right!"), but to start arguments online, for clicks & $$.
good explanation, I never disliked math I was always challenged I loved it. I always found out a different way to solve equations and I was always told "no that is not how we learn it" was never taught away from a systematic perspective. However I did not care I always went against teachings, the schools I attended definitely does not show this side of math.
You can evaluate this expression at least 6 different ways but you still get the only correct answer 9
RS can get his head dropped 1 or 9 times as a baby and he’d never graduate elementary school
A curious person like you would be amazed from the mathematical universes that unravel, after these stupid problems, equation, and expression disappear
Hi, can you consider my solution?
from 6/2(1+2),
= 6/(2+4),
= 6/(6)
Therefore, 6/2(1+2) =1.
I reverse the process for checking the equation 6/2(1+2) =1.
to prove that x=2 and satisfying the equation 6/2(1+x) =1.
6/2(1+x) =1
6/(2+2x)=1
6=2+2x
Then , 4=2x
x=2
Therefore, 6/2(1+x) =1
I would like to know how to prove the equation 6/2(1+2) =9? because the division sign is confusing to the process, and i think it will give us an idea of how the equation should be express instead.
Nevertheless, my point is that math concepts must be applied in reality.
So, i will choose the process that is useful, especially in our profession/career.
PEMDAS 6/2(1+2) parentheses first (1+2)=3
Now the equation is 6/2(3) or 6/2×3 then you go left to right so you divide first 6/2=3 so now the equation is 3×3 now you multiply 3×3=9
Or to simplify 6/2(1+2)
6/2(3)
Or 6/2×3
3(3)
Or 3×3
=9
Something people don’t understand is how the order of operations works. When we say division and multiplication or the other way around, it goes left to right regardless. Example: 10 / 2 X 3, the answer is 15, because 10/2 is 5, and 5 X 3 is 15. The same applies here. Let’s remove the bracket and put 3 in its place: 6 / 2 X 3. You do 6/2 first because it is before anything else. 6/2 = 3. Then 3x3 = 9
Their actually is rules for mathematicians by the American mathematical society because book publishers follow rules. And when computer languages process symbols unless you want to create additional halting problems you have to have a default order of doing things. You can always add parentheses and brackets for clarity.
In algebra, they usually talk about identifying the terms and then the associative, distributive, and commutative properties when they ever talk about PEMDAS. Also, the student would look at the division symbol as a slash. If the constants are rewritten as a, b, c, d, then it will be a/b(c+d). If you do the parentheses first, multiplication second, and finally the division, you will get 1, which is all could get taking PEMDAS literally. The problem would have to be written as 6(1+2)/2 to get 9.
Everybody agrees what the result would be if we were to take PEMDAS literally. The question is if we should break with tradition, rewrite the text books and start taking PEMDAS literally. Who died and made PEMDAS king, suddenly? PEMDAS is a simplified mnemonic for teaching the order of operations to children.
Best answer ever.
Yeah. If you want to last in the college environment, you can't ruffle feathers. It's taught wrong by his fellow educators, like the model of the atom, as a solar system. It's not until college they really teach what an atom looks like. Not a solar system. If he were to publish that equation, The Physics Style Guide would say the answer is 1. They do the implied multiplication over division. Here's the likely context too:
1 = 1
1 = 6/6
1 = 6 /(2+4)
1 = 6/2(1+2)
"So what do you, as a mathematician, think?"
"I do not care."
Precisely.
Typical mathematician, talking to oneself again. 🤔🤓🥴🙂
Who uses that symbol these days? Even calculators have clear numerators and denominators now, and fractions are much easier to work with than... *that*.
i remember when i was a child and i used to put brackets everywhere and the teacher said that there is no need...
I wish your teacher had focused on something more interesting!
The Order of Operations and the various properties and axioms of math were developed to eliminate ambiguity and to minimize the need for excessive parentheses...
So, while you could use extra parentheses to make the expression more clear for confused individuals. They are not required accept as a crutch for those confused individuals...
The way i was taught i use bimdas and i substitute the 1 with x and since i know the value of x already i know its 3 but i put the x there bc theres an invisible multiplication sign there which makes it easier for me then i solve both of them (both 3) then multiply them bc of that invisible multiplication sign. Idk if this sounds crazy to people but i got 9
That’s correct too the answer is 9 so there is nothing wrong you solve it correct
I think it’s 1 because I’ve seen people replace the brackets with a multiplication sign, but I’m pretty sure that follows different rules. You need to do the 2(3) and get 6 to then get 6/6 = 1 (I believe, confidently)
I'm not trying to change your opinion this is just how I processed it,
1+2=3 so the problem turns into 6÷2(3), next you divide 6 by 2 which equals 3, so the question becomes 3 (3) and 3 x 3 equals 9. It would only equal 1 if you used the math strategies used before 1917
And you are correct. the 2(2+1) is one set = to 6, all other explanations are beyond me. 6/6=1 always has been always will be
@@mokooh3280 they are wrong and so are you...
There is no mathematical difference between 6÷2(1+2) and 6÷2×(1+2) They both equal 9
When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses. TERMS are separated by addition and subtraction not multiplication or division. 6÷2 is a SINGLE TERM juxstaposed to the parentheses as a whole not just the numeral 2....
Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
6/2y = 3/y by Algebraic Convention
6/2(a+b)= 3a+3b by the Distributive Property
Convention doesn't trump LAW and the Distributive Property is a LAW...
@@vi7033 prior to 1917 some text book printing companies pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math so the ERROR was corrected post 1917.
This ERROR i.e. misuse of the obelus means that 1 is not and has never been the correct answer...
Dr Trefor is saying that it is our fault if we write "6 / 2 (1 + 2)", but I think that this is unfair. How are we to know that this is ambiguous, if we are able to type this exact line into a calculator? And indeed, different calculators will give different answers (9, or 1). It is NOT our fault for not realizing that this is ambiguous. We learned these kinds of symbols in elementary school; we can type them into our calculators; and our calculators give no indication that what we typed in is ambiguous. We are led to believe that this is unambiguous and well-defined mathematical language.
Blame the institutions that teach us and use this language; don't blame us.
It's not ambiguous when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended. The correct answer is 9
Thank You Sir. Really appreciate your work.
You are most welcome!
The thing that annoys me about this, is the people who insist that pemdas is a rule. However you can easily break it as long as you know what you are doing
I don’t blame them though, that’s how many schools taught it
Subscribed. Learned a ton from this one video. Your description of how I view mathematics is spot on. And that's probably the reason why I'm never good at mathematics. The moment I first appreciated mathematics, particularly algebra, is when I was working as an analyst. When I found a real life application of the basics. I can't really describe what struck me back then but the way you mentioned "heart" of mathematics was the right word for it.
The way you describe how this expression is ambiguous also applies to my limited coding experience. If I want my program to arrive to a specific answer or output, say 9, then I would "tailor" an expression that will arrive to that desired result. Not sure if my analogy is correct though.
an analyst? you're a smartie, and you know it. it's always been my exp that folks who hate algebra are good at geometry and vice versa! diff sides of the brain i heard!
@@melissalynn5774 not me, I excel in both
@@melissalynn5774 Don't see how someone not good at algebra could excel at trig. Algebra is required to understand trig.
The difference between the two is you're either multiplying by 1/c in the first solution, or you're multiplying by c/1 on the other. The problem doesn't make it clear which of the two. It's like saying the word "bear" without context- do you mean the verb or the animal?
Trying to explain this to anyone who just does math by rote is an exercise in losing brain cells. They furiously exclaim that their way is the only way to interpret the expression.
Yup bingo
100%
I agree with him that the math problem is posed ambiguously & that's caused the viral argument over order of processing. He DUCKED giving the 'Once & For All' answer to the math problem, like the video title SAID!
Haha math clickbait?
1 is the answer.
BODMAS
B - Bracket
O - Orders
D - Division
M - Multiplication
A - Addition
S - Substraction
6÷2(1+2)
6÷2(3)
Here, you still have bracket, so clear the bracket first, then go for division according to BODMAS
6÷6=1
Knowingly or unknowingly, the girl is correct.
Multiplication and division have the same value so you have to do it left from right.
That's just NOT how brackets work though?
The rule is that you do everything INSIDE the bracket, not around it
So when it says 6÷2(3) that's the same as typing 6÷2x3 which explained by the first arguement is 9
What is says is that at some point mathematicians decided to change the order of operations. Why? thats the real question, this problem was created my mathematicians. My theory some grad students were getting high one night and thinking of ways they can mess up math, and thereby get tenure. This is a problem that never should have happened. There are much bigger problems to solve, but weed does crazy things to the brain.
I totally agree with your assessment in this video. The question is now made ambiguous because of this so-called new rule that makes the answer 9. At the end of the day, the mathematical expression should not be ambiguous or not intuitive. The so-called new way feels like a rule someone picked out of their rear. The old rule makes sense, and it is intuitive, hence the correct way. At the end of the day, the expression shouldn't be to trick you.
Thank You, Dr. Bazett!
I agree with your viewpoint 100% (including that I hate people posting problems like this with the intent of causing arguments and going viral).
As a graduate with a BS in physics, I don't have your expertise, but I do have a unique perspective.
There is no equation - EVER - that simply pops into existence like this "as-is".
In my experience, equations start by someone (a mathematician/scientist/engineer) working to solve some specific problem.
That person moves values around following the rules of math, until the equation is solved, and enters values to get the specific result.
At this point in the solving process, 6/2(1+2), the person working the equation SHOULD know without a doubt whether the (1+2) value is in the numerator or the denominator. If they don't, I think they have poor skills at keeping track of their equation.
It's not that this equation should or should not have one specific answer. It is that In The REAL World, anyone who works a problem to this resulting equation will always know how to complete the solution.