Math Prof answers 6÷2(1+2) = ? once and for all ***Viral Math Problem***

[@DrTrefor]

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!

[@deadseymour]

I thought you explained it well in the video already- I'm honestly baffled that people continue to argue which answer is "correct" 🤷

[@deadseymour]

@@NeoiconMintNet he most definitely explained but, maybe you didn't understand his explanation?

[@deadseymour]

@@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.

[@deadseymour]

@@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

[@deadseymour]

@@NeoiconMintNet Since you're bringing 5th grade math into this, that implies mathematical standing does actually matter to you.

[@GanonTEK]

I'd easily give this video a 6÷2(1+2) out of 10

[@digambarnimbalkar8750]

It means 1 out of 10.

[@JustVezix]

@@digambarnimbalkar8750 Nah, they gave this video a solid 9.

[@GanonTEK]

@@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.

[@GanonTEK]

@@JustVezix Schrödinger's rating 🤔😋

[@severeaura6540]

In other words 6÷2(1+2)/10...?

[@DarinBrownSJDCMath]

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."

[@DrTrefor]

Indeed. Heck, I haven’t even used that symbol in at least 15 years!

[@DarinBrownSJDCMath]

@@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.

[@DrTrefor]

Thanks for mentioning, always like hearing they are being used. Hope your students find them helpful:)

[@ActuatedGear]

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.

[@LudusYT]

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.

[@PJ-ts7uz]

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))

[@GanonTEK]

100%

[@Krise-g2k]

It's a normal 5th grade question in asia

[@ttthttpd]

On top of that, people who use algebra or have studied higher maths (even lightly) will tend to answer using implied multiplication. People who use arithmetic regularly, or who never studied higher maths will get the "left to right" answer. In the end, both audiences will think the other is stupid. For either "forgetting" a basic rule, or for having never learned the more general rule.

[@bjornfeuerbacher5514]

@@ttthttpd I have studied higher maths, and I indeed would have used implied multiplication. But I would not have thought that the other side is stupid for never having learned / having forgotten a rule; instead, like Dr. Bazett, I would have said that the question is ambiguous.

[@casualpeen]

yes but the axiom people refer to is over 100 years old and was used in such a rare niche use case, people who openly use division in that way are totally wrong and I wouldnt trust them to calculate anything for me lol

[@CeceNorman]

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.

[@calebfuller4713]

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... 😂

[@harrymatabal8448]

Mr Norman you are also correct so 6×3÷2=9

[@pokemonfanmario7694]

​@@calebfuller4713fairly certain some teachers skip that part, like mine.

[@zakelwe]

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.

[@CeceNorman]

@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.

[@AnthonyOliverio]

If coding has taught me anything, just put parentheses around everything.

[@DrTrefor]

haha right? Computer programmers just don't have this issue:D

[@michaelbauers8800]

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.

[@RemunJ]

The problem with all those extra parentheses is readability, especially with inline expressions.

[@Delirium55]

..and that's how we got Lisp.

[@RS-fg5mf]

@@RemunJ 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...

[@kobusswart554]

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

[@GanonTEK]

100%

[@makenzimedlin4328]

My exact thought process thank you

[@lyvectra6270]

As a mechanical engineer, I 100% agree.

[@Milesco]

As the son of an electrical engineer, I agree, too. 😊 It troubles me that *_so many_* people think otherwise!

[@RS-fg5mf]

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....

[@jayjpepedreamer]

As a civil engineer, my instinct is to change that devision sign into a diagonal slash and get the answer 1 too. 😅

[@user-by7hj4dj9s]

It’s the same, ÷ should not be used. But in essence ÷ = / = : Yes : is also used for division.. and it’s all the same.

[@Milesco]

@ 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.

[@adamwalker8777]

@@Milesco no! a/bc = a/b*c!!!!!!

[@masterblaster3653]

Shame on you how did you became civil enginner

[@taoliu3949]

@@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.

[@Sindraug25]

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.

[@jaysonkmendoza]

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.

[@jamesschaaf612]

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.

[@MrGreensweightHist]

The correct answer is 9

[@ZS-bg7jo]

@@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.

[@wrrsean_alt]

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.

[@deadseymour]

So basically, both answers are correct. It's the question that's wrong. Just a sloppy set up

[@RS-fg5mf]

WRONG

[@Kage-jk4pj]

Definitely wrong, there are a bunch of questions like this in my text book. Here in Australia.

[@RS-fg5mf]

@@Kage-jk4pj can you post pics of your textbook so we can see what it says...

[@RS-fg5mf]

@@filename1674 No you can't. 🙄🙄🙄

[@tommy8290]

@@RS-fg5mf Argue with a maths professor on this one? You are unbelievably up your own rear end

[@jguo]

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...

[@mokooh3280]

Well bring it

[@skiddadleskidoodle4585]

What is 77 + 33

[@opticalmouse2]

@@skiddadleskidoodle4585 "What is 77 + 33" Easy, it's 7733.

[@geirmyrvagnes8718]

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!

[@foxfactcheck]

kzbin.info/www/bejne/on2mdZaXa8mMpqM

[@markcash2]

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).

[@GanonTEK]

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.

[@RS-fg5mf]

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

[@AudriusN]

@@RS-fg5mf stop spamming your stupidity

[@shaunpatrick8345]

@@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.

[@RS-fg5mf]

@@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...

[@nsn5564]

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.

[@trwent]

It is an expression, not an equation.

[@manzerm7805]

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.

[@zerxilk8169]

pemdas vs the bs

[@xybersurfer]

the problem is indeed the implicit * sign

[@kimf.wendel9113]

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.

[@manzerm7805]

@@kimf.wendel9113 The 2 is outside the bracket. If it was 6÷(2*3) no confusion would arise.

[@kimf.wendel9113]

@@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

[@KevinKuo]

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.

[@DrTrefor]

Exactly! I should hire you to be my script writer:D

[@physicsmathsworld2033]

@@DrTrefor lol 🤣🤣🤣

[@donaldthomas7070]

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.

[@kirkspreiter6444]

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!!

[@gustavo9758]

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.

[@isovideo7497]

I use equal precedence when explicit * is used, but give implicit multiplication higher precedence.

[@larsholmstrand7579]

so. No answer is right, right?

[@carlhartzell6054]

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.

[@kurtka8720]

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.

[@murattanyel1029]

After all, math is a language, too.

[@jeremy5602]

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)".

[@wrrsean_alt]

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.

[@carlhartzell6054]

@@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.

[@habacue713]

I forgot how much I hated math. Him explaining math to me is like the equivalent of a warm glass of milk.

[@trwent]

Yuck.

[@mirkotorresani9615]

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.

[@AtomicExtremophile]

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.

[@jianxiongRaven]

Ya man . Now the tricky thing is identidying questions like this and when its (a+b)

[@kimf.wendel9113]

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.

[@Joe_Narbaiz]

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.

[@kimf.wendel9113]

@@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.

[@Andrew-it7fb]

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.

[@tingkagol]

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?

[@Steponlyone]

As a mathematician and an engineer, I love that this problem became viral because it shows the fundamental differences between rules and conventions.

[@bernardgome5564]

You said it all and so few likes

[@melissalynn5774]

but us folks for whom math has always made me feel stupid, i i need rules!

[@enysuntra1347]

​@@melissalynn5774The rule is called "#PEJMDAS": Parenthèses - Exponentiation - Juxtaposition - explicit mult/div - addition/subtraction.

[@plumber1337]

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.

[@MrGreensweightHist]

You are incorrect.

[@chrisgriffith1573]

So your point is: This problem communicates badly, or was designed to go viral knowing what limited understanding people have surrounding mathematical rules, and why they are applied.

[@panjak323]

Normal people use either fractions or division symbol followed by bracketed expression. No ambiguity in that.

[@detroittigersandotherbaseb7220]

The only rules involved here is the commutative and distributive principles. Why are people doing 6/2*3? Multiplication and addition are commutative so the numbers can be swapped. And using distributive principle you have the answer of 1 It should be 6 6 ----- --------- 2(2+1) or 4+2

[@maxxiong]

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.

[@manzanajoemerj.9849]

I'm with the 2nd argument as well. Since it makes more sense when you think about algebra. Along with distributive property of Multiplication

[@jshad1074]

@@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

[@olblue3478]

@@jshad1074 always do parenthesis first and open them... Its argument 2

[@no0bjago900]

@@jshad1074 when you start to use / , I'd say any numbers come after that would be as one denominator

[@SeanMaxhell]

@@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?

[@LudusYT]

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.

[@stevecolour8010]

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.

[@nickjunes]

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.

[@Jry088]

The problem is what if this actual problem shows us on the test. We all know test are there to be tricky

[@nickjunes]

@@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.

[@RockinRack]

@@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.

[@CiscoWes]

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.

[@kimf.wendel9113]

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.

[@doughendrie5468]

You are correct. This is a simple term divided by a term. Monomial division.

[@Archimedes_Notes]

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

[@DrR0BERT]

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.

[@txheadshots]

I went to a birthday party with the strippers, JFK and Stalin

[@keekwai2]

PHD in WHAT, you clown?

[@ibarskiy]

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

[@txheadshots]

@@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

[@keekwai2]

@@ibarskiy Just repeat 5th grade, and this time, stay awake.

[@Darkev77]

Your excitement got me excited xD!

[@Ligierthegreensun]

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.

[@SoraRaida]

Yup bingo

[@GanonTEK]

100%

[@doughendrie5468]

Well it is just a term dividing a term.

[@rachelcolomb]

This video went somewhere far more exciting then the viral problem, glad I watched and have subbed.

[@davidhuber6251]

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.

[@impos1ble32]

I liked your points at the end on how society views mathematics. Would love a whole video dedicated to that!

[@DrTrefor]

This is actually a great idea and a BIG topic imo

[@justdoit2585]

kzbin.info/www/bejne/m32wlJailN9sjsk

[@PuzzleAdda]

Viral Math Equation 6÷2(1+2) = ? Watch this video for answer - kzbin.info/www/bejne/sKK7p3WCjdxoisU

[@popeyelegs]

How society views math doesn't solve the problem.

[@DrTrefor]

Ok which you all just sent this viral again:D

[@craigquann]

Doc, the year is 2025... the world has decended into chaos... it's back again.

[@Of_UnCommon_Sense]

Given: Physicists, Scientists, Mathematicians, Educators, etc., follow the international mathematical standard ISO 80000 - 1. Within this, there’s a rule against division followed by multiplication or another division without parentheses... as it’s seen as indeterminate or ambiguous. So not only is the expression 6/2x3 the wrong solution it is also "verboten", and definite not equal to anything. Sorta puts the lie to saying there are two solutions. The correct solution is that 2(3) is a single value. There is only one operator not enclosed in parenthesis so one operation to be done... it is... a / b where a=6 and b=2(3)=6 If indeed you are an educator, how can you not know this ??

[@smartmanapps5588]

"Mathematicians, Educators, etc., follow the international mathematical standard ISO 80000 - 1" - no, we don't. You'll find none of them pay any attention to ISO, especially since most of what they wrote was based on them not understanding how division works (they also say not to use an obelus, without giving any reason why not). "Within this, there’s a rule against division followed by multiplication or another division without parentheses" - in fact it specifically says a Multiplication SIGN, and there is NO Multiplication sign in this expression, so there's actually no problem with it as far as ISO are concerned. 6/2(1+2) is fine by ISO, 6(2x(1+2)) is also fine by ISO - since we added Brackets when we added the Multiply sign - what ISN'T ok is rewriting it as 6/2x(1+2), which is AGAINST what ISO says, unless of course that is explicitly what was meant in the first place. "If indeed you are an educator, how can you not know this ??" - well, not knowing what ISO says is fine, not knowing about Terms and The Distributive Law is another matter altogether though! 😂

[@Of_UnCommon_Sense]

@smartmanapps5588 Precisely my point. Tho I could not state it as elegantly. I quoted the ISO standard to illustrate that 6/2(3) could not be 6 / 2*3 with a multiply sign, which is not per ISO even allowable. That 6 / 2(3). Has no solution with an exposed multiply sign. How could a math educator not know this,, is what's was actually intended.

[@suhrrog]

This was the best explanation for this problem I've heard so far. Essence: Don't write your problem in an ambiguous form!

[@peterthomas5792]

Except it's not ambiguous to anyone competent in maths. The answer is 1, and that's it. All other answers are wrong.

[@theonethatsabovetoaa5645]

@@peterthomas5792ion see your PHD so ur wrong

[@markprange4386]

With no multiplication sign, the only indication that (1 + 2) is multiplied comes from its being grouped with 2.

[@johnsciara9418]

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.

[@devkird6069]

thata numbers right there

[@axelmac7856]

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

[@axelmac7856]

On this operation

[@georgearnold841]

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.

[@bambajoe1721]

Too much wordas for 1 math problem my friend

[@slavdog3180]

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)

[@vi7033]

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

[@mokooh3280]

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

[@RS-fg5mf]

@@mokooh3280 they are wrong and so are you...

[@RS-fg5mf]

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...

[@RS-fg5mf]

@@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...

[@Cocreatewithus]

Yes it is a very poorly written expression. BUT, even though the parentheses also stand for multiplication, one must also solve the parentheses before doing anything else, correct? So, while (1+2) does equal 3, the parentheses aren't yet gone. It is still 6 (oh crap where is the stinkin division sign on this new computer??? There isn't one, ok fine...); it is still 6 / 2(3). In order to eliminate the parentheses, you have to solve 2(3), which is 6. Good, now the parentheses is officially removed via the distributive property, and we can resume with order of operations left to right. So now, we have 6 / 6, which is 1. But, no mathematician in their right mind would write something this way in order to solve something.

[@smartmanapps5588]

"even though the parentheses also stand for multiplication" - no they don't, they stand for a Factorised Term, solved via The Distributive Law, a(b+c)=(ab+ac), 2(3)=(2x3),

[@krnisa.karim30]

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 😌

[@mirkotorresani9615]

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".

[@akosualynn6469]

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!

[@mrkitloin]

People: ITS 9! ITS 1! ITS 9! ITS 1! me: Its both

[@digambarnimbalkar8750]

It's 1.

[@scottlamontia5622]

So basically if you have a mathematical expression that is poorly written, it seems as if the answer you’re trying to achieve doesn’t matter Especially when you could argue that two possible outcomes can be true in spite of the fact logic usually dictates that only one possible answer is supposed to be the only correct answer and the other one must be false

[@wes9627]

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.

[@GanonTEK]

100%

[@DLBozarth]

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.

[@gavco115]

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.

[@JorJorIvanovitch]

In symbolic logic, this is why brackets matter. I was always taught that coefficients immediately preceding parentheses were operators to execute immediately after what was solved inside the parentheses. The answer should be: 1. Brackets should be used to avoid ambiguity. If they want 6 to be divided by 2 first, then it should be written (6÷2)(1+2). Or, 6÷2×3. In this case, it would be left to right without the inclusion of any parentheses. But that is a different problem than when you have a coefficient in front of a parentheses. I can not remember encountering issues like this in school because brackets were used to avoid this.

[@lastchance8142]

Absolutely correct. I don't see how 2(3) is any different at this point in the process than encountering 2x. Forget PEMDAS. Let the rules of algebra apply.

[@melissalynn5774]

thanks. i thought so!

[@doughendrie5468]

Yes, let the rules of maths apply. No one has mentioned the definition of factors. Factors must be whole numbers. Factors cannot be fractions. Factors are separated by multiplication and division. The term “6 divided by ”has no connection to the factor 2. And to prove the answer, use the Golden Rule of Algebra. There is only one specific division to remove, by multiplying both sides of the equation by 2(1+2).. 6/2(1+2)=1 multiply both sides by 2(1+2) 6=1*2(1+2) simplify 6=1*6 proven

[@seanclark6438]

Sir it is my understanding that when you distribute the coefficient to the parenthetical expression you distribute it to both numbers in the parenthetical expression and then complete the expression in order to remove the brackets

[@BboyTurtlefresh]

Why don't we factor the 2 back into the parentheses before adding the parentheses?

[@refilwe9954]

My answer is 1 But for chaos' sake because why not: Why don't we factor the 6÷2 back into the parentheses because the left to right rule of the pemdas? Again just for chaos' sake😁

[@doughendrie5468]

⁠@@refilwe9954Can’t do that according to the rules of maths. 2(1+2) or 2*(1+2) are both single terms, made up of 2 factors. The term 6 cannot strip a factor from the term 2(1+2) or the 2*(1+2). This is a simple term divided by a term. Monomial division. Can’t believe the number of people on here, that don’t remember factors multiplied are a single term.

[@refilwe9954]

@@doughendrie5468 true. But let em live and let me enjoy my chaos🤣🤣 some of us have been saying that; they don't want to listen

[@jamesrobbins26]

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.

[@MGmirkin]

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.

[@Of_UnCommon_Sense]

@@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.

[@kimf.wendel9113]

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.

[@Andrew-it7fb]

​@@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.

[@shaunpatrick8345]

@@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.

[@mushtaqkasba8702]

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.

[@tombodmer1424]

Ultimately, this equation is grammatically incorrect in the first place. Math as a language is intended to be written in a way that can be agreed upon. As such, both answers COULD be correct. However, this ambiguity would not exist if the equation was written correctly to begin with. It should either be (6/2)(2+1) or 6/(2(2+1)). The original equation is just an incomplete equation that left it's lunchbox at home.

[@llems1484]

This is like a dangling modifier in grammar.

[@jeffschroeder4805]

It seems like inserting a quote without including the context - doing so can often twist the original meaning entirely. At the very least the meaning is made ambiguous.

[@TenTonNuke]

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.

[@GanonTEK]

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.

[@Pajo25ify]

@@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).

[@simongpunkt]

wow you really didn't get the video you just watched start to finish huh

[@Reinshark]

I'm very happy to see a math professor rightly calling out these problems as fundamentally uninteresting, rather than certain OTHER channels (perhaps pictured in this video) that seem to relish in the clicks generated by viral order-of-operations nonsense.

[@hootax8980]

"So what do you, as a mathematician, think?" "I do not care."

[@DarinBrownSJDCMath]

Precisely.

[@rubennaidoo3939]

Typical mathematician, talking to oneself again. 🤔🤓🥴🙂

[@DRosenberg-w5x]

A couple of questions for you, Dr. Bazett... 1) Given that "x" does not equal zero, what is the quotient of 2x divided by 2x? 2) Does a monomial need to be encased in a set of parentheses to be understood as ONE TERM with a single value which is the PRODUCT of the coefficient multiplied by the variable (factor)?

[@phoenix2634]

I'm not Dr. Bazett, but I can answer your questions. 1. The quotient of 2x divided by 2x is 1. 2. Whether a monomial needs to be encased as one term depends entirely on what notational conventions are used. (Read: defined) If you define a term as the product of a coefficient multipled by a variable, generally speaking no. However, defining a term in this manner is incomplete. ½x is a term ¾x is a term, 5 ----x²y is a term 2 Each of these are a product of a coefficient and a variable. Fractions can be a coefficient. Each example doesn't use infix notation. Fractions written using infix notation become a ÷ b or a/b. The definition of a term you provided does not account for writing a fraction using infix notation. This introduces ambiguity. Infix notation is inherently ambiguous. The inherent ambiguity can be resolved through use of grouping symbols for all non-associative sequences of operations; just consider division, multiplication, addition and subtraction, these sequences are: Successive divisions. (a/b)/c ≠ a/(b/c) Successive subtractions. (a - b) - c ≠ a - (b - c) Division followed by a multiplication. (a/b) × c ≠ a/(b × c) (a/b)c ≠ a/(bc) Subtraction followed by an addition. (a - b) + c ≠ a - (b + c) A multiplication or division before or after an addition or subtraction. (a × b) + c ≠ a × (b + c) Successive multiplications are associative. (a × b) × c = a × (b × c) (ab)c = a(bc) Successive additions are associative. (a + b) + c = a + (b + c) Multiplication(s) followed by a division are associative. (a × b)/c = a × (b/c) (ab)/c = a(b/c) Addition(s) followed by a subtraction are associative. (a + b) - c = a + (b - c) The inherent ambiguity can also be resolved through a defined order of operations. So whether or not 2x divided by 2x needs to be notated as 2x/2x or as 2x/(2x) or considered undefined depends on the order of operations used. In the research I've done, skimming through 150 or so textbooks primarily published in the US, a couple in Canada, from 1860 to 2010, (books after 2010 aren't available to view in digital libraries) I've come across 4 basic order of operations. (They each start the same) 1. Evaluate what's inside grouping symbols. 2. Evaluate any exponents or other orders (Here's where they diverge) 3. (A) Evaluate multiplication/division left to right (B)(i) Evaluate juxtaposed multiplication (ii) Then evaluate division,/multiplication left to right (C)(i) Evaluate all multiplication (ii) Evaluate division left to right (D) use grouping symbols to indicate the order of multiplication and division (Here they return to being the same 4. Evaluate addition/subtraction left to right Using (A) 2x divided by 2x would be 2x ÷ (2x) or 2x/(2x) Using B or C 2x divided by 2x would be 2x ÷ 2x or 2x/2x Using D 2x divided by 2x would need to be 2x ÷ (2x) or 2x/(2x) if it were written 2x/2x or 2x ÷ 2x it would be undefined.

[@cronnosli]

I'm sorry but I never see an engineer resolves a/bc as a/b*c but always as a/(b*c)

[@Araqius]

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

[@Bakadesu-]

@@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.

[@Araqius]

@@Bakadesu- In physics or engineering, there is [unit].

[@Bakadesu-]

@@Araqius Then I guess I'm not an engineer then. Lol

[@RS-fg5mf]

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.

[@omarcedric9193]

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.

[@melissalynn5774]

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!

[@xeroxcopy8183]

@@melissalynn5774 not me, I excel in both

[@Of_UnCommon_Sense]

@@melissalynn5774 Don't see how someone not good at algebra could excel at trig. Algebra is required to understand trig.

[@eknaap8800]

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.

[@valdir7426]

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.

[@eknaap8800]

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

[@universe25.x]

Your answer is 1 just because you find it more aesthetic? Are you friends with your brain?

[@eknaap8800]

I am more inclined to use my right brain hemisphere, but occasionally my left one kicks in...@@universe25.x

[@jetroejuke70]

I was always taught that 2(1+2) is a single term. 2x1 is 2, 2x2 is 4, add together and you get 6. 6/6 is 1.

[@DarkAngelGRM]

You have been taught math correctly and arrive at the correct solution to the equation. This is my second thumbs up. 👍

[@tehpanda64]

No human is writing such an ambigous math problem (unless the goal is to be ambiguous). Therefore this math question is in fact the equivalent to the quintessential english class question "what did the author mean by __"

[@doughendrie5468]

Not sure what your mathematical eyes see. My eyes see a simple unambiguous division. The term 6, divided by the term 2(1+2) Monomial division.

[@Indistinguishable0]

​​@@doughendrie5468 Hm, but why then TI calcs, Wolfram Alpha and Google say 9? 🤔 Is there a reason why they don't see a monomial division here?

[@doughendrie5468]

@@Indistinguishable0Ti 84 supports implied multiplication. As do others. Implied Multiplication The TI-84 Plus recognizes implied multiplication, so you need not press X to express multiplication in all cases. For example, the TI-84 Plus interprets 2m, 4sin(46), 5(1+2), and (2*5)7 as implied multiplication. Note: TI-84 Plus implied multiplication rules, although like the TI-83, differ from those of the TI-82. For example, the TI-84 Plus evaluates 1/ 2X as (1/2)*X, while the TI-82 evaluates 1/2X as 1/(2X) (Chapter 2). Parentheses All calculations inside a pair of parentheses are completed first. For example, in the expression 4(1+2), EOS first evaluates the portion inside the parentheses, 1+2, and then multiplies the answer, 3, by 4. 4*1+2 6 4(1+2) 12 TI86 Implied Multiplication The TI-86 recognizes implied multiplication, so you need not press X to express multiplication in all cases. For example, the TI-86 interprets 2π, 4sin(46), 5(1+2), and (2*5)7 as implied multiplication. Parentheses All calculations inside a pair of parentheses are completed first. For example, in the expression 4(1+2), EOS evaluates 1+2 inside the parentheses first, and then multiplies 3 by 4. 4*1+2 6 4(1+2) 12 TI-30X PlusMultiViewTM Calculator () Implied Multiplication 4図(2+3）Enter 4*(2+3) 20 4(2+3） Enter 4(2+3) 20 Page 15 Operator Manual

[@Indistinguishable0]

@@doughendrie5468 The TI-84 Plus recognizes implied multiplication, but evaluates 1/ 2X as (1/2)*X, so 6/2(1+2) = (6/2)*(1+2) = 9, as is the case with some other models of calcs (including TI and HP), as well as with Wolfram Alpha and Google and Bing online calculators. So what is the reason why all of them give 9 instead of 1?.There surely must be some reason for that, I guess.

[@doughendrie5468]

@@Indistinguishable0The TI84-Plus recognises implied multiplication. The TI-84 Plus recognizes implied multiplication, so you need not press X to express multiplication in all cases. For example, the TI-84 Plus interprets 2π 4sin(46), 5(1+2), and (2*5)7 as implied multiplication. And parentheses. Chapter 2). Parentheses All calculations inside a pair of parentheses are completed first. For example, in the expression 4(1+2), EOS first evaluates the portion inside the parentheses, 1+2, and then multiplies the answer, 3, by 4. 4*1+2 6 4(1+2) 12 So 6/2(1+2) will be simplified first, before (1/2)*X due to parentheses.

[@pali0123]

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

[@anshulanand02]

My whole life has been a lie

[@bluesparrow-hx5qf]

This "problem" is invalid, because the calculation is not sufficiently defined. It's all the same with those puzzles. The real problem is do identify the definitions, and there are always two, otherwise this would not be a puzzle

[@remainedanonymous8251]

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.

[@galzajc1257]

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,...

[@theedspage]

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.

[@GanonTEK]

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.

[@maxxiong]

@@GanonTEK It's not c times m. Otherwise mm would be m squared.

[@GanonTEK]

@@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

[@maxxiong]

@@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.

[@xoxoxoxoxoxoxo6921]

I agree with the points in this video. This channel is so underrated though.

[@DrTrefor]

Thank you!!

[@MGmirkin]

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.

[@MGmirkin]

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 & $$.

[@fackarov9412]

i remember when i was a child and i used to put brackets everywhere and the teacher said that there is no need...

[@DrTrefor]

I wish your teacher had focused on something more interesting!

[@RS-fg5mf]

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...

[@theedspage]

I agree, this problem needs clear notation.

[@mokooh3280]

i am moko and think it could be written better however it is correct in it definition the answer is 1

[@ninjaslash52_98]

the issue is that i went through my entire life going through ambiguous problems like this nobody ever explained it clearly and it was always "memorize this formula"

[@robertparker7243]

Frankly if I were a professor of mathematics and I saw a heated debate about the outcome of a simple expression, I wouldn't laugh at it - I'd be worried.

[@petersansgaming8783]

@TheCeilingFanCollectorHD bro missed the entire point of this video lmao

[@catdog1584]

I think it depends on the context from the real world. Math in the old days doesn't use symbols but words, that way mathematicians were able to visualize every problem they were working on. *E.g.* If *6/2(3)* got described in words, it'll probably would be read as "6 numbers in 2×3 table" While 6÷2×3 is more like "six cakes for two people but then these 2 people multiply the cake by cutting each cake to 3 pieces, and now there's 9 pieces of cake".. something like that

[@RS-fg5mf]

The context and notation should match... If the notation doesn't match the context then it's wrong... 6÷2(1+2)=9 and if there were context that says otherwise then the notation needs to be changed.. 6÷(2(1+2))=1

[@catdog1584]

@@RS-fg5mf you're right but I don't think the (1+2) in both first and second equation has different answer therefore I put it as "3" rather than "(1+2)" to shorten the context. But if you're going with the consistency, you can still find alot of context that match the notation. Great analysis mate

[@trwent]

It is NOT an equation, it is merely an expression.

[@RS-fg5mf]

@@trwent True

[@sdlcman1]

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.

[@geirmyrvagnes8718]

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.

[@papato20]

So the answer is that this is "ambiguous"? If that's the case, I don't understand why it's so hard to say that. It's incredible how difficult it is for some people to say that (I've already seen 3 videos that refuse to give a clear answer, even if that answer is that "the answer is ambiguous". They prefer to ignore the topic by talking about the trivialities of mathematics. , like in this video)

[@SoraRaida]

Just look at the replies on this comment section. People still doesn't take "it's ambiguous" as an answer, because they're so fixed on the mentality that there is only 1 correct answer according to a set of rule.

[@papato20]

@@SoraRaida for real!!! is fight in all the comments. but at least in my case i obtained the answer, good luck for the rest XD.

[@tcmxiyw]

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.

[@freddyt55555]

This is why the division symbol isn't usually used in algebraic notation. It's used to symbolize the division binary operation when learning basic arithmetic--or on a calculator. On a calculator, it's a division OPERATOR that divides the first number by the second number. There's no implicit grouping. In algebraic notation, you use the "fraction bar", and in this case, there is an implied grouping even without parenthesis.

[@RS-fg5mf]

The vinculum is a grouping symbol. It's not implied grouping, it's actually grouped within the denominator... 6 -----(1+2)= 6÷2(1+2)= 9 2 6 ----------- = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... _________ 2(1+2) = (2(1+2))

[@MartinBeerbom]

This is something that came from calculators (which is also why the symbol is the way it is -- calculator manufacturers combined the fraction bar and the ":", which was the written division OPERATOR symbol in many countries.) The earliest calculators had only OPERATOR buttons, and on the RPN calculators that actually made sense. Pressing "÷" divided Y by X (on the stack). You also could not use "-" to start a calculation like "-2+5", because "-" was the operator for "subtracting x register from y register", and you mostly had something on the y register that you couldn't see. You had to re-arrange the computation, or use the "+/-" or "CHS" (change sign) button that Hewlett-Packard wisely prepared for you. And actions like A÷B÷C made sense as shortcut entry for A/(B*C). But then the calculators gained algebraic and then textbook entry, and the symbols moved away from being just operators. It's easier to see with the "-". If you have an expression like "2x-x^2", which most would rewrite as "-x^2+2x", the "-" is not an operator anymore, but a modifier for the number following, and "x-y" becomes short for "x+ (-y)". Which leads to modern (non-graphing) calculators having a SECOND minus key (labeled "(-)" on Casios) for the modifier -, which may or may not be used interchangeably with the normal "-" operator key beside the numbers (and yes, this leads to a lot of confusion and lengthy discussions away from maths with students I tutor. Essentially, I must them teach how a calculator works and is used because the regular teachers have no time or inclination to do it.) I also recall that the earliest Casio algebraic calculators retained the "+/-" key from the earlier RPN calcs even though you did not need it. You could enter a calculation like "-2+5" with the normal "-" operator key if you started from a display showing "0". The calc would take pressing "-" as: Take what is currently displayed (0), and subtract the next number entered from that. So 0-2 = -2.

[@RolandOrre]

There are many calculators that support implicit grouping. My Casio fx-991ES PLUS for instance, it returns 1 in this case.

[@starcrossed0]

this problem shows up on twitter every 2 months and honestly it's just annoying at this point lol

[@DrTrefor]

Right!? I've seen it SO many times:D

[@azelynhirano]

The first time I saw it was in 2015, when I still had access to my high school math teachers

[@leonardriceiii934]

I’m surprised you didn’t mention the distributive (at least I think that’s the one) property to clarify. Like this: 6/2(1+2) so you distribute the 2 in front of the () to both the numbers which becomes 6 / (2+4) distributing the 2 doesn’t take away the parenthesis. 6 / 6 = 1

[@GanonTEK]

Distribution is usually defined explicitly as a•(b+c) as the starting point. It's more convenience it's written as a(b+c) and works fine in isolation but in context you have to be careful. Academically, multiplication by juxtaposition implies grouping so 6÷2(1+2) written explicitly is 6÷(2×(1+2)) Using distribution then gives 6÷(2+4) = 1 Literally/programming-wise, multiplication by juxtaposition implies only multiplication so 6÷2(1+2) explicitly is 6÷2×(1+2) instead. Using distribution (6÷2×1+6÷2×2) = (6+3) = 9 So, distribution gives both answers because distribution doesn't interpret implicit notation. The order of operations simplifies, it doesn't interpret implicit notation either, so it can't resolve the ambiguity either and gives both answers also. No rules can help here because the ambiguity happens in interpreting the implicit notation which happens first and before any properties can be used to rewrite the expression or before the order of operations is used to simplify the expression..

[@MrJahka]

a lot of people were (apparently) taught that pemdas is dogmatic universal rule rather than a simple mnemonic for elementary school arithmetic. if we don’t agree on the notation we aren’t having a math debate. Good video

[@lee1874-pringles]

Finally found a sensible answer to this kind of stupid questions 😭! Thank you!

[@GanonTEK]

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.

[@jjh7611]

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

[@RS-fg5mf]

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

[@RS-fg5mf]

@@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...

[@roscius6204]

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.

[@IncredibleFlyinSquid]

The ambiguity of this equation is what's incorrect about this whole situation.

[@SoraRaida]

And yet, so many people still couldn't get it.

[@michaelsanders2655]

This is the best video regarding these “viral” math problems. The issue at hand is how people were taught. I was taught juxtaposition, meaning the answer would be 1. I keep telling myself that I’m going to ignore these videos, but I get drawn in. Then, I like to say “The answer is 42”.

[@Nethezbet]

Hmm, this is the best explanation I've seen. Very logical and having lacked that sort of mathematical knowledge it never occurred to me that math could be vague, it just seemed like "This is the rule, apply it." But then we see why it is not so simple. Thanks!

[@peterthomas5792]

Maths isn't vague. It's just that people assume that PEMDAS/BIDMAS, or whatever your local acronym is, is univerrsally applicable - it IS NBOT,. Just as 'I before E except after C' doesn't work for weird scientific things. a/bc is universally evaluated as a/(b*c) - consult any maths textbook. PEMDAS is WRONG in this scenario and there is no need to add parentheses for 'clarity'.

[@TARASTItheloser]

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

[@teknul89]

That’s correct too the answer is 9 so there is nothing wrong you solve it correct

[@mr.t5585]

Sorry for bad English… but the bracket is still there 6/2(1+2)=6/2(3) you have to finish the bracket or not? So 2x3? The parenthesis is not finished and you can’t just put the brackets away.

[@smartmanapps5588]

"So 2x3" - (2x3) actually - you can't remove brackets until there is only 1term left inside - but otherwise you are correct. 6/2(1+2)=6/2(3)=6/(2x3)=6/6=1

[@joeguadarrama3523]

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.

[@phoenix2634]

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.

[@joeguadarrama3523]

@@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?

[@phoenix2634]

@@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.

[@joeguadarrama3523]

@@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.

[@GanonTEK]

@@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.

[@nathanielsizemore3946]

As in most things in life, good communication is essential.

[@Technium]

The best possible take

[@AbnormalPecker]

Doesn't a division symbol imply a numerator and a denominator. Doesn't the order of operations dictate that you deal with operations in the numerator and denominator before moving forward?

[@GanonTEK]

The issue is with the ambiguous notation used it's not clear what exactly is in the denominator

[@smartmanapps5588]

Not strictly speaking - different terminology - but yeah, similar concept. "Doesn't the order of operations dictate that you deal with operations in the numerator and denominator before moving forward?" - the people who are getting the wrong answer are failing to treat 2(1+2) as a single Term divisor, and thus not doing the whole thing in the first step, which is the correct way to handle it, since order of operations is Brackets before Division. The instead somehow decide to class it as "Multiplication", even though there's literally no multiplication sign, and thus end up with the wrong answer.

[@smartmanapps5588]

@@GanonTEK "The issue is with the ambiguous notation used" - there's nothing ambiguous about a(b+c), it's the standard form of a Factorised Term, as per MATHS TEXTBOOKS - you know, those things that you seem to be allergic to looking at. 😂 "it's not clear what exactly is in the denominator" - it's TOTALLY clear, 2(1+2). Maybe you need glasses? 😂

[@travisnapoleansmith]

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.

[@GanonTEK]

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.

[@johnwagonis]

Yes, use the horizontal line, not the forward slash.

[@kaelyn800]

There’s two ways to get 1 and only one to get 9 you could also multiply out 2(1+2) and do 2x1=2 and 2x2=4 Aka (2+4) so 6 / 6 =1 Along with seeing the (3) as still being in a parentheses connected to 2 and needing to be solved before we can finish the rest of the problem

[@GanonTEK]

Well, same amount of ways for each pretty much. More than 2 for each also. Academic interpretation of 6÷2(2+1)= 6÷(2(2+1))=1 #1: 6÷(2(3))=6÷(6)=1 #2: 6÷(4+2)=6÷(6)=1 #3: 3÷(2+1)=3÷(3)=1 #4: 6÷2(3) = 2÷2=1 Modern programming interpretation of 6÷2(2+1)=6÷2×(2+1)=9 #1: 6÷2×(3)=3×3=9 #2: 3×(2+1)=3×(3)=9 #3: 6÷2×(3)=18÷2=9 #4: 3×(2+1)=(6+3)=9 It's an ambiguous question. Whichever interpretation you pick gives the answer you want to prove is right before you start so using order of operations or distribution can't prove anything here, it's circular. It's a notation issue before you even start to simplify.

[@_kagiex3197]

Bro what is the correct answer you see ??

[@RS-fg5mf]

BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = (1/2) 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....

[@maxxiong]

Multiplying out 2(1+2) is not valid unless you assume brackets around it first.

[@GanonTEK]

@@maxxiong One widely used interpretation of multiplication by juxtaposition implies brackets around it. That's why it's ambiguous. The modern programming interpretation uses explicit notation whereas a lot of academic writing used implicit notation.

[@billingram1622]

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!

[@DrTrefor]

Haha math clickbait?

[@Kelvinllovejr]

I've looked everywhere on the internet and still no one has answered my question. Since we come into 2(1+2) in parenthesis then after you get 2(3) that section should still be considered a parenthesis part. How does it transform from parenthesis to a simple multiplication situation? Obviously, if you come into a problem with 6/2(3) you would automatically assume that the 2(3) needed to be multiplied and would do it right after dividing 6/2.

[@GanonTEK]

P in PEMDAS or B in BOMDAS is for inside parentheses/brackets only, not outside. Otherwise you run into issues like 4²(3). If a P or B step is still present, how can you do it before Exponents? The notation here is ambiguous. It depends on which interpretation of multiplication by juxtaposition you use. Academically, juxtaposition implies grouping and multiplication (1). Literally, juxtaposition implies multiplication only (9). Both use the same order of operations after the implicit notation is interpreted. It's just bad writing. It has no single correct answer without more information.

[@x00g40]

I agree, but a case where it does matter is programming. For example, Python interprets "2/4*3" as "(2/4)*3", read left to right.

[@DrTrefor]

Yup, any calculator or programming language has to impose a choice of how it parses expressions like these

[@notahotshot]

Yeah well, that is the correct interpretation.

[@RealMesaMike]

But what happens if you enter "2/4(3)" into python? You can't, because you get an error: >>> 2/4(3) Traceback (most recent call last): File "", line 1, in TypeError: 'int' object is not callable >>>

[@namewithheld8115]

I loved your answer at 3:30. It's the best possible answer, other than the general statement of "write your equations more clearly."

[@trwent]

It is NOT an equation, it is merely an EXPRESSION.

[@meofamily4]

Thank you. As a high-school math teacher, I heartily approve of your response.

[@RolandOrre]

It was first at university I learned about juxtaposition having higher precedence than division.

[@jef711]

PEJMDAS is better than pemdas

[@Nitrox-.]

Everyone I know in science would give a higher precedence to implied multiplication at the minimum if not multiplication in general. 1/2x is interpreted as 1/(2x) because interpreting it as (1/2)x doesn't make any sense, that would be written as x/2, otherwise you're just using needlessly complicated notation. In general you'd definitely want to write it with a top bottom devide if there could be ambiguity. But you would really never see a single line string written down like this, especially if it is all numbers which in it self is very uncommon.