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

lol, am I really doing this? Ok, fine. There is a **viral math problem** about, uh, order of operations. You know, #BEDMAS or #PEMDAS. The most common form is 6/2(1+2) but it also shows up as 60/5(7-5) and other equivalent forms. What is the correct answer explained by a math prof? Sorry, I don't care. But I'm happy to share a few thoughts on why I think this issue repeatedly going viral says some things about societal views of mathematics.
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[@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!

[@yourmomsfilms]

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

[@NeoiconMintNet]

@@yourmomsfilms he didn't expkian, he blamoved the question for not understanding the answer.

[@yourmomsfilms]

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

[@NeoiconMintNet]

@@yourmomsfilms he definitely didn't explain, he simply repeated what he was told, including the acronym to remember the rules, but he didn't explain how the rules work. he's like someone that didn't know how to cook, was given a recipe for instructions to cook one thing, but still doesn't understand how to cook.

[@MuffinsAPlenty]

@@NeoiconMintNet Are you the same person as R S?

[@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...?

[@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.

[@RemunJ66]

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

[@Delirium55]

..and that's how we got Lisp.

[@NeoiconMintNet]

@@Delirium55 lisp existed before C++ from what I remember, C came before lisp.

[@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.

[@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.

[@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.

[@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.

[@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.

[@yourmomsfilms]

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

[@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...

[@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

[@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%

[@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.

[@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?

[@ShaneGlynDunning]

Bedmas and pemdas are nonsense and I'm glad I wasn't taught them in school. Instead, I was taught how to read an equation. 6/2(1+2) is the simplified version of 6/((2x1)+(2x2)) which equals 1. To get the answer 9, you would have to write a completely different equation, which would be (6/2) x (1+2) or (6/2)(1+2) simplified.

[@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.

[@haramsack]

This "problem" was artificially engineered to cause controversy. It is not a coincidence, that both common interpretations of the expression have integer results. The addition with the paranthesis is just there to not make the implicit multiplication look too much out of place and overall the symbols are used in a combination that we would normally not encounter in pracitse.

[@MartinBeerbom]

No, the problem appeared because there are different calculators that take this exact formula input and output different results. Because some calculators follow strict PEMDAS, the others don't (they give implied precedence). And the "strict PEMDAS" calculators only exist because North American Math teachers (below university/college level) asked for them. And all those calculators claim "textbook entry" as selling point.

[@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.

[@silpheedTandy]

Dr Trefor is saying that it is our fault if we write "6 / 2 (1 + 2)", but I think that this is unfair. How are we to know that this is ambiguous, if we are able to type this exact line into a calculator? And indeed, different calculators will give different answers (9, or 1). It is NOT our fault for not realizing that this is ambiguous. We learned these kinds of symbols in elementary school; we can type them into our calculators; and our calculators give no indication that what we typed in is ambiguous. We are led to believe that this is unambiguous and well-defined mathematical language. Blame the institutions that teach us and use this language; don't blame us.

[@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

[@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.

[@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

[@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?

[@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.

[@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

[@petiobg]

the division sign ➗ is reserved for first graders and R&B bands. i have never seen this used in algebraic expressions for a good reason, it is ambiguous and poor form. thats like saying that 11 = 1 ( implying a multiplication sign between the digits)

[@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".

[@trwent]

This whole discussion is really NOT about math at all. It is about agreements, or the lack thereof. If we agree that implicit multiplication has a higher priority in the order of operations (OOO) than explicit multiplication, then the answer is 1. If, on the other hand, we believe that all multiplication (whether implicit or explicit) is to be given the same priority in the OOO, then the answer is 9. (By the way, there is absolutely NO "mathematical" proof that the answer is either 1 or 9; any attempt at such a proof--for example, using the Distributive Property--involves obvious circular reasoning, and it is absolutely COMICAL that some people in this comment section cannot see that.) The fact is, there is no absolute universal agreement about the key question here. The best advice is to avoid such ambiguities by using additional grouping symbols, or horizontal fraction bars in lieu of ÷ or / (a horizontal fraction bar DOES imply an ordering; it says to divide the entire top by the entire bottom). If you think about it, the reason we HAVE OOO--and the reason they are as they are, generally--is to be able to write expressions unambiguously while using the fewest number of grouping symbols possible, to keep clutter to a minimum. (If there were no OOO at all, then we would need to write grouping symbols every time a mathematical expression had two or more operations, except when the order the operations are performed would not make a difference in the answer. For example, in 17 - 4 x 3^2, we would need to write 17 - [4 x (3^2)], to know which operation to do first, second, and then third, which would be a hassle, and more bulky due to all the grouping symbols.) Keeping in that spirit, if I want to divide abc by pqr, it seems desirable to write abc ÷ pqr rather than abc ÷ (pqr). So, for this reason, it would be cool if the rule for multiplication in the OOO is that implicit multiplication has higher rank than explicit multiplication (but still lower than exponentiation). If we could all come to this agreement, then that would be good. Engineers and scientists pretty much already have. Mathematicians, not so much.

[@GanonTEK]

Glad to see more people, like you, understand the situation properly

[@d.v.t]

you summarized very well. There's another video posted by a French professor with references. kzbin.info/www/bejne/qorJZHaml9Z0jNE

[@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.

[@ogostrich]

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.

[@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.

[@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...

[@jimmcneal5292]

Irl in scientific community(at least math and physics) it would be interpreted as 6÷(2(1+2)), but in science we almost always initially do the algebraic transformation till we get the answer(here it would be a÷b(c+d) ), and only then substitute numerical values. Plus, as the author said, horizontal line is almost always used instead of "÷" or ":" symbols.

[@isovideo7497]

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

[@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.

[@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.

[@Kyanzes]

I think it's lazily put together. 6/2(1+2) vs. 6/2x(1+2). When the multiplication is not presented, by tradition, it is calculated first. Seriously, how do we interpret 1/2Y? And how about 1 / 2 x Y?

[@sivashankarselvam2171]

1 is the answer. BODMAS B - Bracket O - Orders D - Division M - Multiplication A - Addition S - Substraction 6÷2(1+2) 6÷2(3) Here, you still have bracket, so clear the bracket first, then go for division according to BODMAS 6÷6=1 Knowingly or unknowingly, the girl is correct.

[@Hub689]

Multiplication and division have the same value so you have to do it left from right.

[@DrTrefor]

Ok which you all just sent this viral again:D

[@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.

[@charlesward8196]

When something is “implicit” it relies upon the reader making a certain assumption about the intended meaning of the incompletely written statement. When we “assume” we make an “ASS” out of “U” and “ME.” The assumption that is made by the reader depends upon the knowledge and experience of the reader, which is different for each individual. I understand that publishers like to save on ink by eliminating “un-necessary” braces, brackets, and parentheses, but precision depends on being concise.

[@mikestuart7674]

The meaning of "Implicit items" is NOT arbitrary. They are implicit because any darn fool knows that they are there. That is why we don't bother writing them. If the readers experience is such that he is unable to see them, then that is not on the writer. Do you remember in elementary school how the teacher made you write all those unnecessary items in red for a month or so, just so you would know they were there?? Do you remember them harping... "maintain your parenthesis." if you didn't show brackets around your answer... I remember... At the time I thought it silly. Now I see why it was important.

[@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.

[@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.

[@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.

[@Yotanido]

If I see 1/2x I will interpret that as 1/(2x), not (1/2)x and I don't think anyone does this differently. So why is this so special? 6/2(1+2) is, to me, a fraction with 6 on top. Maybe it's the division symbol?

[@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.

[@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.

[@shitfartshorsecockerson7387]

WHAT HAPPENED TO DISTRIBUTIVE PROPERTY???

[@MuffinsAPlenty]

Nothing happened to the distributive property. You can use the distributive property just fine for this expression. The disagreement people have about this expression is fundamentally one of interpretation. Does multiplication by juxtaposition have a higher precedence than explicit multiplication/division or not? If you believe multiplication by juxtaposition has a higher precedence, then you will distribute the 2 into the (1+2). If you believe all multiplication and division have the same precedence, then you will distribute the 6÷2 into the (1+2). The distributive property has nothing to do with the disagreement. It all comes down to interpretation and whether people think the explicit division or the implicit multiplication has a higher precedence.

[@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..

[@vaginalarthritis1753]

Before I watch this, I'm gonna say it has to do with the order of operations after you perform what's inside the explicit bracket. If thats the case, I do not care. Coming from someone doing math degree, I've learned math is about more than getting the right answer. Its about thinking, human ingenuity.

[@DrTrefor]

We are going to agree a lot then!

[@notahotshot]

"Math is about more than getting the right answer" I hope you never change your mind, even when your employer pays you less than you are owed.

[@angelmendez-rivera351]

You are right, but unfortunately, notational conventions still exist, and they have to exist. No amount of thinking is going to eliminate the necessity in using symbols with the agreed upon rules. Even in natural languages, this is true. This is why dictionaries exist.

[@Zephyr-tg9hu]

Never in my life would I have thought I'd agree so much with someone whose name is literally "Vaginal arthritis", yet here we are,

[@RealMesaMike]

@@angelmendez-rivera351 To (mis-)quote a well known adage about standards: "The great thing about notational conventions is that there are so many of them to choose from."

[@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

[@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

[@Antagon666]

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

[@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.

[@mikestuart7674]

@@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.

[@Darkev77]

Your excitement got me excited xD!

[@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.

[@aphextwin5712]

Well, as others have said: Just don’t use the ÷ symbol, use the ‘horizontal divider’ line. The ÷ symbol is a crutch when you don’t have a proper equation editor available. Maybe I’ve been using Excel wrong, but having to type out an equation in an Excel field in linear form always feels cumbersome as you have to use so many parentheses that it becomes harder to read (applies to type of coding as well).

[@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.

[@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.

[@mrkitloin]

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

[@digambarnimbalkar8750]

It's 1.

[@SirPhilosopher]

What bothers me is there has to be a right answer. The correct answer to the viral equation is that it’s wrong from the start because it has no *correct* answer. Because it’s contradictory to have 2 different correct answers. Math teachers in my childhood held fast to the idea there was always and only 1 right answer. Or math is broken.

[@dgkcpa1]

The problem isn't that complicated. You've got two terms, 6 and 2(1+2), separated by an operator (÷). Simplify the terms, then perform the operation, and you get: 6÷6 which equals 1. Do it vertically: 6 Term 6 ----------- Operator Then Simplify -------------- =1 2(1+2) Term 6 Too many people don't want to take the time or effort to simplify the expression before jumping into the "order of operations", so they just key it into a calculator or math engine and expect the machine to do their thinking for them. Garbage in, Garbage out. It is the user's job to know how their calculator operates and format their entry accordingly. If the calculator's default treatment of 6÷2(1+2) is (6÷2) x (1+2), then the operator must be smart enough to add brackets or parenthese to the entry so that the calculator will solve the problem as 6÷(2(1+2)).

[@AJeazy]

The Calculator treats it as 6 / 2 x (1+2) and then uses Pemdas which would just be multiply and divide in order from left to right. So 6 / 2 x (1+2) is simplified to 3 x (1+2) and then 9.

[@matthewstump7563]

Three rules to remember: 1. multiplication and division are dominant to addition and subtraction. 2. Division and multiplication are equal to one another 3. Addition and subtraction are equal to one another. That means this problem 6÷2(1+2) can not ignore that 6÷2=3. That cannot be changed. It is a dominant equation. So it would be 3(3) which means 3x3=9.

[@dgkcpa1]

@@AJeazy The calculator is just a machine, it will not do your thinking for you. You have a choice in the matter. You are not required to accept the calculator's default treatment of 6÷2(1+2) as 6 / 2 x (1+2). After all, the calculator is your tool, not vice versa. In this case, if the calculator treats 6÷2(1+2) as 6 / 2 x (1+2), (as some do), then I would adjust my input by entering the problem as 6 / (2 x (1+2)). The calculator will stll multiply and divide in order from left to right, and 6/(2x(1+2)) = will simplify to 6/(2(3)) and then to 6/(6), and then to 1.

[@dgkcpa1]

@@matthewstump7563 Yes, 6÷2=3 certainly equals 3, but the equation here is not 6÷2=3; it is 6÷2(1+2). We have a numerator, 6, a division operator ( ÷) and then 2(1+2). Now if you want to do 6÷2=3 first, ok, but what happens to the (1+2)? How does that quantity go from being a factor of 2 to a factor 3? (6/2) Answer: It can't. Let's look at the problem vertically: 6 6 x 1 6 1 1 ----------- = --------------- = ------ x --------- = 3 x ------ = 1 2(1+2) 2 x (1+2) 2 (1+2) 3 We do not ignore that 6÷2=3. But we do not ignore, either, that 2 and (1+2) are factors, and that 2(1+2) forms the denominator in the expression 6÷2(1+2). This expression can be simplified to 6÷6 by applying the P in PEMDAS, before we get into the order of operations, or, if you prefer, we can skip simplification, do 6÷2 first yielding 3, and then continuing the division operation by dividing 3 by the sum of (1+2). Either way yields the same answer, 1. In other words, 6÷2(1+2) does not equal 6÷2 x (1+2), it equals = 6÷2 ÷ (1+2). The division sign in 6÷2(1+2) applies equally to both the 2 and the (1+2).

[@matthewstump7563]

@@dgkcpa1 you are completing this problem as if the first product was 6+2 instead of 6÷2. These two do not get treated the same: 6÷2(1+2) 6+2(1+2) The parenthesis in both problems mean these numbers are to be multipled by what is outside of the parenthesis. These two problems would be: 6÷2×(1+2) 6+[2×(1+2)] Using PEMDAS you do parenthesis/exponents in both first. 6÷2×3 Left to right applies here because division and multiplication are equal. 6+2×3 Multiplication first then addition. It's that simple. You can't treat 6÷2 as if it is 6+2.

[@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!

[@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.

[@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.

[@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.

[@noThankyou-g5c]

there isn’t a real answer. Order of operations is a “social construct” more or less. we decide which way to compute something. Outside of highschool and maybe college if you didnt get very far in Math you would never even see an expression as ambiguous like this unless you wrote it yourself trying to work something out. In which case you would probably know the order to compute. Or better yet you would’ve wrote it down in a way that wasn’t ambiguous to begin with

[@AndyCole-nc6dn]

I love that you show better ways to notate this to remove the ambiguity. I think the problem is that math is usually taught as math in a vacuum. The "right" answer is what real world application you are calculating. Instead of focusing on how it's written, the real answer should be to go back to the real problem to clarify the question. Arguing over a poorly notated equation is just silly.

[@GanonTEK]

100%

[@schwarzerritter5724]

A similar example to the hidden multiplication operator is -3². Is it supposed to be read as -(3²) or (-3)²? When you are too concerned with making the formula look nice, you are creating ambiguity.

[@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

[@mikestuart7674]

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

[@elmer6123]

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%

[@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.

[@lazik711]

On the board or in the book, I agree. When programming, it comes out 9 - c language, python, fortran, octave, libreoffice...

[@jamesdelacruz3857]

Hi, can you consider my solution? from 6/2(1+2), = 6/(2+4), = 6/(6) Therefore, 6/2(1+2) =1. I reverse the process for checking the equation 6/2(1+2) =1. to prove that x=2 and satisfying the equation 6/2(1+x) =1. 6/2(1+x) =1 6/(2+2x)=1 6=2+2x Then , 4=2x x=2 Therefore, 6/2(1+x) =1 I would like to know how to prove the equation 6/2(1+2) =9? because the division sign is confusing to the process, and i think it will give us an idea of how the equation should be express instead. Nevertheless, my point is that math concepts must be applied in reality. So, i will choose the process that is useful, especially in our profession/career.

[@TheArchytech1047]

PEMDAS 6/2(1+2) parentheses first (1+2)=3 Now the equation is 6/2(3) or 6/2×3 then you go left to right so you divide first 6/2=3 so now the equation is 3×3 now you multiply 3×3=9 Or to simplify 6/2(1+2) 6/2(3) Or 6/2×3 3(3) Or 3×3 =9

[@J0shibo]

Something people don’t understand is how the order of operations works. When we say division and multiplication or the other way around, it goes left to right regardless. Example: 10 / 2 X 3, the answer is 15, because 10/2 is 5, and 5 X 3 is 15. The same applies here. Let’s remove the bracket and put 3 in its place: 6 / 2 X 3. You do 6/2 first because it is before anything else. 6/2 = 3. Then 3x3 = 9

[@hootax8980]

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

[@DarinBrownSJDCMath]

Precisely.

[@rubennaidoo3939]

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

[@bojiguy2005]

The correct answer is 1. If you’re fuzzy.

[@larsholmstrand7579]

so. No answer is right, right?

[@danielgoodman3578]

I appreciate and agree with both: -your very valid point--BE UNAMBIGUOUS. BE CLEAR in the math you're writing--especially when you involve division, because it changes the result so dramatically depending on what you mean! -your pinned comment where you default, as I do, to interpreting this as taking care of 2(3) before the division. Though it is nonetheless, as originally presented in the problem, an ambiguous expression.

[@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

[@lidular]

The thing that annoys me about this, is the people who insist that pemdas is a rule. However you can easily break it as long as you know what you are doing

[@Lonewulf321]

I don’t blame them though, that’s how many schools taught it

[@petersisler1398]

Before calculators this problem would be poorly framed, since it was considered to be ambiguous. So there was no answer. The observation that there is wide disagreement suggests ambiguity.

[@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%

[@xoxoxoxoxoxoxo6921]

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

[@DrTrefor]

Thank you!!

[@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

[@matthewkendall5235]

I think the issue comes down to complex parsing - and most folk only knowing at most 5-6 of the - is it now over 20 rules of precedence? Also many of these rules don't arise from field theory so ask does a base change precede a tower precede a concatenate precede a conditional precede a factorial and they can't even answer a simple question like is twelve concatenate 3 factorial i.e. 12 | 3! parsed as 123! or as simply = 126. Worse the "rules" of precedence aren't written in stone in any one place and any one times for all of pure and applied mathematicans to agree with. If someone asks why is a tower parsed top down and not bottom up - well PEMDAS isn't much help... So the authors of Cayley and the Feymann lectures point to the brackets creating a grouping on the implied multiplication immediately preceding it - which elevates its precedence about 6 levels above a normal multiplication - so folk see it as akin to asking what is 6 divided by 2x for x = 3? Fact that no one providing an answer goes beyond PEMDAS - the simplest of all sets of precedence laws and states that there are many, many more - nor knows where to find them - to me tells a very clear story.

[@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.

[@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

[@nathanielsizemore3946]

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

[@kimobrien.]

Their actually is rules for mathematicians by the American mathematical society because book publishers follow rules. And when computer languages process symbols unless you want to create additional halting problems you have to have a default order of doing things. You can always add parentheses and brackets for clarity.

[@TBoy205]

TIL you need a PhD in mathematics to solve a first grade level math problem

[@jamescollier3]

Well in 5th grade, you can say 9. But if you want to publish with the intellectuals, The Physics Style Guide says 1

[@anshulanand02]

My whole life has been a lie

[@kenevans233]

Thank You, Dr. Bazett! I agree with your viewpoint 100% (including that I hate people posting problems like this with the intent of causing arguments and going viral). As a graduate with a BS in physics, I don't have your expertise, but I do have a unique perspective. There is no equation - EVER - that simply pops into existence like this "as-is". In my experience, equations start by someone (a mathematician/scientist/engineer) working to solve some specific problem. That person moves values around following the rules of math, until the equation is solved, and enters values to get the specific result. At this point in the solving process, 6/2(1+2), the person working the equation SHOULD know without a doubt whether the (1+2) value is in the numerator or the denominator. If they don't, I think they have poor skills at keeping track of their equation. It's not that this equation should or should not have one specific answer. It is that In The REAL World, anyone who works a problem to this resulting equation will always know how to complete the solution.

[@ddgyt50]

Ask any physycist or engineer if 9 is the coorect answer and you wiil receive am emphatic No. The expressiin, a/bx is always understood to be a times the inverse of bx.

[@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.

[@STUDENT-wb6ux]

This question is more like the English statement - "I saw a man using binoculars " it can either the man under observation is using a bino, or I observed the man with the help of a bino.😌

[@ZeHoSmusician]

Yes, that's why punctuation was invented: "I saw a man using binoculars" != "I saw a man, using binoculars"

[@daddiesgurl1968]

You would be an interesting math teacher. I excel in other classes & while I do pretty good with math, I believe that my problem was how it is explained to me. I had accelerated classes & I had a Geometry class...I still shudder...I learned more about my teacher than I did about the class. I took regular Algebra 2 & Trigonometry. I actually understood the concepts of Trig but I mixed things up. I did get a B on my final & my teacher said she checked my test twice. 🤨 Anyway, I will check out your other videos. I know I am not dumb when it comes to math, I just need the right teacher.

[@blahblah3859]

ChatGPT, Cerebras, Claude say the correct answer is 288. Claude: If I had to provide an answer for this expression, I would say 288. Here's why: In modern mathematics, multiplication and division are typically treated with equal precedence and calculated from left to right. Most modern calculators and mathematical software follow this convention. Many mathematical institutions and textbooks have adopted this interpretation as standard. The calculation process would be: 48/2(9+3) = 48/2 × (9+3) // First, evaluate inside the parentheses = 48/2 × 12 = 24 × 12 // Perform division first (going left to right) = 288

[@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 >>>

[@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”.

[@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