6÷2(1+2) = ? Correct Answer Explained By Mathematician

What is 6÷2(1+2) = ? This problem went viral and generated millions of comments on Facebook, Twitter, KZbin and social media sites. I explain how to get the correct answer by using the modern interpretation of the order of operations. I also explain how you would get a different answer under historical usage of the division symbol. (Note: some people write 6/2(1 + 2) = but this has the same answer.)
I studied Mathematics and Economics at Stanford. Press coverage of my work:
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*I get many, many emails about this problem and am unable to reply to them.
Blog (text explanation): wp.me/p6aMk-4OV
Slate explains the history of the division symbol
www.slate.com/articles/health_...
Here is a 1917 article from "The American Mathematical Monthly" that explains the usage of the division symbol as an exception to the order of operations
www.jstor.org/stable/2972726?s...
Google evaluation
www.google.com/#q=6%C3%B72(1%2B2)
WolframAlpha evaluation
www.wolframalpha.com/input/?i=...)
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[@MindYourDecisions]

20 million views!

[@idontclickbait8453]

Really a good video!

[@lavion8090]

lesgo

[@gandolph999]

It has millions of views because the problem initially looks too simple to have a video (and it is an excellent video). I wondered if I missed something and chose to watch. So, the order of operations rules were revised and both 9 and 1 are correct answers. I thought it was 1 (the algebraic grouping of terms as you noted). Great to know the rules changed. Thanks for making the video.

[@mlas1308]

@@idontclickbait8453 It is a good video

[@alfredoooooooooooooooooooooooo]

Great video, still slightly confused because I am taught that x(y) is one term and should be treated as 1 number but glad to learn that there are 2 different systems

[@danielrushing55]

1960 we will have flying cars in the future 2020: world debate over 5th grade math

[@javin9743]

daniel rushing this is probably 5th or 6th

[@danielrushing55]

Ya true but either way

[@adomnibest5099]

Nope, it's 2nd (in my country)

[@alexreede2555]

adomnibest lol I learned it at 5, #homeschooliscool

[@indevious9659]

5th grade too

[@ankitahuja3791]

This is not a math problem... this is a rule problem....

[@RS-fg5mf]

The rules support the correct answer 9

[@blubber0_0]

Yeah, technically this rule doesn’t have to be a thing. Just for convenience.

[@jude3426]

Blubber Beast um it definitely DOES have to be a thing. It’s there for a reason

[@RS-fg5mf]

@@jude3426 it doesn't give priority to multiplication over division... It is for convenience and less clutter....

[@LunaEllis]

@@RS-fg5mf Can we just agree to use the fraction sign when diving? It makes the intended outcome a whole lot clearer

[@ronnycook3569]

IMO the real answer is "don't write expressions this way." You write expressions so that they can't be easily misinterpreted. If there's any possible confusion, add parentheses to resolve the confusion. This is particularly so when dealing with non-commutative operations such as subtraction and division. This probably comes from a background in computer programming; you don't want to make assumptions about order of operations, because while the people who implement the languages usually have standards to work from, they sometimes screw up. (To make things even more complicated, in computer programs sometimes operations have side effects - like "x++ + x" in C - "x++" evaluates x then increments it, so the value depends on whether the left or right side of the addition is interpreted first. The solution is "don't write it that way.". Eliminates confusion and therefore bugs.)

[@GanonTEK]

100%

[@northwestclasspnw7974]

Amen

[@benvergus1573]

Its just the correct way of writing it, there isn't much to be confused about

[@benvergus1573]

Basically you just read from left to right and solve brackets first, whats so confusing?

[@ronnycook3569]

@@benvergus1573 The fact that it's debated at all indicates that it's found confusing. This whole kerfuffle would not EXIST if people didn't find it confusing. The implicit multiplication when concatenating terms - as is done in this particular problem - complicates matters further. What is 4x/2x ? Left to right you get 4x, divided by 2, multiplied by x, so 2x^2. But I read it habitually as 4x divided by 2x and get 2, and I'm betting that 90% of people trained with strict left to right in evaluating PEMDAS/BODMAS do the same. Writing with clarity in mind, and avoiding any possible confusion, is never a bad thing. Particularly when dealing with a field where precision matters, such as mathematics or engineering.

[@IILiamHD]

as an engineer who has done advanced university level maths for about 7 years now, I would get 1. its the convention usually followed in physics/engineering textbooks to solve as terms and let implicit multiplication (brackets esp) go first

[@afsdfsadhasfh]

I'm in a similar position and I 100% agree. It's disingenuous by the video to imply there's one correct order, when so many physics and engineering books do operations in the 2nd way. The video is also wrong in stating that calculators all calculate in the same way. Mine doesn't. I guarantee that if any engineers I know saw something like "6÷2x" they'd calculate the 2x first. It has nothingo to do with the division symbol. Implied multiplication (for example, 2x rather than 2*x) in all the engineering I've learned always takes priority over normal multiplication. If you write it as 2(1+2) instead of as 2*(1+2) there has to be a reason for it, and common sense (mine at least) dictates it's because you mean the order of operations to be different. Real world math isn't a puzzle designed by someone to fool you, it's an objective way to state things and should be written accordingly. The problem here is the question, not the answer. Just write it as (6÷2)(1+2) or as a fraction and the problem is solved.

[@IILiamHD]

@@afsdfsadhasfh absolutely spot on. couldnt put it better myself

[@IILiamHD]

@@afsdfsadhasfh the question is deliberately misleading

[@thepsychologist8159]

@@afsdfsadhasfh Conclusively, the experts say this. The equation is ambiguous and indeed, it can yield two different answers. Like the use of language, to convey something such that it can't be misinterpreted, it must be delivered with clarity, the intention should be made clear. The same with maths equations. To yield only one result, the equation should not be written with ambiguity and the intention of the writer must be clear. If it does or can, the equation should be re-written.

[@WobblePizza]

Jesus. This is dangerous. Hopefully you stay in school and don't go actually build something one day.

[@Shade7x]

The programmer's wife sends him to the store. She says "Get one carton of milk, and if they have eggs, get a dozen". The programmer came home with 12 cartons of milk, because they did have eggs.

[@burnerjack01]

And that's why I used to hate computers so much.

[@brianfedelin7015]

It would be 13 cartons because of the and boolean logic instead of or.

[@keesdenheijer7283]

I'm truly thankful for the opportunity to give thumbs-up #42.

[@4_youtube_is_dead]

tf

[@Dan.Bielecki]

Agreement with Brian Fedelin. 13 cartons of milk. One carton, and if there are eggs, get a dozen. So 1 + 12 = 13. And the issue with computers is not the logic of them, it is how a human evaluates a human expression and then programs the computer. In this case, the issue was with the wife, since the expression was not clearly defined from the start by defining a dozen of WHAT was desired, the milk or the eggs. See, it is actually a trap by wife against the husband. No matter what he were to bring home, it would be incorrect since she could then change WHAT was the dozen to be of.

[@molo2793]

The correct answer is 9 but the way I was taught math makes me keep saying 1

[@tclauk79]

MOLO 27 yes 🤔

[@dbzayan]

MaxMisterC Both of them state that multiplication and division have the same importance, and are some left to right. Put it in a calculator if you disagree.

[@berdyie]

@@MaxMisterC Heck it wasn't even that for me. I always just ASSUMED (don't remember if it was actually in the education I got) that anything next to a bracket was in itself inside "invisible" brackets. So if you had 2(1+2), it would simply be read as (2(1+2)) = 6 regardless of what was put in front of it. I guess I never really bothered searching up if this was wrong, OR that my teacher might have been in the same group that still insists on this sort of thinking. Either way the new method (answer of 9) is correct and there just isn't much you can do about it. That's the rule and that's how maths works I guess.

[@raynatumbeva780]

@@berdyie there is no such a rule in Maths.

[@berdyie]

@@raynatumbeva780 For my incorrect way of thinking or for the correct method in the video?

[@netherworldofmind7402]

I can ensure you that in most Stem environments the symbols ÷ and / are pretty much forbidden, every division must be written as a fraction, so all formulas and expressions are just sequences of products of franctions, and the length of an horizontal line is clearer than any pemdas rule

[@RS-fg5mf]

6 ------(1+2) = 6÷2(1+2)= 9 2 6 ---------- = 6÷(2(1+2))=1 2(1+2) WHY?? Because the vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator... ________ 2(1+2) = (2(1+2)) two grouping symbols each Objective facts...

[@realGBx64]

What about implicit multiplication? I know many people would interpret 1/2x as 1/(2*x) while 2x^3 is never understood as (2x)^3

[@RS-fg5mf]

@@realGBx64 Most people confuse and conflate an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity by this convention to Parenthetical Implicit Multiplication. They are not the same thing... 1/2x = 1/(2*x) by Algebraic Convention 1/2x^3= 1/(2*x^3) by Algebraic Convention 1/2(x)= (1/2)(x) by the Distributive Property 1/2(x^3)= (1/2)(x^3) by the Distributive Property... 1/2x and 1/2(x) are not the same thing.

[@WobblePizza]

No they aren't. There is nothing wrong or ambiguous with the equation as written. Not everyone is using a whiteboard.

[@davidbrattain1446]

How on earth do you operate a calculator without those two symbols available?

[@user-qd4yc4vf7c]

Clearing the parens is not simply performing the operation within but also performing the operation dictated by the parens. Therefore the operation requires multiplying 2x3 to get 6 prior to the next operation. If the equation was: 6 divided by 2y there would be no ambiguity that it would be 6/(2y)not (6/2) x3.

[@wadabid6165]

Nope, if you got 6÷2y you do 6/2 times y. Its just the current rules, i agree its weird and maybe confusing because we never use "÷", we always use fractions, but the rules are the rules and they say that if theres no parenthesis, you only divide by the first number, the closer to the "÷" symbol. Which is 2, therefore 6/2 × 3 = 9

[@snekarmy4559]

The parentheses are just around 3 though, not 2*3

[@markprange4386]

@@wadabid6165: 2y is grouped. Just like 2π, 2π, or 24.

[@whoff59]

You are using PEJMDAS like in some calculators (not all of them). J meaning Juxtaposition. But this is not PEMDAS which is the official math rule for instance in USA.

[@markprange4386]

@@whoff59: PEMDAS is not an official rule anywhere. It's not even a rule.

[@LeoH3L1]

We shouldn't change things like the order of operations, it's incredibly dangerous in things like engineering to have two different people unknowingly using two different standards.

[@awesomemccoolname7111]

Explain that to a teacher. Go ahead.

[@walkingman8943]

They only thing common about school now is that every child is getting left behind.

[@rileyjeffries1282]

That’s why no mathematically inclined individual worth their salt uses the division symbol.

[@justingray8161]

order of operations never changed, it's always been the same. He just explained that that specific symbol for division meant something very specific other than just division over 100 years ago but the actual order of operations has never changed.

[@jgt2598]

That's why for any serious communication of mathematics you have to be more explicit than this ambiguous problem. Hence why peer-reviewed papers use fractional notation and make copious use of parenthesis to remove ambiguity.

[@jamesgcrawford]

the correct answer is this is a poorly written problem.

[@revdarian]

You are 100% correct, this is the proper answer.

[@jamesgcrawford]

What is the correct answer? I don't know man, math isn't typically fully divorced from reality, let's look at the reasons why you're crunching these numbers and we can re-write it so it makes sense!

[@matlohn9381]

+James Crawford yeah realistically equations would never be written this way but I think the majority of math rules indicate the answer is nine

[@Theaksten]

In mathematics, there is no such thing as bad problems. Only bad rules and the misuse of good ones.

[@Theaksten]

No, the first premise in PEMDAS, is to solve for the answer within parentheses. You never distribute into parentheses first because you would then misapply the order of operations. PEMDAS: Parenthesis, Exponents, Multiplication And Division, Addition And Subtraction (IF the same precedence, then left to right). Any order with And in between has the same precedence! Since the problem is 6/2*3 or 6/2(3), we must follow the premise regarding left to right because the problem involves only multiplication and division, orders of the same precedence. Parenthesis is only a symbol of multiplication when a number or expression is adjacent to it. If the problem were 6/(2*3), then the logical answer is 1, because we solve for the answer within parentheses first, as according to the first order of the order of operations. The answer to 6/2(2+1) is not 1.

[@AaronProctor.]

I graduated with my Bachelors in 2019, the answer according to the way I was taught throughout my education is 1. Because I was instructed by my professors to visualize this problem as 6/(2(1+2)) or 6/6 which equals 1. The person who wrote this did so in a way that is designed, purposefully or ignorantly so, to cause confusion. Dr. Trefor Bazett has an insightful video on this topic

[@trickortrump3292]

Are you saying that you took university level math within the past 10 years and your professors taught you that in the case of 6➗2(1+2) you’d make 6 the numerator with the 2(1+2) being the denominator? Ima have to throw the bs flag on that one. It doesn’t even make sense that your professors would have even been instructing you on this when this is just basic math that young kids learn. It’d be like saying “When I was pursuing my master’s degree and my professor was teaching me my times tables…” If you took this stuff recently, you’d have been taught to solve left to right 6/2x3 =3x3 =9

[@RS-fg5mf]

Dr. Trevor Bassett is wrong and so are you... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ---------- = 6÷(2(1+2))= 1 2(1+2) The vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator. ________ 2(1+2) = (2(1+2)) Two grouping symbols each ________ 2(1+2) has two grouping symbols (2(1+2)) has two grouping symbols

[@RS-fg5mf]

​@@trickortrump3292the bigger question would be why a University would be using the grade school obelus to teach higher level math... We have reviewed the video and the penalty flag stands... Good call Ref....LOL

[@trickortrump3292]

@@RS-fg5mf Yeah I deserved that. When I first looked at it, I solved it your way and then the video told me I was wrong. I bought into the reasoning for why I was wrong. This question is just a mess! I went down the rabbit hole yesterday after my comment. It’s insane to me that so many experts seem to say that the right answer is “there is no right answer” because it can be correctly solved two different ways, yielding two different answers. I can’t accept that. If both answers are correct, that makes both answers wrong too. I’ve removed the bs flag I originally threw. 👍😉

[@RS-fg5mf]

@@trickortrump3292 don't remove it. LOL The red flag stands on the play because you are absolutely correct... The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended is 9 I was agreeing with you. Don't let these mathematical numpties change your mind. Those who understand and apply the basic rules and principles of math correctly as intended will get the correct answer 9 Those who fail to understand and apply the basic rules and principles of math correctly as intended will get the wrong answer 1 Those who can't prove 1 and can't accept 9 will argue ambiguity... Failure to understand and apply the basic rules and principles of math correctly as intended doesn't make the expression ambiguous and isn't a valid argument against the expression...

[@3Deez3]

I graduated in 2003. I was and am pretty good in math subjects. I was taught to solve this with the answer of 1. The brackets are to be dealt with b4 other division of multiplication occurs

[@qwenettadixon6911]

Same here. Aleays remembered pemdas, but didnt hear bodmas til recently

[@3Deez3]

I never heard bobmas b4 lol I said brackets to be inclusive I guess lol.

[@mikehuston2132]

I was taught to expand the products in parenthesis first. So 2(1+2) would be 2+4 which is 6. 6/6=1

[@diegovaca8026]

exactly mate@@mikehuston2132

[@mirandahotspring4019]

@@qwenettadixon6911 We learned it as BEDMAS (E = exponent)

[@mattmanhero2375]

The problem isint the equation itself, it's whoever wrote it.

[@MrGamecatCanaveral]

No the problem is teachers changing they way they teach things.

[@babyyoda7749]

@@MrGamecatCanaveral Not teacher the education system.

[@MrGamecatCanaveral]

@@babyyoda7749 no teachers are the ones. Teachers are teaching about gender and sexuality for example. The education system is not telling them to teach that.

[@aeroljameslita4975]

@@MrGamecatCanaveral Teachers change the way they teach things because we discover new things over time. Before, it was thought that the earth is the center of the solar system. Copernicus discovered that it is actually the sun. Now, do we need to change the way we teach about the solar system? Yes. It's crazy that what we believe in the present will never be entirely 'true' as it could be proven false in the future.

[@aquaneko14]

@@aeroljameslita4975 from a subjective point of view, isn't the point of perception the center of your reality? So the Earth is the center of the universe for everyone on it.?

[@Pepa14pig]

As a math student, I’m mad at the way this is written. My teacher said he would fail anyone who wrote math problems like this 😂😂😂

[@RS-fg5mf]

Your math teacher has issues but as long as he is grading you I suppose you need to do what is expected... There is nothing wrong with the way the expression is written just the ignorance people have about parenthetical implicit multiplication...

[@justcheck6645]

@@RS-fg5mf Isn't that the whole point - it is perfectly valid but makes it unclear and you have to think about it - are you mad because you got it wrong? I would be a bit concerned about your math's teacher.

[@RS-fg5mf]

@@justcheck6645 I am a math teacher and I didn't get it wrong. LMAO When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly you get the correct answer 9.... Did you get it wrong??

[@RS-fg5mf]

@Christopher Butler if you don't care, why bother replying?? LOL Enjoy your show. 😁😁

[@violinplayer3518]

I graduated math in 80s, cannot believe people are discussing primary school math

[@realGBx64]

I don’t think the problem here is the division sign. I think the problem is the “implicit multiplication”. In my experience as a scientist and teacher, many people would say 5/2x = 5/(2*x), similar with the number in front of the parentheses, without the multiplication sign.

[@antelope6125]

Algebraic equation vs. simple arithmetic. 2x is a variable and it's coefficient, which implies multiplication. You don't have any implied multiplication in arithmetic because there are no coefficients since you have no variables.

[@nickmcginley4570]

@@antelope6125 The number in front of the brackets is a coefficient.

[@coreyramstein9778]

Yes!!!

[@backtoschool1611]

I understand both methods. It will be interesting what my books say when I get to this type of problem. I am studying from pre-1900 Maths books right now.

[@briannadeleon4672]

Just graduated and I was legitimately taught that "historical way" all through school

[@mansuryayas]

Same here

[@kayking2536]

I graduated high school 15-16 yrs ago and got my associate degree in '08, that's how I was taught, the old way.

[@fiveoctaves]

That's terrible.

[@sikksotoo]

Ditto, and I aced math in school

[@pentowerchannel5306]

Old technique is correct because a=6/2(1+2)is not equal to. b=6/2*(1+2) In this video he is using second technique No doubt 1 answer

[@ammarmar3628]

That's why nobody uses the division symbol. It's confusing and it leads to errors. Both in math and physics, fractions are the way to go.

[@MarioLandscape]

I prefer using ÷ over /. I only use / with fractions, but use ÷ when dividing numbers. Using improper fractions instead of using the division symbol is something that I rarely ever do. I never find it confusing when using ÷, and it never confuses me.

[@MarkQub]

@@MarioLandscape nope

[@MarioLandscape]

​@@MarkQub. What do you mean 'nope'? I just stated that I prefer using this: *÷* of this: */,* when dividing. The person said that nobody uses ÷, because it's confusing, so I said that I do use ÷, and that it doesn't confuse me.

[@MarkQub]

@@MarioLandscape you see how you lost. You got mad/serious over someone who just simply said “nope”

[@MarioLandscape]

@@MarkQub. What on Earth? Lost what? What did I lose? Please explain. All I did was ask you what you meant by nope. How was that getting mad?

[@ricflair9717]

If you substitute all numbers with variables: a÷b(c+d) = You would get a/b(c+d) = a/(bc+bd) Not a/b × (c+d) That's how algebraic functions are ordered.

[@spamspamspamspam3459]

That's using implicit multiplication vs bodmas. You're coming with the assumption that b(c+d) multiplication holds higher priority than a/b part. That's all it is

[@adrianmcbride1666]

​@@spamspamspamspam3459it does because you first need to resolve the parenthesis. They are not resolved until you distribute them out (by multiplying the coefficient into the brackets).

[@spamspamspamspam3459]

@@adrianmcbride1666 That is absolutely not true, () in bidmas only applies to whats inside the parenthesis.

[@adrianmcbride1666]

​@@spamspamspamspam3459my father studied to masters in mathematics, he has stated that in different areas both methods are correct. Within South Africa, and from what I have gathered much of the rest of the world) that is exactly how it is done. Simple reason, it should not matter in which order one does the multiplication or division whether right to left or left to right as long as it does not combine any addition or subtraction.

[@adrianmcbride1666]

@@spamspamspamspam3459 again, differs from location. However in my country, that is the correct method to resolve the parenthesis. Notably from the logic my father gave me, this method means that it does not matter in which order you do the multiplication and division once the brackets are solved. The method you use requires that one solves the equation in a specific order lest the answer be different (I. E. If one first does multiplication before division). As was said, the purpose of the bodmas is so that regardless of the order one solves the equation (within each relative Order) in that it will arrive at the same answer. The method we use, the order in which division and multiplication is done matters not.

[@MickJacobsen]

I do not recall the "same precedence" rule. So, I have always taken PEMDAS to be performed, literally, left-to-right. But, knowing the rule now, I think a better mnemonic would be (P)(E)(MD)(AS). It will be mine from now on! Thanks for the lesson.

[@clowncircus-dm3fd]

BODMAS

[@InsanityoftheSanities]

@@clowncircus-dm3fd WHO CARES AMERICA SAYS PEMDAS.

[@hat_cbw6972]

@@InsanityoftheSanitiesjokes on you, some American schools use BODMAS

[@InsanityoftheSanities]

@@hat_cbw6972 Jokes on you, who cares and who asked?

[@hat_cbw6972]

@@InsanityoftheSanities Jokes on you, we can assume that who asked is an imaginary number. An imaginary number that we know of is sqrt(-1) which is i. Conclusion: I asked.

[@erickbangle3781]

think of it like a fraction. there's a reason why in higher math '÷' isn't used.

[@erickbangle3781]

dont forget that 2(3) is one term

[@FusedAtoms]

I know right... I'm suprised the education system is failing this hard to teach math.

[@thomasgronek6469]

but it's not a fraction. there is the discrepancy in the solution

[@DrenRawr]

all division is a fraction lol

[@supervegetto1087]

Exactly Erick!! we use the division sign just to teach kids but it is wrong in advanced math.

[@indy1435]

Its gonna go viral again because its in my recommended

[@miguelgm808]

6 / (2*1 + 2*2) = 1

[@frozenx1337]

@@miguelgm808 did u even watch the video?

[@devannatesan5053]

Ya

[@danielje97]

Nobody care

[@Araqius]

@@miguelgm808 (4/2+2/2)(3) = 9

[@quantum2330]

sounds like the problem here is not what the answer is, but who wrote the math's problem incorrectly

[@MrTSFunny]

I'm pretty sure you missed a different confusion. I get that some would interpret the division sign as you did, but there is also the belief that implied multiplication has priority over other division and multiplication, because it was implied, it has to be resolved. You can't just change 2(3) into 2x3, because they are bound. 2x(1+2) does not just equal 2(1+2), because 2(1+2) = (2(1+2)). I realize it doesn't make a difference until (÷) gets put in front of them. Say we wanted to divide energy by 12. Would we write 12 ÷ mc^2 or would we write 12 ÷ (mc^2). We all recognize mc^2 as energy, m and c^2 are bound by implied multiplication. a completely different thing that 12 ÷ m x c^2. Or divide 12 by the area of a circle: 12 ÷ πr^2. π x r^2 is implied and therefore bound. this is the real argument implied multiplication has priority of not.

[@DJ_Dopamine]

I am 45 years old and have honours degrees in Engineering and Science. We were always taught that the answer should be 1, because of the order of operations rule that we were taught to use. If you change the rule, you change the answer. I was not aware that the rules had changed!

[@bigbadlara5304]

I just graduated high school and my answer to this problem was 9. I guess it's taught correctly now atleast :)

[@GanonTEK]

It seems that multiplication by juxtaposition, ab or a(b) etc., may impliy grouping, or it may not, so the notation is ambiguous making both answers valid. It depends on context (e.g. academic or programming). It's just bad writing. Modern international standards, ISO-80000-1, mention that brackets are required to remove ambiguity if you use division on one line with multiplication or division directly after it. The American Mathematical Society's official spokesperson literally says "the way it's written, it's ambiguous" even though they use the explicit interpretation. Wolfram Alpha's Solidus article mentions this ambiguity also. Microsoft Math gives both answers. Many calculators, even from the same manufacturer, don't agree on how to interpret multiplication by juxtaposition. No consensus. Other references are: Entry 242 in Florian Cajori's book "A History of Mathematical Notation (1928)" (page 274) "The American Mathematical Monthly, Vol 24, No. 2 pp 93-95" mentions there was multiplication by juxtaposition ambiguity even in 1917 (and not the ÷ issue) "Common Core Math For Parents For Dummies" p109-110 addresses this problem, states it is ambiguous. "Twenty Years Before the Blackboard" (1998) p115 footnote says "note that implied multiplication is done before division". "Research on technology and teaching and learning of Mathematics: Volume 2: Cases and Perspectives" (2008) p335 mentions about implicit and explicit multiplication and the different interpretations they cause. Other credible sources are: - The PEMDAS Paradox (a paper by a PhD student on this ambiguity) - The Failure of PEMDAS (the writer has a PhD in maths) - Harvard Math Ambiguity (Cajori's book above is talked about here) - Berkeley Arithmetic Operations Ambiguity - PopularMechanics Viral Ambiguity (AMS's statement is here) - Slate Maths Ambiguity - Education Week Maths Ambiguity - The Math Doctors - Implicit Multiplication - YSU Viral Question (Highly decorated maths professor says it's ambiguous) - hmmdaily viral maths (Another maths professor says it's ambiguous) The volume of evidence highly suggests it's ambiguous.

[@nixboox]

@@bigbadlara5304 The answer is one because this video makes a mistake by ignoring that these equations require the distributive property. If you "just graduated" I'm not at all surprised that no one taught this...

[@nixboox]

That is correct. This video made a mistake when it ignored the distributive property. The entire problem is wrongly represented here.

[@GanonTEK]

@@nixboox Distribution can give both answers as it is a notational ambiguity. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. I.e. does 2(1+2) = (2×(1+2)) or 2×(1+2)? Implicit: 6÷(2×(1+2)) = 6÷(2+4) = 1 which is used by academic writing. Explicit: 6÷2×(1+2) = (6÷2×1 + 6÷2×2) = (3 + 6) = 9 which is used by modern programming and also by the American Mathematical Society according to their statement on the matter. That's why it's ambiguous. The rules can't help when the problem is the notation which has to be interpreted first. It's just written poorly and not in line with modern international standards. It should be (6/2)(1+2) for 9 or 6/(2(1+2)) for 1. Those are unambiguous and follow the guidelines.

[@chris-jn2wu]

oh sorry i’m late, KZbin just recommended me this video 3 YEARS LATER

[@tristansgs7739]

chris yeah me too

[@sanpiaupiau6093]

same

[@onurakar3694]

isnt it 2 years

[@chris-jn2wu]

Onur Akar technically, yes

[@everythingever9037]

Me2

[@nelsonojserkis219]

I agree with the fact that this question is written horribly. If someone asked me what’s 6÷2(1+2), I’d say 9, but if someone asked me what’s 6÷2x when x is 3, I’d say 1.

[@RS-fg5mf]

You would be correct... 6÷2(1+2)= 9 y=1+2 THEN 6÷2y= 1 but 6÷2(y)= 9 2y has no delimiter, it's DIRECTLY prefixed and forms a composite quantity by Algebraic Convention. 2(y) has a delimiter and separates the TERM outside the parentheses from the parenthetical sub-expression.

[@GanonTEK]

Yes, it is indeed horribly written. It all depends on which interpretation of multiplication by juxtaposition you use. Academically, multiplication by juxtaposition implies grouping, but the more literal/programming interpretation does not. Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. You'll even find a large proportion of scientific calculators give 1 and a large proportion give 9 if you type exactly 6÷2(1+2) into them. (6/2)(1+2) and 6/(2(1+2)) are acceptable notations but 6÷2(1+2) is not.

[@RS-fg5mf]

@@GanonTEK you keep telling lies... giving priority to parenthetical implicit multiplication is not an Academic interpretation implying grouping... You're WRONG.

[@GanonTEK]

Nope, the evidence is clear showing that interpretation has been in use. You're the one telling lies saying rules describe notation which it utter nonsense and easily disproven. Maths rules are language independent. ​@RS-fg5mf

[@RS-fg5mf]

@@GanonTEK you're wrong and willfully ignorant... Keep believing your own lies.. Academically, directly prefixed (no delimiter) coefficients and variables form a composite quantity by Algebraic Convention. Parenthetical Implicit Multiplication is NOT the same thing... 6÷2y = 3÷y by ALGEBRAIC CONVENTION 6÷2(a+b)= 3a+3b by the Distributive Property regardless of your mathematical incompetence... Two different types of Implicit multiplication that are handle in two different ways... 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.

[@spinning_data2974]

To avoid confusion, be clear. If I solve this using computer or ask a person to do it, I’ll make sure my intent is clear by putting more parenthesis.

[@RS-fg5mf]

Then you are disregarding the purpose of the basic rules and principles of math and the Order of Operations... Parentheses are meant to force a change to the STANDARD Order of Operations not to define or enforce the rules...

[@InsanityoftheSanities]

​@@RS-fg5mfNot all people are delusional like you.

[@GanonTEK]

Exactly. Most people would write this properly like (6/2)(1+2) for 9 or 6/(2(1+2)) for 1 Or even better, two line fractions, which are best practice and reduce required brackets too. Good writing is a great skill and well done on making sure you would be clear and avoid confusion.

[@ggibby0909]

After all this debate and discussion, I think we can all agree that this is why we use fractions instead.

[@jamesfiddler1976]

Eventually, yes. This is a fifth-grade expression used to teach and reinforce the order of operations. This is pretty much ground zero. From there, we stop using the obelus in favor of the solidus and vinculum and go into fractions, as well as teaching reciprocals and the multiplicative inverse. People just forget how to evaluate expressions using the order of operations due to lack of practice. Sometimes, all they remember is an acronym and then convince themselves that there are six steps instead of four and that multiplication always comes first when it doesn't.

[@godlikefish1193]

All my homies hate ÷

[@pirilon78]

@@jamesfiddler1976 how the hell do you not use order of operations im highschool? You need to use them for literally any equation

[@jashickey]

@@godlikefish1193 - Me too my friend. All forms.

[@jamesfiddler1976]

@@pirilon78 Who says I did? I never even hinted that we don't use the order of operations beyond junior high. It should be common knowledge that we do.

[@Marc_NL666]

I'm 40 y/o and was taught the historical way in school. I don't feel historical though. I feel f*cked over because somewhere along the line people decided to change the rules of the game (and didn't inform me!!)

[@Marc_NL666]

Good ol' "Meneer Van Dalen Wacht Op Antwoord" for the Dutch viewers...

[@Marc_NL666]

@@manofculture9051 Your mom sends her regards! And dinner's at six, be on time please.

[@kennethmiller2333]

The problem isn't that the rules were changed; the problem is that they are being misapplied.

[@HQBergeron]

I hate order of operation squabbles. That is not math, it is convention. If there is a governing body for math they should get together and design a convention that is definite, obvious, and universally agreed upon and taught. I was taught the historical method, but knew the current method, so I knew there were two possible answers depending on which system you used. (Not counting the latest anti-racist belief that every answer is correct because saying there is a definite answer would be racist.)

[@michaelangellotti4773]

@@HQBergeron Agreed. It's mathematical semantics. #planetpluto

[@rangersun2177]

Boy this had me stumped! Thanks for the explanation!

[@canasya]

Your reference to the history of typography where a division sign was preferred over multiple parentheses is interesting. But today such problems do no service to math literacy or math education. They only create confusion, chaos, and distraction and alienate students sitting on the fence. Math is beauty. There is no scope for ambiguity in it. Only sadistic teachers who derive pleasure in putting others down so as to guard their positions design such problems. I think one easy way to write an unambiguous arithmetic expression is to enter it into an Excel (or Google) spreadsheet which is now easily accessible around the globe. Excel will not accept the division sign, or the parentheses without a multiplication sign preceding it. Thus, you will have to write 6/2*(1+2) if you want 9 as answer, or 6/(2*(1+2)) if you want 1 as answer.

[@Mariam-kg7fr]

I wish you could have been my math professor in college:)

[@SuperChuckRaney]

Math teachers are invariably one step below orangutans in critical thinking. Math teachers drool, Physics rule. A "math teacher" can easily prove that 2+2=5. And be correctly following "math rules". Here with 6:2(1+2)..... These "math teachers" incorrectly apply pedmas by only doing HALF of the 'p'. Do you see a Multiplicative Operator hanging around between 2 and ( ...? no? Then it's "Implied" hence it's the Coefficient of the (1+2). The 2 is the coefficient of the (1+2) variable. It achieved the place in the expression as a reduction of (2+4). Since (2+4) simplified is 2(1+2). 2x+2y is 2(x+y), right? Soo, to insert the Google example here ... some/most of the math calculsators are incorrect for the reason you mention..... and further THE PARENTHETCAL EXPRESSION includes it's Coefficient Modifier. It's the literal definition of Parentheses. YOU ARE ONLY DOING HALF THE PARENTHETICAL EXPRESSION by only solving inside the (). Common solution isn't following pedmas, and just cause a drunk guy in 1920 provided a solution in a bar on a napkin, doesn't make it true. 2+2 still don't equal 5. Sounds like he had good marketing tho.

[@BlueDragon-gv3ky]

I was taught that when there isn’t a * between a number and a parentheses, you had to do them first! Just as if it were 2x

[@SuperChuckRaney]

@@BlueDragon-gv3ky I've repeated that to be shouted down by the chorus. Distributive Property of multiplication, it is. They rip the coefficent off the parentectical. You would NOT do it with an Exponent.

[@harrymatabal8448]

Mr canasta forget parenthesis. If you multiply first then we have 6×3÷2=9.

[@pwalker1360]

As a trained engineer in his forties, I immediatey turned the expression into a fraction. I also have to say I don’t think I’ve ever seen that division sign used anywhere after fourth or fifth grade.

[@RS-fg5mf]

And in 4th or 5th grade arithmetic the correct answer is 9 .... The symbol is found on almost any calculator. Best to understand it than to be confused by it...

[@Superdada]

Funny, I just left a similar comment. I’m an engineer (39 yrs old) and did same as you. That’s the reason engineers and physicists don’t use that silly division symbol.

[@alxlej]

@@Superdada i don't understand the debate about the division symbol. what difference does it make whether you use : or / ? they do mean the same, don't they?

[@borismuller1086]

@@alxlej They mean using fractions instead of a symbol and having everything next to eachother.

[@alxlej]

@@borismuller1086 but the meaning and therefore the resulting operation are still the same, aren't they? what am i missing?

[@ute.fritzkowski]

I used the historical version all my life. I must be rather ancient.

[@michellevanzyl]

Seems we all historical and the new version only rules in special areas, clearly the areas where I’m not. I live in South Africa and here the answer is still 1🤣 should you want the answer to be 9 it would be written as a fraction not a division sign(which can’t even be found on my keyboard, so let’s just all retire the devision symbol and I’d be happy to concede that the answer is 9😂

[@willwalker24601]

Basically the answer isn't "wrong" if you use the historical version... they're just asking different things... in modern math, it you wanted to ask the exact same question as the historical you would have to write is 6÷[2(1+2]

[@ute.fritzkowski]

@@willwalker24601 It comes down to "just use brackets to make clear what you mean". Mathematics is supposed to be a universal language, but there are still a lot of dialects, aka different notations. I see that a lot lately as I am german but using english youtube videos to review some things since I am studying for a new profession. They are doing a lot of things differently than I learned them at school 20 years ago. Maybe they do them that way in schools now too, I don't know. But since such differences exist, one should strive to write expressions as clearly and unambiguously as possible. Most of those "puzzles" thrive on their ambiguouty.

[@CorgiCorner]

I think the difference is people forget about the brackets so they just disappear

[@abhisheksilverrrrr]

✔️✔️✔️👍👍 Correct answer is surely 1 To those who are telling it 9 Dont know how? For this xy ÷ xy = 1 But Its not y²(according to those who are telling answer to be 9) Similarly, 6÷2(3)=6/(2*3)=1 As simple as that...

[@devonm042690]

The answer is 1. The 2 just to the left of the left parenthesis is part of the 'parentheses' or 'grouping' portion of the Order of Operations because of the Distributive Property.

[@RS-fg5mf]

WRONG... The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division. The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them... The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication... The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral... 6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property Parentheses removed... 6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property. Inner parentheses REMOVED This can be further demonstrated using the vinculum.... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ------------ = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.... ________ 2(1+2) = (2(1+2)) two grouping symbols each That over bar (vinculum) is a grouping symbol _______ _________ 2(1+2) = 2×1+2×2 one grouping symbol each Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation...

[@noland3147]

Classic example of overthinking a 7th grade question. Search of math calculator on google and type the question in its simple 3(3) is always multiple and brackets are always done first in grade 7

[@bernie9728]

I grew up in the 1960's and back then 1 would have been the answer. I think one of the main reasons that kids struggle with math is because of problems just like this. 90 + percent of all people will never encounter this type of problem in their lifetimes outside of school. I think kids today to be taught a more practical math Kids today can't make change. If you want to know where we are headed as a society, rent the movie Idiocracy. We are closer to it than you think.

[@philp3512]

Please excuse my dear Aunt Sally. I thought everyone was taught that.

[@RS-fg5mf]

PEMDAS = 9 and some people were taught other acronyms that mean exactly the same thing, like BODMAS

[@godelnahaleth]

That's how I learned it in high school, class of 1998, and then in college in the early 00's... to do it in the exact order of the sentence.

[@RS-fg5mf]

@@godelnahaleth No, you were not taught to follow PEMDAS as 6 exact steps... SMDH Own your mistakes and stop blaming your teachers for your failure to pay attention in class and learn correctly...

[@achyuththouta6957]

@@RS-fg5mf Nope. 2(3) is not the same as 2*3. Anyway it's been 4 years since I came across ÷ sign. I only use fractions and never had to come accross controversial problems like this one.

[@hannahdewert]

Exactly its easy

[@anvesha5406]

Next viral problem.. 1+1 = 2 or 11.. 🤔🤔🤔

[@jdpace4371]

1+1 = 10 ;) 1+1+1 = 11, 1+1+1+1 = 100.

[@parzival8786]

@Jure Lukezic binary smh

[@jdpace4371]

@Jure Lukezic That only works for very large values for 0. I was representing numbers in base-2; however, if we're talking string concatenation then yaaaaaaaaassssss!!!

[@jdpace4371]

@Jure Lukezic So how does it feel that your joke went over our heads? Don't you feel bad for us smug little pedantic bastards? We could have strung that out, like "I was writing in binary" ... "no you weren't" ... "yes I was" ... "no" ...

[@whod]

10

[@RtsFps1]

After learning calculus, this answer is 1. Visualize the division line, 6 is the numerator, the 2(1+2) is the denominator. From there solve the denominator however you want, you’ll end up with 6. 6/6 = 1

[@paulblart7378]

Incorrect, and this has nothing to do with calculus, it's fundamental algebra. Division is division, not an implied fraction. If anything, it's the other way around: a fraction is implied division

[@admiralvirhz]

If you write it as algebraic equation you can clearly see how it’s supposed to be done. X/Y(A+B) = X/(YA+YB), since you need to distribute property of Y among entire parenthesis first, and fully evaluate that before going back to division of X. Using numbers it’s: 6/2(1+2) = 6 / (2*1 + 2*2) = 6/6 = 1 This is how math works. People outside of America aren’t thought any of PEDMAS, BODMAS or whatever bdsm acronym is used. People are thought how order of operations works in practice, often explained by definitions, and orders, and with a help of algebraic equations, since when you remove numbers it’s clearer to see how things are evaluated.

[@paulblart7378]

@@admiralvirhz Incorrect. The distributing property is multiplication, which has no precedence over division. It would be wrong to distribute 2(1+2) before doing 6/2. The first operation would leave you with 3(1+2) and then you can distribute to get 3+6=9

[@admiralvirhz]

@@paulblart7378you’re making logical error here. Multiplication doesn’t have priority over division, you’re right about this and it’s set in stone, but to fully value what’s inside parenthesis you need to distribute 2 over it. There’s no sign of multiplication, so you need to understand that it is 6 divided by double parenthesis. You see your logical mistake here? It’s not 2 multiply parenthesis since there’s no multiplication sign. It’s double parenthesis. It’s really bad written problem to deal with, I no wonder why so many people get this wrong.

[@paulblart7378]

@@admiralvirhz It's an implicit multiplication. It can be rewritten as 6/2*(1+2), the fact that there isn't an explicit sign doesn't change the problem. I don't know what you mean by "6 divided by double parenthesis", but there is no rule that implicit multiplication groups the operands together. You would do 6/2 first, then multiply that by (1+2)

[@AshokSingh-vv8wi]

When braket operation is completed its bracket symbol is removed by multiplication sign automatically and after wards by applying the bodmas rule division is done firstly and multiplication afterwards to get the final result

[@eliashowe8418]

Maybe that’s why no one past fifth grade uses that division symbol

[@nielsheirbaut4120]

Finally someone with logics

[@supertron6039]

Yup, you're right.

[@alonso3184]

Yes thank you

[@sageight818]

Looks like someone didn’t pass grade 3 English. passed*

[@fatkidinc.3058]

@@sageight818 if you're gonna roast someone on spelling, please atleast be right next time. Thanks

[@geekforu8574]

Just came to check if my brain was working and it is, I’m tired of fighting over the right answers through Twitter 🏃‍♀️

[@blazecraze3652]

Just ask them if they've taken advanced functions or calculus, and then tell them if they ever used the ÷ symbol instead of /. I think thats some pretty solid evidence I should say

[@katie-hy6ii]

NOT ALL OF US COMING FROM THAT TWEET BYEAKSJSKSJ 😭

[@Araqius]

@@blazecraze3652 ÷ and / are both division sign.

[@carterangliss9288]

Same lol

[@paisleigh1989]

@@blazecraze3652 UK don't take calculus or advanced functions as a separate subject so that wouldn't work

[@rafvs69]

I haven't read any comments yet, but you could also be wrong in your interpretation of the order of operations. once you add 1 and 2 you're left with a 3 inside parentheses, and a convincing case may be made that since it's inside a parentheses, your're supposed to resolve that operation before the division, since parentheses trump multiplication/division... so again 1

[@adrianmcbride1666]

Yeah, that's the logic my father taught me. It should not matter what order you do multiplication and division so long as one does not cross addition and subtraction (once parenthesis have been solved). The method being shown means that it is neccesary to solve the division and multiplication in a specific order because you will either end up with a 6/6 or a 3*3 depending on which order you do multiplication and division.

[@OLaro1483]

Answer is 1 Multiplication by juxtaposition always before division, they’re connected. 6 over 2(1+2) 6 over 6 is 1

[@_Just_Another_Guy]

Wrong. Because in PEMDAS/BODMAS (notice the M and D placement) multiplication and division are on equal level. Write the ÷ in fraction form and it becomes less ambiguous: (6 / 2)(1 + 2) 6 (1 + 2) / 2 6(3) / 2 18 / 2 9 This also works with PEMDAS where × or ÷ is done on a first come first encountered basis from left to right after doing the parenthesis: 6 ÷ 2 (1 + 2) 6 ÷ 2 (3) 3(3) 9

[@OLaro1483]

@@_Just_Another_Guy try that with money. Let say you have 6$ in your pocket « a », You want to give it to your 2 friends that are with you that day « b », Each one of them you give 1$ in the left hand « c » and 2$ in the right hand « d ». Answer is how many times can you do that on that day. a ÷ b(c+d) So the answer is?

[@HolzAlaska]

@@OLaro1483 making a text base question which fits your interpretation isn't an argument

[@azavo2502]

dont worry, this issue will never show up in important engineering situations because the division symbol would never be used. instead using a fraction would make everything a lot more clear

[@maximilianomolina5975]

Yep i remeber i got so used to fraction that when i saw the division simbol at first i thought it was percentage xD

[@GanonTEK]

100%

[@anonymes2884]

The real-life solution, as per the ISO recommendation, is just to use brackets to disambiguate. (6/2)(1+2) is totally clear regardless of division symbol used and works for handwriting, calculators, typed documents etc.

[@GanonTEK]

@@anonymes2884 100% and ISO-80000-2 says that ÷ should no longer be used also.

[@wallytruman9433]

I don't know algebra

[@pacha7977]

Oh, I had the 1917 math class, then

[@TheHerothief]

Me too 😂 honstly, not surprised, though

[@MistyHelloKittyWorld2]

Me too.

[@mathfullyexplained]

Try my channel mathfullyexplained

[@cluix6260]

I guess I didn’t.. I got 9

[@daystar862003able]

Me too ✋

[@Jojocl87]

When did they stop teaching this “historical” way because I was in elementary in the 90s and was taught PEMDAS and got 1 as my answer. You can’t all of a sudden change the way math is done because people feel like it!

[@RS-fg5mf]

PEMDAS = 9 NOT 1 You fail to understand PEMDAS correctly

[@Jojocl87]

@@RS-fg5mf and you’re a prick. Congratulations!

[@coreyramstein9778]

Some people were taught that multiplication by juxtaposition takes precedence over explicit operations… hence why 3/2n is 3/(2n) and not (3/2)n The same juxtaposition glue applies to parenthetical coefficients… and in this case, 2 is that parenthetical coefficient. So using PEMDAS, but assigning multiplication by juxtaposition a higher priority than explicit division, the answer is 1. Additionally, if you use the distributive property from the get-go to resolve the parentheses, you get 1.

[@abvolts-animation6008]

I did this in 10 seconds... how can it be viral?

[@ghislainmaury2065]

Check Wiki on the order of operation, it is indication that there is an ambiguity/confusion with expression like 1/2x for some it is (1/2)*x = x/2 and for other it is 1/(2*x) Here we have the same type of problem : a/bc, so same problem : is it (a/b)*c or a/(b*c) If for you it is not confusing, then you do not know math enough, because to remove the confusion in that sort of expression, there is a rule that apply to in-line math expression : "Always add parentheses to delineate compound denominator" So here the first thing to say is that "that expression do not follow the rule for in-line math, so It can't be solved using the order of operation; It has to be corrected first" And the problem is that it seems that a lot of people do not know that rule, so they give the result corresponding to one interpretation or the other ... making it viral Should all of those people go back to school ? Or should only the one that wrote that ambiguous expression go back to school ?

[@j.a_]

@@ghislainmaury2065 English translation please?

[@davidesguevillas]

This is so easy, I solved it in 5 seconds

[@abvolts-animation6008]

Defaulty Boi lol.

[@abvolts-animation6008]

Letucces Satan yeah... I’ll pass thanks I’d gladly take yours though

[@ericsantiago6851]

Let me guess...next viral thing is 1+1

[@onezaproductions3998]

Answer is obviously 0

[@GatsDomineaux]

Lmao

[@romanblecha1102]

1+1=11 no? :D :D

[@andreflores4486]

There are no possible solutions to that equation.

[@zeuxlaught2797]

wrong

[@user-si6fd6lw6e]

If you think about it obediently, the answer is 9, as you can see in the video. The reason why the answer is 1 is that the range of the effect of "/" is considered as "ー". The ー used for fractions is a grouping symbol, and it is visually obvious that the numerator and denominator are distinguished to the extent that the ー is written. But if you don't or can't put it in fractional form, using / instead means that you need to use parentheses as this symbol has no grouping effect. The confusion continues because this point was not taught in school education, or because some textbooks treated it as a "special rule".

[@joshuakehl5891]

Thank you. That makes perfect sense to me now. May I copy your answer to use as a explanation and teaching device?

[@user-si6fd6lw6e]

@@joshuakehl5891 様(sama) Actually, this is my opinion based on information I got from Wikipedia, other videos and all like that. Therefore, I am not confident that the information is absolutely correct. However, I don't think it's wrong, and I think it's easy to understand when explaining to students and others who don't fully understand mathematical signs, so please feel free to use it. It seems to be true that mathematical signs are used in a surprisingly careless way in science and engineering subjects other than mathematics.

[@MrGreensweightHist]

The answer is not 1. In fraction form it would look like this... 6 ----- (1+2) 2 You are erroneously grouping the 2 with the (1+2)

[@joshuakehl5891]

@@MrGreensweightHist erroneously grouping the 2- in fraction form, now we're making sense!

[@chumbue6537]

yes it is dumbass haha@@MrGreensweightHist

[@generator6946]

I never got so tired of something as Algebra. Along with this and too many other horrors made me stage an escape! Still don’t regret it to this day.

[@markbacu7949]

So it boils down to the old "I didn't give the wrong answer. You didn't ask the right question".

[@invalidnoun4746]

@MATHEMATICAL FRAUD HUNTER RESPECT TO MATHEMATICS that is not how math works bro...

[@GanonTEK]

Pretty nice way of saying it. It's like "A union B intersection C" in sets and expecting a certain answer. You can't write that either because it's ambiguous.

[@Owen_loves_Butters]

@@GanonTEK PEMDAS is not ambiguous. 6/2(1+2) Parentheses first 6/2(3) Multiplication and division left to right 3(3) 9 There is no ambiguity. The ambiguity is people not recognizing 6/2(3) = 6/2*(3) = 6/2*3 = 9. Implied multiplication is treated the same as regular multiplication. The it’s the same as “I didn’t read the question correctly, therefore I am not wrong”

[@GanonTEK]

@@Owen_loves_Butters There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. I.e. does 2(1+2) = (2×(1+2)) or 2×(1+2)? Both are widely used. 6÷(2×(1+2)) = 1 (using PEMDAS) 6÷2×(1+2) = 9 (also using PEMDAS) PEMDAS isn't the problem. The notation used is. That's the cause of the ambiguity. That's why there is such a large disagreement and even calculators from the same manufacturer don't agree. You shouldn't write a/bc or a/b(c) anymore. It's not acceptable notation. ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove any ambiguity. A PhD student wrote a paper on the ambiguity called The PEMDAS Paradox if you want to look it up.

[@Owen_loves_Butters]

@@GanonTEK You don’t add parentheses if there are none to begin with.

[@vulcan5176]

Me: Answer is obviously 1 "Answer is 9" Me: Well fuck me.

[@kaicluster6783]

It's actually 1 :) The person explaining made a crucial mistake thinking that () is the same as x. As many explained, it's not

[@thegreatchinesedragon4610]

Atomicninja - It is really 1 the ( ) go first. So it's 1+2 first which equals 3 obviously. Then the equation is 6/2 x 3 (the / is a division symbol). Then you multiply 2 into 3 then it's 6/6 and then your final answer is one. Simple to learn in school easy math.

[@belalmangrio383]

Kai Cluster I don't agree I think the answer is 9 because you add the parentheses P, then you go from left to right so 6/2=3 and 3x3=9

[@marving.5436]

first off 6/2=3 is already wrong because there is no multiplication 6/2(3) is not the same as 6/2*3 or 6/2*(1+2) if you want to elimate (1+2) the equation should be (6/3) / (2(1+3)/3) then you get 2 / 2 =1 or simply just 6/6=1 The correct answer is 1 because 2(3) is somewhat like y(x) which means the value of y is multiplied by x time.. going that approach 2(3) is interpreted as 2+2+2 = 6 6 / 6 = 1 the algebraic expression is z / y(x)=

[@sam9s]

Divide should precede Multiply so 6/2x3 should be 3x3=9

[@GamestaMechanic]

Say the equation out loud or type it: "six divided by two times, in parenthesis, one plus two." That would be 6 / 2 * (1+2) You cannot say the equation to get it to equal one UNLESS you specify that there's a bracket like this: 6 / [2(1+2)] Since there are no other extra brackets specified, order of operations say left to right and you get 9. Some school calculators get 1 because they automatically put assumed brackets because that is what is most likely meant by the user, however, what is orginally typed would equal 9, not 1. Most calculators take why you typed literally and give you 9, which is what is supposed to happen. Unfortunately people use calculators that make assumtions and then the people think that's what is supposed to happen. This is why writing code needs specifications, you cannot just let the computer make assumptions for you or else it would be wrong A LOT, you'll be wrong.

[@GanonTEK]

The issue with the first line there is that it is not clear how to read it as we don't know what interpretation of multiplication by juxtaposition they used. It's like how "What is 6 divided by 2 times 3?" written like that is also not clear. It's bad writing. You need commas for clarity. So, "What is 6 divided by 2, times 3?" This would be (6/2)×3 = 9. You could also write "What is 6, divided by 2, times 3?" for the above answer. Or "What is 6, divided by, 2 times 3?" This would be 6/(2×3) = 1. Proper notation and writing makes all the difference. This is just a badly written expression. That's all.

[@markprange2430]

The grouping of 2 and (1+2) is what told you that multiplication was involved. 2(1+2) and 2(3) are groups. The group remains as (2 × 3). The original expression showed 2(1+2) as a group.

[@Dojibu]

I was under the impression that multiplication came before division in the order of ops. Glad to see my math wasn't completely off since I ended up with 1 in the end. Still, need to brush up on that stuff.

[@Arcessitor]

It does, depending on the system. Video maker is wrong.

[@tahmidali603]

funny how i got both answers and then blamed my math teacher for making me so indecisive

[@stephensteinhauer3346]

Same. I got both answers and then watched the video to see what obscure rule they would pull out of their hat.

[@andinbriwel1092]

The "1917" example is exactly how I was taught both in high school algebra and in college algebra. That was in the 80's, not 100 years ago, lol.

[@Nga_Babaye]

What does attending school in Appalachia have to do with it? Yes, I did and I was taught the 1917 way I guess from 1996-2013. WCU was still using in it in 2013, and so was all the other kids from other parts of the US.

[@realisticdan3302]

same here...thank god for comments almost gave up on my math.

[@Henry-mq2fv]

honestly i dont know how long ago people didnt use the order of operations but im sure that in the 80s all mathematicians used it. id go as far as to say it probably existed at least a thousand years ago

[@sanfrand22]

Don't be jackass Steve

[@Nga_Babaye]

realistic dan Appearantly my wife was retaught the correct way when she went to UMiss. Guess that why my kids always come home with the wrong answers when I help them do their school work

[@frankhooper7871]

The simple fact that different so-called mathematical experts have different interpretations proves that there is no correct answer.

[@MisterGeezy]

I'm 67 and old school math. I came up with 9.

[@RS-fg5mf]

Good job.

[@normalhuman7969]

History biggest question: How did a 5th grade math question create a riot on the media

[@Mobilizes]

It's not even 5th grade, it was 3rd or 4th grade

[@medicine8470]

I learned this in 5th grade and solved this in approximately 5 seconds.

[@TannerK.]

Please read this comment, thank you. Solve for the 2 in parentheses, it is not a bracket [ ] 6/2(1+x)=9 3(1+x)=9 3+3x=9 3x=6 X=2 The problem is that people who think that it is 1 believe that after simplifying 2(1+2) is that they think it is the denominator of the fraction. For that to be true, there must be a parentheses in front of the 2.... (2(1+2)). You will do that first if that was in the problem, but it isn’t. 6/2(1+x)=1 6/2+2x=1 Now you see that there is a fraction, but what can it be. If it is 2+2x, you get 2 as your final answer, which is correct. If it is just 2, you get -1, which is incorrect. However, you do division before addition, so you do 6/2 to get 3, eventually getting -1 as the solution. 3+2x=1 2x=-2 X=-1 This is incorrect, because we are trying to solve for 2 in the parentheses... 6/2(1+x)=1 X should equal 2. People think that after distributing the 2 into (1+x), the whole thing stays in the parentheses. It disappears after you distribute. Thank you for your time.

[@zaherramashaa3762]

@@TannerK. literally the best explanation thank you

[@conscious5122]

Lol

[@frozenzenberry4101]

Congrats, this just became topical again. Expect another influx of views my man.

[@darlinn7195]

i just came to check if im braindead turns out nah

[@kolowar6600]

I dont know why people think this is hard

[@Nash-]

@@kolowar6600 because they failed second grade

[@qc7511]

Anf another influx of ignorants disliking this video again.

[@sadbird6644]

I’m here to see if I’m brain dead this is basic I’m not through the video yet so I’m pretty sure it’s 1

[@AndreMachado-lb8cp]

In physics, when we write units, we use a dot to specify these multiplications.

[@Mrjpisgreat]

The answer is 1. Especially since the distributive property applies to the expression in parenthesis. 2(1+2) = (1x2+2x2) = (2+4) = 6

[@RS-fg5mf]

The Distributive Property supports 9 not 1 The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division. The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them... The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication... The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral... 6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property Parentheses removed... 6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property. Inner parentheses REMOVED This can be further demonstrated using the vinculum.... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ------------ = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.... ________ 2(1+2) = (2(1+2)) two grouping symbols each That over bar (vinculum) is a grouping symbol _______ _________ 2(1+2) = 2×1+2×2 one grouping symbol each Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation...

[@tylermartin9225]

in short don’t use the outdated division symbol, just use the typical numerator and denominator removes all uncertainty

[@charliedallachie3539]

It still becomes 6/2*3. So if it’s multiplication before division that’s 6/6 =1

[@o_sch]

@@charliedallachie3539 thats not using the numerator and denominator, when you use an actual numerator or denominator you would have a certain part be under it. Either 6/(2*3) or (6/2)*3

[@charliedallachie3539]

@@o_sch yea I understand the two answers but in other problems which is which? I’ve always wondered PEMDAS in general I’m sure there’s a complex mathematical proof of it out there somewhere Edit* there is no proof it’s a convention.

[@JakobSchade]

@@charliedallachie3539 but it isn't multiplication before division. they are equal, so it is left to right.

[@Wexexx]

@@JakobSchade Sure, but that's if you use PEMDAS or whatever else. There's still plenty of books where they don't use PEMDAS and have a difference between implicit and explicit multiplication. 2*3 is explicit (a * sign) and 2(3) is implicit. In that case, implicit is many times higher of importance than explicit. So 6/2(1+2) would simply be 6/6=1.

[@lena__speaking7080]

I solved this in 5 seconds this shouldn't be a problem for anyone who attended school.

[@lailaaa24]

Same

[@jessetarpley4]

Exactly, I always loved doing really long order of operations math problems.

[@_mahiii]

Same, idk why they are making fuss over this? I mean this is taught in school....

[@suboryoulldiein24hours48]

@@_mahiii lol he got 14 million views it served its purpose

[@thejaramogi1]

Unless your History teacher doubled as your Maths teacher!

[@mrei7119]

I think this question is more about the convention of math, rather than the essence of the math itself.

[@thepsychologist8159]

Conclusively, the experts say this. The equation is ambiguous and indeed, it can yield two different answers. Like the use of language, to convey something such that it can't be misinterpreted, it must be delivered with clarity, the intention should be made clear. The same with maths equations. To yield only one result, the equation should not be written with ambiguity and the intention of the writer must be clear. If it does or can, the equation should be re-written.

[@noland3147]

The brackets are just grouping things together, if there is no operation to the left of the brackets, assume a multiplication, otherwise the operation tells you what to do. Truth is math doesn’t have 2 answers he gave a answer from 1917 rules which is when we figured out how to make a light switch😂

[@ivan0912]

People: nine. Me, an intellectual: *nein*

[@rydh6zgjhbfvrwhb259]

Ivan09 Ja Ja! Ich spreche auch Deutsch Translation: Yes yes! I also speak German!

[@Brontok]

@@rydh6zgjhbfvrwhb259 Grüße aus NRW XD

[@derechtepilz]

@@Brontok Heyyyyyyyyy! Ich grüße dich zurück! Natürlich auch aus NRW!

[@derechtepilz]

@survival pete ??

[@Kochoplays]

Ivan: posts a meme containing a fraction of german The comments: iCh bIn dEmEnT

[@ZoeTheCat]

As you climb higher in math, virtually 100% of physicists, engineers and mathematicians will interpret the answer as 1. There is no debate over this at all. The implicit multiplication of 2 on the bracket is a SINGLE quantity that takes precedence prior to division. Most physicists/engineers/mathematicians would never even write such a potentially ambiguous expression. They would instead write 6/2(1+2) where the / is a horizontal line. Alternatively, they would write 6/(2(1+2)) leaving NO ROOM FOR AMBIGUITY. PEMDAS is NOT universally accepted. The implicit multiplication on the bracket does indeed take precedent. You are doing a disservice to kids trying to learn mathematical protocols. PEMDAS isn't the total protocol.

[@Araqius]

Totally wrong and nonsense. What is the inverse of "implicit multiplication"?

[@NeelDhar]

I agree with ZoeCat

[@TomJacobW]

It is really funny indeed, because it gives a hint from "where people are coming". I studied physics for some time and it was completely obvious to me, that a juxtaposition has a higher order than "read left to right". It's "obviously" 1. As mentioned; 6/2y with y=1+2 is 3/y, not 3y.

[@ZoeTheCat]

Tom Yes. You give a great example. According to PEMDAS, x/yz = xz/y which is OBVIOUSLY unconventional. The implied multiplication of yz binds the two components of 'y' and 'z' together.

[@cameronparkes6629]

Zoe TheCat Agree with you 100%. I'm an engineer and my first response was to say the answer is 1. Everything on the right is just a factored 6.

[@Man_Cave]

These often confused me in college, for once the parenthesis were added together, the parenthesis would often be dropped. Without the parenthesis, multiplication is no longer implied.

[@RS-fg5mf]

6÷2(3) and 6÷2×3 are mathematically the same. When there are no OPERATIONS left INSIDE the parentheses to evaluate you can remove the parentheses and replace with an explicit multiplication symbol or leave them to represent implicit multiplication which is simply multiplication without the need for a physical multiplication sign...

[@Man_Cave]

You jogged my memory. I used to insert a dot to the left of the parenthesis once it was solved to remind myself it was still multiplication. It's a shame that no job I had, after college, required me to know much algebra. It's not like riding a bicycle. You can and do forget.@@RS-fg5mf

[@nickmcginley4570]

@@RS-fg5mf Haha, here you are after all these years, still lying, still being ignored, and still devoting your life to perpetuating your miseducation!

[@mikosan77]

Yay! I still remember how to do these problems.🙂

[@Thedoctv]

Imagine using division symbol instead of fraction.

[@mrpickle8959]

Yeah that's how that goes...?

[@hochigaming14yearsago90]

It's 5th grade man

[@sensei5668]

@@mrpickle8959 fractions is also division 1/2 is the same as 1÷2, and this works for anything.

[@Langweiler11]

@@sensei5668 sure it is the same thing. but with fractions the error couldn't happen as the order is directly visible. I personally haven't seen that operator once in university. If you are forced to write in one line (e.g. in programming) people use "/"

[@sensei5668]

@@Langweiler11 that's my point

[@zkyz735]

In France i've been taught it in a way, that this equation equals 1. Basically 6/2(1+2) has brackets. We were taught that brackets were always a priority with the number infront of it. So what we would do is first 2*1 + 2*2 = 6, and once we got the brackets completely gone, we can finish the equation which would be 6/6 = 1. Also even if i added the numbers, it was always important to clear the brackets. Here 6/2(3) still has a bracket and doesnt just dissapear. So i would multiply 2 and 3 to get rid of the bracket. Thus we still receive 6/6 = 1 I was always taught this way and was surprised seeing that the correct answer was 9. This blew my mind

[@lastannahme]

I'm pretty sure that in germany we were taught the second answer as well (equasion equaling 1) for the exact same reason you describe here (getting rid of the brackets first) and then finally dividing anything on the left side by what is left on the right side. From my point of view the answer 9 is "wrong". And even if it's just a "rule" thing, we'd better universalise that rule. To me, somehow, the answer "1" also makes more sense in a mathematical- asthethical way.

[@j.r.arnolli9734]

People in Europe, born before 1970, learned, that multiplication goes before division. Just a fact. I mentioned 1917, because in that year, in the USA it became official that multiplication and division are equal and You start from left to right. In 1980 is was commpn practice all around the globe. ( In the Netherlands it took till 1992 to use the 1917-method). Mathematics is about agreements and those changed over the years to an (new) international standard...

[@lastannahme]

@@j.r.arnolli9734 Thank you for this insight. Anyhow I was born in 1980 and I'm pretty sure that if I showed this "problem" to my old schoolmates/ peers here in germany 99% would come up with the anwer "1". Yet again maybe I'm wrong. If this really is new international standard it still doesn't make a whole lot of sense to me in terms of logical usage of mathematical language.

[@antsfur]

Yes you are correct the answer is 1. You solve the brackets first to get a number on its own then you finish off by 6 ÷ the answer in the brackets. If your answer is 9 then you are inventing your own mathematics !

[@barbarashirland9078]

I was taught that way also. In the US, but three generations ago. I’m old.

[@sarahmaerosa2712]

i saw this once on a meme and got curious so I searched the answer up, and now i feel smart and educated by being a part of a comment section full of people who have done math their whole lives. XD

[@user-bt9gs1yt5v]

I would have thought the presenter of the equation asked us to "Divide six by twice whatever value is within the brackets". The equation can be written as A divided by B where B=2(1+2). So if [A divided by B] = 1 and A =6, then B=6, This implies 2(1+2)=6 which is correct. If [A divided by B] = 9, and A=6, then B= 6/9or 2/3. This implies 2(1+2)=2/3 which is incorrect. The rule 'brackets first' is short for "Solve the brackets first" This implies the removal of the brackets from the equation first. In adding 1+2, you do not get rid of the brackets! You are still left with 2(3), and one cannot remove brackets from the equation without solving them. In this case you must multiply 3 by 2 to the brackets. You cannot just add in or take out brackets ad lib! In applying values, 2(3) is a single Value and is expressly implied in the equation. 2 x (1+2) is two values and is not implied in the equation. It appears to me the problem lies in the computerisation of the equation. To me the equation is simple and unambiguous and means the same now as it did 50 years ago!

[@SPACKlick]

Not sure where you're getting your "modern interpretation" from but certainyl in the UK 6 ÷ 2(1+2) wouldn't be treated as (6 ÷ 2)(1+2) because the implicit multiplication where no dot or multiplication symbol is used takes the same priority as the bracket. So, 6 ÷ 2(1+2) would be read as 6 ÷ 2y where y=1+2 If the original were written as 6 ÷ 2 x (1 + 2) then 9 is the correct answer but when written as 6 ÷ 2(1+2) 1 is still the correct answer.

[@SPACKlick]

I have and did.

[@UnderMan]

You are wrong. It's the same thing whether it's written as 6 ÷2(1+2) or 6 ÷ 2 × (1 + 2). The multiplication symbol is implicit. The only way it could be written to equal 1 is 6 ÷ (2(1+2)).

[@SPACKlick]

UnderMan UnderMan So are you saying, you would read 6 ÷ 2y as 3y? Because if so, I've no idea where you're doing mathematics.

[@Araqius]

SPACKlick 6 ÷ 2y = 3y

[@SPACKlick]

Yep very sleepy there when I wrote that. I meant that in order of operations 2y is treated there as a single unit, 6 ÷ 2y = 6/(2y) rather than (6/2)y ie 3/y vs 3y

[@StuartLynne]

When programming, the correct answer is to never leave any ambiguity, so always add enough parenthesis to ensure that anyone reading it will understand your intention. So write 6 / (2 * (1 + 2)) OR write (6 / 2) * (1 + 2). Both are correct, but only one would be correct depending on what your intention is. So always make sure that you enter something that cannot be misinterpreted.

[@RS-fg5mf]

So you are saying that we should be forced to write 5+(2×10) because too many people fail to understand the basic rules and principles of math and incorrectly believe that 5+2×10= 70

[@StuartLynne]

@@RS-fg5mf Yes, for the same reason we add comments. Make sure that we know what is happening and that people reading it in the future know that we know what it means.

[@RS-fg5mf]

@@StuartLynne then the Order of Operations and the various properties and axioms of math become redundant if you're going to add crutches for people who fail to understand and apply the basic rules and principles of math correctly

[@onemorelisa3785]

@@RS-fg5mf When math is being taught completely differently between generations, it’s bound to be misinterpreted. I’d rather them be in-depth and redundant so that people in the future won’t have to just assume anything. Assuming things causes a lot of problems.

[@RS-fg5mf]

@@onemorelisa3785 math is only being taught differently if the prrson teaching it is incompetent.... The Order of Operations and the various properties and axioms of math were established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses... The basic rules and principles of math have been the same for over 400 years... Math is based on rules not popularity or personal opinion. Failure to understand and apply the basic rules and principles of math correctly as intended is not a valid argument against them...

[@jmm1233]

all division is multiplication , so it 6*0.5*3 which = 9 by doing this makes it easier to read in reverse as (2+1)6/2 as that makes (3)*6*0.5 which = 9 1/2 * 3 * 6 also = 9 can also write the equation out as 3/2(3+3) = 9

[@easy_s3351]

The problem remains....do you consider 6÷2(1+2) to be 6*1/2*(1+2) or 6*1/*(2(1+2)). What if it was written as 6÷2a? Would you consider that to be 6*1/2*a or 6*1/2a? Does a÷bc=a/bc or a÷bc=ac/b? The problem is ambiguous and should have been written as either 6÷2*(1+2) to get 9 or as 6÷(2*(1+2)) to get 1.

[@redcrewmate1813]

6 ÷ 2(1 + 2) * Parentheses come first. 6 ÷ 2(3) * Rewrite with ×. 6 ÷ 2 × 3 * Multiplication and division are the same precedence, so solve left to right. 3 × 3 9 is the answer.

[@GanonTEK]

It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4

[@TrevorKeenAnimation]

So, the problem that yielded an answer of 1 in 1917 yields 9 today. Wow - inflation is everywhere!

[@TeachUBusiness]

Trevor Keen You are correct. I was taught to do the the parenthetical expression, multiply and then divide.

[@tozendai840]

The world parenthesis should be eliminated from all human language

[@claudiosass]

Trevor Keen j

[@RemingtinArms]

you were taught wrong. the end.

[@DustinSilva]

It's because the creator of this video is a moron

[@apocalypse_-_0076]

Watching 5th grade math at 2AM even though I'm 15🤦‍♂️

[@jctorres6112]

well theres only one truth. math is attractive 😂😂

[@anelaknezevic5680]

Sameee 😂

[@melodymartinez7003]

Same

[@asher7124]

Im 11

[@blinkcatmeowmeow8484]

Im 15 too

[@aram5642]

Could be that a distinction should be made between ÷ and / as well as between 8y and 8•y. Today we are using those notations (within each pair) interchangeably, but maybe it shouldn't be so and each variant should actually affect the order of operations in its own way.

[@MrGreensweightHist]

"a distinction should be made between ÷ and / " They are the same sign. "as well as between 8y and 8•y." These are different, and the problem, as written, would be more like 8•y

[@aram5642]

@@MrGreensweightHist they are the same sign -> this is exactly what I am pointing at. Maybe they shouldn't be.

[@heroinwaltz5267]

I'm quite certain people did not argue that correct answer was 1, but because of poor educational system. I was taught correct order around 25 years ago and still remember it.

[@alaynac1610]

It’s 9.. how is this even viral, it’s 5th grade math.. Also, I’m referring to PEMDAS which is taught in 5th grade. Watch the video if the answer you got wasn’t 9..

[@rohmanatasi1771]

It's 1

[@angie___nn7352]

It’s 7

[@PankajKumar-ts9hq]

Noob, its one

[@dominatinggamerguy5795]

Not even 5th grade math. It’s like 2nd grade math

[@realhi144]

It depends on pemdas or bodmas

[@_MrMoney]

For things like this we don't use the ÷ symbol anymore, it's just easier to use a fraction so you can differentiate (6/2)×3 from 6/(2×3)

[@vostokk]

That's exactly what I was thinking.

[@Lucian24]

It's not its 1

[@Lucian24]

You work left to right

[@_MrMoney]

@@Lucian24 What I meant is that we now use fractions so we don't need to specify which goes first. You operate all on the top and divide it by all on the bottom. So you can differenciate: (6/2)×3 = 3×3 = 9 From 6/(2×3) = 6/6 = 1

[@Lucian24]

That makes sense

[@LaReganto]

This is really interesting because in another video someone told me the answer of 9 is incorrect because that way is learned in preschool and isn't correct because it would be only a way to make math easier for children. I personally have learned that way you get 9. But I like math and I understand the logic behind the 1 answer completely. Even the 2 answer is understandable.

[@RS-fg5mf]

The only correct answer is 9 and the person who told you it was 1 is mathematically incompetent...

[@GanonTEK]

Yes, it's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. So both answers are valid. Plenty of external evidence too. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4 The people who are wrong are those who think there is only one correct interpretation here. Like Presh in this case, unfortunately, and the other guy replying to you too.

[@nickmcginley4570]

@@RS-fg5mf Ignore RS, he is a psychotic first grade teacher.

[@elouanlahougue]

@@RS-fg5mf Are you in first grade? Aren't you able to understand a video? 1 is logical according to one meaning of the confusing division symbol.

[@RS-fg5mf]

@@elouanlahougue 1 is NOT logical when you actually understand and apply the Order of Operations and the various properties and axioms of math... If 1 were logical based on the premise of the obelus that this video entails 6÷2×3 would equal 1 and 6÷2+4 would also equal 1 I think you need to pay more attention to what's being said in the video...

[@gshalabama]

To make this easy look at the division symbol, it’s a dot over a line over a dot. This tells you that what is on the left of the symbol goes on top and what’s on the right goes on the bottom.

[@shadowfoxx14]

Interpretation is the key word. The problem should, in my opinion, always be written as: (6÷2)(1+2) so there is no more confusion on interpretation

[@nox_cadit]

But if this is on a test, you want to know that your students actually paid attention and learned correctly, writing the way you did removes the so called "ambiguity" (which there is none) and then there will be no way to actually know if they have learned correctly

[@RS-fg5mf]

When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended you get the only correct answer 9 If you don't apply the basic rules and principles of math correctly then you are already confused.

[@lolmom3590]

@@RS-fg5mf the answer is 1…

[@RS-fg5mf]

@@lolmom3590 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....

[@RS-fg5mf]

@@jonnel4038 x÷2y = x÷(2y) by Algebraic Convention... BUT x÷2(y)= x÷2*y by the Distributive Property... Parenthetical implicit multiplication does not have priority over division. 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 not just the number next to it. The correct answer is 9

[@lohphat]

The term 2(1+2) = x(a+b) = (xa+xb) = (2x1+2x2) = (6) Thus: 6÷2(1+2) = 6÷(2x1+2x2) = 6÷(2+4) = 6÷(6) = 1 There are 4 terms in this equation, not 5: 1) the 6 2) the 2(a+b) containing: 3) the 1 4) the 2 If it were written as 6÷2*(1+2) THEN there would be 5 terms and answer would be 9. 1) the 6 2) the 2 3) the (a+b) containing: 4) the 1 5) the 2 The 2(1+2) is a SINGLE term which must be resolved first before being divided by 6. The issue is that 2(1+2) --> (2x1+2x2) --> (6) RETAINS THE PARENTHESES thus resulting in 6÷(6) = 1

[@Tsskyx]

You distributed. Distribution is a property of multiplication and division. So essentially you multiplied first. But you can't do that! Parentheses go first, as stated in PEMDAS. So first you must reduce (1+2) to (3).

[@lohphat]

Is 6÷2x = (6÷2)x or 6÷(2x)? 2(3) is an unresolved term.

[@Tsskyx]

lohphat 2(3) is not a single unresolved term, it is two different resolved terms multiplied together. You can't just divide 6 by it, you must remove the brackets first and turn it into 2*3. Then you divide, so 6/2*3 = 3*3, which is 9. Remember, just because there is no multiplication sign (it's called implicit multiplication), it does not make it a single term - after you solved 2(1+2) into 2(3), you can (and HAVE to) remove the brackets, so 2(3) = 2*3.

[@lohphat]

***** By replacing the () with * you've changed the structure of the equation by changing operators. It didn't start as 6÷2*(1+2)

[@Tsskyx]

lohphat You're not changing anything. Who taught you that 2(3) is not 2*3? It's called implicit multiplication. It is used when the multiplication sign can be substituted by a pair of brackets. It does not change the value of the equation at all. 2(3) = 2*3 in all cases. Just because the multiplication is implicit, it does not hold any priority over the normal (explicit) multiplication, unless you want to invent a new rule or something.

[@theodavies8754]

Both. Probably because I don't know much about the subject. I just fix things that people break, which keeps me very busy.

[@kittylearns1867]

basically making homework tougher by 'rules' instead of solvng actual maths

[@razvanruse68]

Fifth grade math: im gonnna end America's whole career

[@glitzyx4x852]

Răzvan Ruse literally

[@Name-ru1kt]

Anthony Lo hey guess what he deleated his comment now yours makes no sense

[@glitzyx4x852]

Fallout Mods who deleted what

[@razvanruse68]

Um what?

[@jacobschoenbaum7689]

Just the people that aren’t in school anymore

[@stefans4562]

srsly. That's just an example for bad notation. Of course the answer is 9. It's still bad notation. This division symbol just shouldn't be used anymore.

[@bythetimeyoufinishedreadin9083]

EXACTLY! It's a matter of confusing notation, but in *that* notation, the answer is 9. If you rearrange it using the "/" sign, the answer will be different! That's because (1+2) is actually a coefficient of 6/2!

[@stefans4562]

nice name.

[@stefans4562]

***** I am aware of that. But I guess even at my university there are some people who would get this wrong. This is terrible notation that is not useds in real life.

[@stefans4562]

***** lol

[@stefans4562]

I want to make this clear: This is NOT a math problem. This is a notation problem. There are many differnt mathematical notations.

[@vwood2]

I suspect it’s an age thing. The method we were taught in Canada in the 1960s & 70’s results in 9.

[@RS-fg5mf]

It's still 9 in Canada and anywhere else in the world that actually understands and applies the Order of Operations and the various properties and axioms of math correctly as intended...

[@janiturunen4059]

When I had math at school we were taught that implicit multiplication is done before explicit multiplication or divison so that equation would be read like you would read 6/2x and not like 6/2*x.

[@MrGreensweightHist]

You were taught incorrectly.

[@noland3147]

You don’t remember what you where taught or was taught wrong😂

[@boopster4795]

1:59 “And this gets us to the correct answer of dine”

[@chickenfeet6123]

Nicolas T-R 😂😂

[@spookbarkee9345]

You brainwashed me and now that’s all I here instead I of nine

[@kingblue8868]

lol

[@ZJC_9]

Speechless of laughter 🤭😂😆

[@someguy5354]

lol

[@heikounsold4920]

I came up with 1 as my solution. After listening to the explanation I found it logical that 9 would be correct. Now I tested my calculators to see what their solution would be and the first one (used as a standard calculator at schools) came up with 1. The second one (a more sophisticated graphic calculator also used at schools) showed me two different solutions, depending on the writing: 1. 6÷2x(1+2)= 9 2. 6÷2(1+2) = 1 but it changed the writing into 6÷(2(1+2)) It's such a simple arithmetic problem but even calculators are challenged. I love it!

[@RS-fg5mf]

When you understand and apply the Order of Operations and the various properties and axioms of math correctly you get the only correct answer 9. Some calculators are not programmed to handle parenthetical implicit multiplication correctly....

[@gajarajmaharjan881]

I'd still say 1 is the right answer because PEDMAS is skipping the priority of multiplication by juxtaposition. TI calculator use PEMDAS but Casio and hp have returned to PEJMDAS. It's mainly only North American "teachers" (note that, teachers not mathematician) who insist on PEMDAS.

[@RS-fg5mf]

@@gajarajmaharjan881 you and many others are confusing and conflating an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing... 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... 6÷2y the coefficient of y is 2 BUT 6/2(y) the coefficient of y is 3 ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM 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

[@gajarajmaharjan881]

@@RS-fg5mf I think, what's going on here is how you understand your question. When I see 6÷2(1+2) I see as the question that if there are 6 apple and you have to divide amongst two groups in which each group has one adult and two children. So how many apple do each person get. Now you tell me how do you interpret this question in real life? You are taking PEMDAS too literally and forgetting that the multiplication by juxtaposition takes priority over multiplication and division. The answer that 6÷2(1+2)=9 doesn't make sense in reality. What is the situation?

[@RS-fg5mf]

@@gajarajmaharjan881 you don't interpret a math expression you evaluate a math expression following the basic rules and principles of math... Your word problem would be correctly written as 6÷(2(1+2))= 1 You have 6 bags divided between 2 groups and each bag contains 1 red apple and 2 green apples. How many apples did each group get?? 6÷2(1+2)=9 apples per group.

[@mrsquid_]

I've just always calculated bracket multipliers as part of the parentheses/brackets step. since the b in a(b) could be any calculation, not just one number(example: 2+1), and it doesn't always equal to just one number(example: {2x+1}, i tend to calculate the bracket with it's multiplier a first. in this case, however, you could use the Assoziativgesetz to remove the brackets, and turn it into just a multiplication, in which case the answer would be 9.