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

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

Ok, you ACTUALLY want my answer? I can't just clickbait you all and not tell you which I ACTUALLY prefer? OK fine, but I can see from the comments I'm going to upset a lot of you:D If I wrote this type of thing on the board, my natural inclination is to write division as a big diagonal dash instead that lumps the 2(1+2) on the bottom. That is, when I take this algebraic string of symbols and write it out - without using any brackets - the way I would write typical calculus expressions in my classes, then I would habitually write it in a way that use spatial relationships that interpret it as being 1. If I wanted it to be 9 I'd be explicit and put brackets around the (6/2), when writing on the board. Using spatial relationships (i.e. not a strict left-to-right application of BEDMAS) is extremely common in math, it's just that normally you don't have as your starting part a character string like this because, as I say in the video, the most important part is to be explicit about what you mean when there is a possibility of ambiguity!

[@yourmomsfilms]

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

[@NeoiconMintNet]

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

[@yourmomsfilms]

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

[@NeoiconMintNet]

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

[@MuffinsAPlenty]

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

[@AnthonyOliverio]

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

[@DrTrefor]

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

[@michaelbauers8800]

Especially with Smalltalk, which I don't think has normal procedural language precedence. I have programmed in C++ for a few decades, and I mostly know the rules, but as you say, when in doubt, write parenthesis, and people will say this in code reviews if they don't think it's intuitively clear.

[@RemunJ66]

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

[@Delirium55]

..and that's how we got Lisp.

[@NeoiconMintNet]

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

[@GanonTEK]

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

[@digambarnimbalkar8750]

It means 1 out of 10.

[@JustVezix]

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

[@GanonTEK]

@@digambarnimbalkar8750 The question is ambiguous and badly written to modern standards so it is both 1 and 9 at the same time (depending on which interpretation you are using - academic or programming) which is the joke 😋. If I wanted 1 I'd write 6÷(2(1+2)). If I wanted 9 I'd write (6÷2)(1+2) or 6÷2×(1+2). These would be unambiguous and the joke wouldn't work then and we wouldn't have the video either as there would be no discussion.

[@GanonTEK]

@@JustVezix Schrödinger's rating 🤔😋

[@severeaura6540]

In other words 6÷2(1+2)/10...?

[@Sindraug25]

My understanding is that "multiplication by juxtaposition" is a separate step in the Order of Operations that comes before the "multiplication and division" step, and PEMDAS leaves it out for some reason; and that mathematicians, engineers, anyone who does math for a living, does the juxtaposition first and would solve the problem in question as 1. We really just need to clear this up by changing PEMDAS to PEJMDAS.

[@jaysonkmendoza]

A lot would follow this rule, but it isn't actually a universally accepted rule of math. The problem here is that the mathimatical community hasn't bothered to settle this for a good reason. No matter what rules you make its always possible to poorly communicate a math problem. This is the same as saying when writing a sentence in english I can misscommunicate by using unclear verbs, sentence structure, or grammar. The point of mathimatical expressions is to clearly communicate an idea just like in any other language. Using ambiguous structures that can have multiple inturrputations is just poor math and you wouldn't find any formal math proof submitted for peer review using them. Math papers avoid the old division symbol because it had two different inturrputations over time. They also clearly communicate the term breakdown using brackets. This question and others like it failed to do that and that leads to multiple correct answers depending on inturrputation used.

[@jamesschaaf612]

PEMDAS leaves it out because PEMDAS is a simplified version of the order of operations that is taught to young kids. The real question is why the order of operations isn't revisited in the US after concepts like functions, multiplication by juxtaposition, and unary operators are understood.

[@MrGreensweightHist]

The correct answer is 9

[@ZS-bg7jo]

@@MrGreensweightHist The 'correct' answer is "fix your notation". 1 and 9 are both right and both wrong depending on if you respect juxtaposition. 1 ÷ 2x vs 1 ÷ 2 * x are two different operations.

[@wrrsean_alt]

I 100% agree! AND....the most important thing is bringing PEJMDAS to primary teachers/education authorities' attention. It is here that most people learn and take PEMDAS as being the correct rule without any other consideration. Even calculator companies need to be consistent. For example, using a CASIO Scientific calculator [Model fx 82AU] gives an answer 9 for this problem. While a CASIO Scientific calculator [Model fx-83GT PLUS], gives an answer 1. The first calculator obviously is programmed to use PEMDAS and the second [same company different model] uses PEJMDAS. So, this means one person in an exam is getting the 'right' answer and the other the 'wrong' answer depending on a teacher's preferred answer/interpretation. This doesn't mean more than that for two students of equal ability (but with different calculators) one gets a mark or two more/less in the test. A little unfair, but this I can cope with. BUT....what if two nurses are in a hospital (with the two calculators I mentioned above), and each calculates (via the formula given by the drug company re the dosage) a medicine dose. They both type in the exact same information, and one (even if she/he checks two or three times) calculates the dosage as 9 units, while the other that 1 unit is required. This is not trivial anymore. Whether they learnt PEMDAS (or know of PEJMDAS) their trust in the calculator is sort of 'Russian Roulette' for their patient. We all need to become consistent. This is not a trivial misinterpretation of one way of looking at expressions compared to another, but an extremely important issue that needs attention.

[@omarcedric9193]

Subscribed. Learned a ton from this one video. Your description of how I view mathematics is spot on. And that's probably the reason why I'm never good at mathematics. The moment I first appreciated mathematics, particularly algebra, is when I was working as an analyst. When I found a real life application of the basics. I can't really describe what struck me back then but the way you mentioned "heart" of mathematics was the right word for it. The way you describe how this expression is ambiguous also applies to my limited coding experience. If I want my program to arrive to a specific answer or output, say 9, then I would "tailor" an expression that will arrive to that desired result. Not sure if my analogy is correct though.

[@melissalynn5774]

an analyst? you're a smartie, and you know it. it's always been my exp that folks who hate algebra are good at geometry and vice versa! diff sides of the brain i heard!

[@xeroxcopy8183]

@@melissalynn5774 not me, I excel in both

[@DarinBrownSJDCMath]

As another math ph.d. myself, my answer is simply, "I would NEVER write such an expression. And I don't think most mathematicians would write such an expression, either."

[@DrTrefor]

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

[@DarinBrownSJDCMath]

@@DrTrefor BTW, thanks for all your great calculus videos! I've used them as supplementary viewing for Calc 1, 2, and 3 this summer and fall with distance learning.

[@DrTrefor]

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

[@ActuatedGear]

Well, it's wrong. The habit has become to write a number next to a parentheses, but between the '2' and the '(' should be an 'x'. No one uses divisors, but if you use them its... formatting that is only used to teach pemdas and in that -- very specific -- formatting, you are required to use every mathematical operator. This skips one, and thus we don't know what else it decided to skip. It's a "wrong" formula.

[@LudusYT]

What about textbooks? I can pull examples from nearly any textbook (math or physics) I own that has a/bc in it, and you're supposed to interpret that as a/(bc). Yes, it's quite obvious in that context to interpret it that way, but I think that definitely casts doubt on the idea that mathematicians and physicists don't use implicit multiplication when writing symbols in-line. This is not to say that one or the other is "correct", but just to cast doubt on your claim.

[@davidhuber6251]

A zillion years ago when I actually did math, I had an RPN (reverse polish notation) calculator. I think using both helped solidify the relationships in my head. At the time I really thought RPN was superior, but had limitations. You had to think to decide which order to type things in. This thinking gelled the thought process of how the numbers related to each other. I think many math students could benefit from learning RPN as a side project. I would often do a problem with both, and if my answers disagreed, it let me know that I had some more thinking to do. I really like how you described this as an English communication problem. Bravo.

[@DLBozarth]

Dr. Bazett, I really appreciate your comment about making sure that we write math problems in an unambiguous manner. This applies to many different aspects of business today, such as contracts, reports, articles, and much more. The biggest problems I have encountered in business have been related to this specific matter, ambiguity. Thank you for this video.

[@remainedanonymous8251]

Sir.... You have solved a war in my house. Not in the way you think! You explained an issue with how my parents communicated with me in general! I did math differently with my step dad and how you explained the 2 differences explained to my logic prone step dad how I function and learned as a creative individual. Thank you.

[@jayjpepedreamer]

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

[@985476246845]

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

[@Milesco]

@ Jose: Moreover, when you have implicit multiplication as a result of the 2 being juxtaposed right next to the (1+2) like that, anybody with any knowledge of math -- or at least, algebra and higher -- will treat that as a single, indivisible (no pun intended) expression. It's basically a ÷ bc (or a/bc), where a=6, b=2, and c=(1+2). And everybody knows -- or damn well _oughta_ know -- that a/bc is a/(bc) and *_not_* (a/b) × c.

[@trwent]

Then textbooks should STATE that EXPLICITLY in the Order of Operations.

[@adamwalker8777]

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

[@masterblaster3653]

Shame on you how did you became civil enginner

[@CeceNorman]

I'm 28 years old and just now learning I was taught PEMDAS wrong. For me it wasn't the parentheses that were the issue. Every math teacher I've had said you have to do the multiplication before division. I was never taught that they were on the same level, and we could just do left to right. If I did, they said the answer was wrong.

[@calebfuller4713]

It is generally accepted that explicit multiplication and division are both on the same level nowdays. If it makes you feel better though, there was a time, back in the 18th or 19th century, when doing all the multiplication first was the more accepted convention. So you're not wrong per se, just a bit out of date... 😂

[@harrymatabal8448]

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

[@pokemonfanmario7694]

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

[@zakelwe]

There is no left to right convention as the video presenter said. When on one line you have to use brackets to replicate both possible answers that the two line notation shows you. If you do left to right you can only ever get one of the two possible answers. With 2 lines left to right is not needed of course, hence why no left to right convention.

[@CeceNorman]

@zakelwe I never said there was. I was saying I could go left to right. My point was that he said it doesn't matter what order the multiplication and division was. My teachers taught me the opposite (outdated way) so therefore there was only one answer with that method vs the current accepted way.

[@rachelcolomb]

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

[@kobusswart554]

As a computer engineer, my instinct is to think of the 2(1+2) as similar to (1x+2x) which is "simplified" to x(1+2) and more clearly written as 6/(2(1+2)) = 1 - Rather use many brackets to provide clarity than leave the next engineer pondering what you meant

[@GanonTEK]

100%

[@makenzimedlin4328]

My exact thought process thank you

[@lyvectra6270]

As a mechanical engineer, I 100% agree.

[@Milesco]

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

[@RS-fg5mf]

You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol... 6÷(1x+2x)= 6÷(x(1+2)) NOT 6÷x(1+2) 6÷x*1+6÷x*2+6÷x*3-6÷x*4= 6÷x(1+2+3-4) as the LIKE TERM 6÷x was factored out of the expanded expression.... 6÷(1x+2x+3x-4x)= 6÷(x(1+2+3-4) as x was factored out of the expression WITHIN the grouping symbol... You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol....

[@impos1ble32]

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

[@DrTrefor]

This is actually a great idea and a BIG topic imo

[@justdoit2585]

kzbin.info/www/bejne/m32wlJailN9sjsk

[@PuzzleAdda]

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

[@popeyelegs]

How society views math doesn't solve the problem.

[@AtomicExtremophile]

In my early years I was taught that the number preceding the bracket was part of the bracket - so 2(1+2) = (2*1) + (2 * 2) = 2 + 4 = 6. This was because I was taught algebraically that a(b+ c) has to have the brackets removed, so this becomes ab + ac.

[@jianxiongRaven]

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

[@kimf.wendel9113]

That is correct. And a parenthesis isn't "solved" until you complete the multiplication or division of it. All rules states parenthesis (or brackets) are to be solved first and foremost.

[@JoeNarbaiz]

So, according to you, a(b+c) is the same as (a(b+c)). I was taught that only the contents within the parentheses are evaluated. Sure, a(b+c) is the formula used to describe the distributive property but the expression of 6÷2(1+2) is composed of only one term and must be evaluated as such because terms are defined by the presence of addition and subtraction and not multiplication and division. You need to evaluate the entire context of the expression and not just part of it. Also, the obelus (÷) does not imply grouping where what is before the sign is the numerator and what is after it is the denominator. That is the function of a vinculum or horizontal fraction bar where what is above the bar is the numerator and what is below is the denominator. If you desire an answer of 1 for the given expression, you must add an additional set of parentheses. 6÷(2(1+2))=1.

[@kimf.wendel9113]

@@JoeNarbaiz a(b+c) is the same as as (a(b+c)) yes. The outside parenthesis is redundant since it is a regular + parenthesis and thus is solved as soon as you solve what is inside. Given there are no terms outside the parenthesis it offers no change. Let's say you want the content to be the 6÷2×3 where 3 is a sum of 2 numbers, you will need to put in those extra parenthesis like (6÷2)x(1+2). Otherwise a multiplicative parenthesis will always take priority. Actually use this quite often in economics, due to the fact that a lot depends on factors.

[@Andrew-it7fb]

I was taught that there is no difference between 2(1+2) and 2*(1+2) and that it's just a shorthand way of writing it.

[@opo010]

Great answer! And ... love your t-shirt :) Where can i get this?

[@carlhartzell6054]

Very happy to see this nonsense described as a language problem and not a math problem. And I know my hard-science colleagues would throw a fit at the comparison to soft science; but when something is ambiguous in the English language the sentence is written in a different way. Thanks for the explanation that the mathematical expression should simply be written in a different way as well.

[@kurtka8720]

agreed, I'm currently trying to explain this to a friend and he's still refusing to believe that it's a language problem. and that onyone who views it the other way is simply wrong.

[@murattanyel1029]

After all, math is a language, too.

[@jeremyroland5602]

There is still an objectively correct answer. It can be shown here: "6 / 2(1 + 2) = 6 / 2(3) = 6 / 6 = 1" because "6 / (1 + 2) = 6 / 1(1 + 2) ≠ (6 / 1) * (1 + 2)", therefore "6 / 2(1 + 2) ≠ (6 / 2) * (1 + 2)". There is no ambiguity because "n(m)" always implies "(n(m))" just like "m" implies "1m" or "1(m)".

[@wrrsean_alt]

Carl, I agree it is a language problem but maybe more..... For example, I just took my CASIO Scientific calculator [Model fx 82AU] and typed in the problem and it gave me the answer 9. I then took another calculator, CASIO Scientific calculator [Model fx-83GT PLUS], and it gave the answer 1. The first calculator obviously is programmed to use PEMDAS and the second [same company different model] uses 'implied multiplication precedence over division 'Juxtaposition' (PEJMDAS)'. So, this means one person in an exam is getting the 'right' answer and the other the 'wrong' answer depending on the teacher's preferred answer/interpretation. This doesn't mean more than that, for two students of equal ability (but with different calculators) one gets a mark or two more/less in a test. A little unfair but I can cope with that. BUT....Now I have two nurses in a hospital, (with the two calculators I mentioned above) they calculate, via the formula given by the drug company, the dosage for a medicine. They both type in the exact same information, and one (even if she/he checks two or three times) calculates the dosage as 9 units, and the other that 1 unit is required. This is not trivial anymore. EVERYONE needs to be taught orders of operations in a consistent way that gives the 'right' answer. As a scientist I use PEJMDAS, but primary students are usually taught PEMDAS, and brackets are often not used if there is a chance of ambiguity. This, I feel, is the main reason why there is a problem - two (or more) ways of interpreting the same 'piece of language'. When does this first come up? In primary school So.... I feel it is very important that primary teachers are trained 'correctly', because it is here that this/these problem(s) are first encountered and can be tackled. Also, by doing this hopefully trust in our health practitioners, and calculator/computer company can be restored.

[@carlhartzell6054]

@@wrrsean_alt so this has been a very long ongoing and thoughtful discussion. What I find most interesting is that some people still believe there is an objectively right answer. With the calculator issue you've expressed there is to me an obvious time when people believed one way to be right and excepted it. Then some evolution happened and a new algorithm was accepted. What makes the version now right and the previous wrong? Also, usually I view math as an explanation for some process in the universe that the series or expression represents. And I'm not saying I disagree with anything or any ones point of view here. But objectively something seems to be changing in the foundations of math.

[@donnamills4169]

Keep up the good work! Your passion for math is like an artists love🎨 of colors! I never saw numbers in that way!

[@geoffclarke1974]

I'm conflicted. Your point about communication being the key is excellent. The conflict is that I clearly remember multiple teachers in primary school saying that equivalent operations are processed left to right and marking any exception as wrong - that is, there is no ambiguity and saying the teacher communicated badly is not reasonable for a 10 year old in the 1970's - real world consequences. After thinking about what you said, it would be more helpful if a teacher communicated about removing ambiguity like an adult would. Great video.

[@suhrrog]

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

[@peterthomas5792]

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

[@theonethatsabovetoaa5645]

@@peterthomas5792ion see your PHD so ur wrong

[@markprange4386]

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

[@Darkev77]

Your excitement got me excited xD!

[@akosualynn6469]

I needed this video when I was in school 18,000 years ago, for my high school teachers. I hated math, and to this day still struggle with it. Don't get me started on comprehension questions!

[@covingtoncreek]

I love this. You turned it on its head in a way and gave us a reality check. Liked and subscribed.

[@txheadshots]

Thank you for the ambiguous discussion at 4 minutes or so… that’s what I’ve been trying to tell people for a long time, but in the end, I’ve added it to my list of things I won’t discuss on Facebook (which includes Politics, Religion, Vague Riddles with more than one answer, and The Last Jedi)

[@yourmomsfilms]

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

[@RS-fg5mf]

WRONG

[@Kage-jk4pj]

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

[@RS-fg5mf]

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

[@RS-fg5mf]

@@filename1674 No you can't. 🙄🙄🙄

[@tommy8290]

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

[@KevinKuo]

I agree. This controversy shows that society thinks of mathematics as a machine, full of operations and devoid of creativity. When in fact it is one of the most creative and beautiful fields, and requires extreme levels of ingenuity, creativity, and abstract thinking.

[@DrTrefor]

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

[@physicsmathsworld2033]

@@DrTrefor lol 🤣🤣🤣

[@donaldthomas7070]

For most people, mathematics is a set of numerical expressions or questions, each of which (usually) has 1 right answer & many wrong answers (most of which, fortunately, are highly implausible). The goal is to find the right answer-or answers, for those comparatively rare cases in which there are 2 or more correct answers.

[@kirkspreiter6444]

Math is a science how you use it as a function is an art but you can't change the scientific elements of the math. Smh!!

[@gustavo9758]

I indeed see Math as a complex machine with very specific rules, maybe because of my background (Software Engineer). So that makes me always see "6 ÷ 2(3)" as "6 ÷ 2 × 3", which is unequivocally 9. I can see the confusion on this being interpreted as "a ÷ bc" which, for what I understand, would be 1. HOWEVER, if you, the guys who really know this stuff, say it's ambiguos, then I believe you and I'm ok with that.

[@LTTheeArtist]

I'm not a mathematician but I tried to explain to people that the statement is too ambiguous to explain. Then proceed to show 2 answers from this statement base on perspective. Thank you for posting

[@Steponlyone]

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

[@bernardgome5564]

You said it all and so few likes

[@melissalynn5774]

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

[@enysuntra1347]

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

[@plumber1337]

Not only that, but following some rules and conventions over others breaks some of the arguments, imo at least. It's easy to confuse people with this type of notation because the results are usually integers... But, if you apply juxtaposition before Order Of Operations then a decimal value can never be represented as its fractional equal without being inserted in brackets because the juxtaposition will enter in effect without applying it to the entire fraction, but the other part of the expression is already inserted in brackets. Eg. 0.25(2+2)=x. You can, according to the concept of equality, replace the 0.25 for 1/4 or, since "/" is equally representative to ":" , as 1/4(2+2) or 1:4(2+2) . However, in any of the latter two, by applying juxtaposition before OOO you will not get x=1 but x=1/16 if the fraction isn't in brackets. But following OOO instead of juxtaposition 0.25(2+2) can be represented as 1/4(2+2) or 1:4(2+2) without any confusion. That example can be replaced with anything similar, like 0.x(a+b)=y being replaced with 1/z(a+b)=y . But we can't forget that 1 is also 2/2, 3/3, 4/4, 5/5, or x/x , and any (a+b) can be written as 1(a+b) or x/x(a+b) . That is how I look at it, I don't know if my argument is valid or invalid since I'm not a mathematician though.

[@MrGreensweightHist]

You are incorrect.

[@arnavbhavsar8567]

Your videos are always fun to watch prof. Your excitement gets me excited

[@LudusYT]

I think this problem is a bit more relevant than you make it out to be. For example, I can pull - from nearly any of my textbooks - an equation written in-line that looks something like a/bc. We are of course supposed to interpret that as a/(bc). Yes, it is obvious in that context what the correct interpretation is, but I don't think we can have the attitude of "I don't care" when expressions like this are written frequently in textbooks and they MUST be interpreted a certain way. I think a better answer would be that the "correct" interpretation depends on the context, but I believe that was implied in your video anyway, so I'm probably nit picking. Love your content! Your vector calc visualizations are amazing.

[@stevecolour8010]

I agree that the problem is just that there is no context. a/(bc) is probably the more useful interpretation for a/bc but these textbooks kinda suck then as our textbooks were unambiguous and wrote fractions vertically when grouped together. When using standard text signs I always Parenthesis in abundance. I also agree that maybe a debate could be interesting but fundamentally the point of the video is that the equation isn't written correctly or consistently which is why there is no need to come to a conclusion when the input is the problem.

[@nickjunes]

There was an explicit choice to NOT include a multiplication sign but they included the division sign in the original problem so it strongly suggests that the right side is the denominator and the answer is 1.

[@Jry088]

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

[@nickjunes]

@@Jry088 I have seen problems like this written in text books although with a / instead of a ÷. In those cases it's usually to save space because they are trying to get the whole thing on one line and then in that case the right side is the denominator. I would not expect a trick. Also if I saw this in a notebook found somewhere I would guess the author left out the multiplication sign because they want the whole right side to be solved first otherwise they would have written X or * just like the wrote ÷ on the other side. Not writing X or * would be inconsistent with the style unless they meant it to be a denominator so if found in a notebook it would be very safe to assume the right side is solved first.

[@RockinRack]

@@nickjunes that's why 1 seemed so obvious to me also. At least the way I learned a(b+c) is all included in the P in pemdas. Otherwise it would be easily separated.

[@jugamath]

Thanks for your response to this type of video/social media post. I groan every time I see another of these posts. I think many people experience math notation in grade school and are drilled on it with strict rules. Sadly, notation for arithmetic problems is what a lot of people remember of math. I don't know whether it would be better to teach students that ambiguous expressions relying only on BEDMAS or PEMDAS are poor style or simply declare all such expressions to be incorrect notation altogether. In language there are rules of style as well as syntax. People should realize that there are stylistic faux pas in math as well.

[@sdlcman1]

In algebra, they usually talk about identifying the terms and then the associative, distributive, and commutative properties when they ever talk about PEMDAS. Also, the student would look at the division symbol as a slash. If the constants are rewritten as a, b, c, d, then it will be a/b(c+d). If you do the parentheses first, multiplication second, and finally the division, you will get 1, which is all could get taking PEMDAS literally. The problem would have to be written as 6(1+2)/2 to get 9.

[@geirmyrvagnes8718]

Everybody agrees what the result would be if we were to take PEMDAS literally. The question is if we should break with tradition, rewrite the text books and start taking PEMDAS literally. Who died and made PEMDAS king, suddenly? PEMDAS is a simplified mnemonic for teaching the order of operations to children.

[@jguo]

Another PhD in math and engineering here. If any of us wrote an expression like that, we failed our education. Unless we walked into a bar and just wanted to start a bar fight...

[@mokooh3280]

Well bring it

[@skiddadleskidoodle4585]

What is 77 + 33

[@opticalmouse2]

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

[@geirmyrvagnes8718]

However, we still understand 1/2x as 1/(2x), since if we meant it the PODMAS way, we would have written x/2. And if there is ambiguity, there is context to clear that up. Six letter acronyms are for children!

[@foxfactcheck]

kzbin.info/www/bejne/on2mdZaXa8mMpqM

[@daddiesgurl1968]

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

[@manzerm7805]

I think the confusing part is the use of the parenthesis without the explicit * sign, so the problem is not 6÷2*(1+2) which would unambiguously be 9, given BODMAS and L to R execution. To examine further, , let us put (1+2) as x, so the expression is 6÷2x which is not the same as 6÷2*x. Although we normally think of 2x as 2*x but in the context of 6÷2x, 2x would mean 6 and the answer would be 1. I do think the expression is ambiguous and the author must rewrite it as (6÷2)(1+2) if he wants 9 to be the answer.

[@zerxilk8169]

pemdas vs the bs

[@xybersurfer]

the problem is indeed the implicit * sign

[@kimf.wendel9113]

No bodmas says it is 1. B is for brackets, so in 6÷2(3) you have to calculate brackets first, aka you get 6÷6. Now all of your reversals works aswell.

[@manzerm7805]

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

[@kimf.wendel9113]

@@manzerm7805 yes, and that means the contents of the parenthesis is shortened by a factor. And to remove the parenthesis you need to multiply is expression inside. All logic in maths says you solve the parenthesis first, that is why the first letter in those order of operations starts with a that. It doesn't matter what is inside, you solve it first until there are no parenthesis

[@mrs.stimsonartkids6951]

The best answer I have seen on the internet! Great job!

[@truewarrior3646]

Thank You Sir. Really appreciate your work.

[@DrTrefor]

You are most welcome!

[@pali0123]

I didn't realize this was a thing. Myself (American) and my British classmates surprisingly had different answers and I did not understand how when I learned it clearly one way. Turns out there's a different method

[@philmonat9352]

Well said ! Sometimes I would answer a math question and was allowed to have either one of two answers, because the structure of the equation/formula was not specific enough.

[@JefferyHunt]

Conventions like operations orders are in a constant state of flux. The transcendental nature of numbers is fixed and constant. I think the contrast of these two things couldn’t be more stark and so it gains attention easily.

[@gentlemandude1]

Even though its subject is insignificant, this is by far the most important video that you have produced thus far . Indeed, It's the way that many people think about mathematics that causes so many problems. Unfortunately, many so-called mathematics teachers reinforce and defend the "arbitrary list of rules" model of mathematics education. Thank you for continuing to fight the good fight. When the pandemic is over, I'd like to visit Victoria just so that I can shake your hand.

[@popeyelegs]

Still doesn't answer the question of what he answer is. Please don't ever work in payroll.

[@quantumgaming9180]

@@popeyelegs Because the answer doesn't matter anyway if the question is wrong in the first place, it's ambigous.

[@maxxiong]

Argument 2 wins for me, because of this: how you rewrite 1/f(1+2) as a fraction should not depend on whether f is a function or a number.

[@manzanajoemerj.9849]

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

[@jshad1074]

@@manzanajoemerj.9849 distributive property doesn’t apply here.. 6/(2(1+2)) is distributive property which equals 1.. 6/2(1+2) isn’t distributive so the answer is 9

[@olblue3478]

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

[@no0bjago900]

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

[@SeanMaxhell]

@@jshad1074 2(2+1)/6 = 1 do you know what does it mean when a result of division is 1? that the operators before and after the division sign are equal. so 6/2(2+1) = 1, not 9. I don't have to add any futile brackets. I don't have to write 6/(2(2+1)) to get 1. I didn't write (6/2)(2+1) to get your stupid 9. could you fix your stupidity please?

[@Ligierthegreensun]

Trying to explain this to anyone who just does math by rote is an exercise in losing brain cells. They furiously exclaim that their way is the only way to interpret the expression.

[@SoraRaida]

Yup bingo

[@GanonTEK]

100%

[@AndresFirte]

GanonTEK, it’s nice to see you in this comment section too. I can see we share the same “obsession” with answering comments hahaha. I’m currently monitoring like 9 different comment sections. Anyways, I just wanted to say hi and thank you for the time you spend spreading your knowledge

[@GanonTEK]

Nice to see you too! Thank you also for spreading your knowledge too. It's difficult when so many misinformed people are spreading either narrow minded or downright false information. The latest one who I see on a lot of videos now spreading the same information has a really poor understanding of a range of things, not just maths. They don't know what a mnemonic is (they think 2 is a mnemonic in 2(3)). They don't know what terms are (they think 2×3 is two terms when it's one). They don't know what multiplication by juxtaposition is and think 2(3) isn't an example of it even though though by the dictionary definition of juxtaposition the 2 and ( are beside each other, aka juxtaposed, and this notaiton does indeed imply multiplication. They think 3(2)² is 36 as they think the juxtaposition has higher priority than Indices (caused by them not understanding brackets). To top it off, they called me as dense as Uranus which is the 2nd *least* dense planet in our solar system (Saturn wins that honour). I'd laugh if it wasn't so sad.

[@AndresFirte]

@@GanonTEK yeah, I get what you mean. I’m dealing with guys that genuinely think they solved the Collatz conjecture (and Goldbach, and twin prime conjecture), and when I explain why they’re wrong they claim that I’m “gatekeeping math”. Also people that have their own weird fuzzy definition of what infinity is and think Cantor’s theorems are wrong. People that think 1/0 = 0, and people that say it equals ∞. It’s a bit frustrating, but every once in a while some of them actually change their mind and get it. I myself once argued that 0.99… was not equal to 1, because I didn’t understand what 0.99… truly meant. Also, some of them can have genuinely interesting questions. And it helps me remember why mathematicians are so rigorous and precise with their definitions, why they have to think so carefully of every step of a math proof.

[@brauggithebold7956]

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

[@MartinBeerbom]

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

[@markcash2]

LOL, my wife is an astrophysics professor and I am an economist. She quite succinctly told me the error was with the person who wrote the original equation allowing for ambiguity to exist. Personally I think the law of distribution must be obeyed before we talk PEMDAS. There is more to math than just PEMDAS. Since there isn't an operator between the 2 and the (1+2) then you have to assume the 2 was factored out of (2+4).

[@GanonTEK]

She is right. The question is badly written to modern standards. ISO-80000-1 mentions about fractions on one line and how brackets are needed to remove the ambiguity now. Back in the early 1900s this would not have been an ambiguous question but with modern programming it now is.

[@RS-fg5mf]

You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol... You fail to understand the Distributive Property correctly. It amazes me how otherwise very intelligent people fail to understand and apply very basic rules and principles of math... The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9

[@AudriusN]

@@RS-fg5mf stop spamming your stupidity

[@shaunpatrick8345]

@@RS-fg5mf 6÷2 is not a single term like (1+2) is. By juxtaposition, it is the 2 which is multiplied by the bracket. "The How and Why of Mathematics" has a couple of videos on this topic where she looks at periodicals to see how professionals would approach it; they all use juxtaposition and get the answer to be 1.

[@RS-fg5mf]

@@shaunpatrick8345 you're wrong and so is she. Every example she gives is in the form of a/bc NOT a/b(c) There is a distinct mathematical difference between 6÷2y and 6÷2(y) despite your misguided beliefs and subjective opinions... 6÷2(1+2) is a single TERM EXPRESSION with two SUB-EXPRESSIONS. 6÷2 is a single TERM sub-expression juxtaposed outside the parentheses as a whole to the two TERM sub-expression inside the parentheses 1+2 There are two types of implicit multiplication and they are not mathematically the same.... Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed (NO DELIMITER) and forms a composite quantity by Algebraic Convention... Example 2y Type 2... Implicit Multiplication between a TERM and a Parenthetical value or across each TERM within the parenthetical sub-expression... Terms are separated by addition and subtraction not multiplication or division.... 6/2(1+2) is a single TERM expression with two sub-expressions. The single TERM sub-expression juxtaposed outside the parentheses as a whole 6÷2 and the two TERM sub-expression inside the parentheses (1+2) In the axiom A(B+C)= AB+AC the A represents the TERM or TERM outside the parentheses not just the numeral next to it. The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y. This is an inaccurate comparison... These two expressions utilize two DIFFERENT types of Implicit multiplication... 6÷2y = 6÷(2y)= 3/y by Algebraic Convention 6÷2(a+b)= (6÷2)(a+b)= 3a+3b by the Distributive Property... All variables have a coefficient written or not. Constants can be coefficients but constants do not have coefficients. There are no coefficients in the expression 6÷2(1+2)... 6÷2y the coefficient of y is 2 BUT 6÷2(a+b) the coefficient of a and b after simplification is 3 not 2 Correlation does not imply Causation. Just because both expressions utilize implicit multiplication doesn't inherently mean they are treated in the same manner... The phrase "correlation does not imply causation" refers to the inability to legitimately deduce a cause-and-effect relationship between two events or variables solely on the basis of an observed association or correlation between them. For people who argue 6÷2(1+2) and 6÷2y should be evaluated the same way, their argument is circular and is an informal fallacy that is flawed in the substance of their argument...

[@Nethezbet]

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

[@peterthomas5792]

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

[@nsn5564]

The correct answer is that YOU NEVER FRAME AN AMBIGUOUS EQUATION LIKE THAT. YOU HAVE PARENTHESES. USE THEM!! THE EQUATION DOES NOT NEED TO BE AMBIGUOUS AND SHOULD NOT BE WRITTEN THIS WAY.

[@krnisa.karim30]

This is why whenever there is a viral question related to science or math, i would look for professionals answer..bcos there is too much unprofessional people answered this question and arguing as if they already finished the whole books of mathematics and start to be judgy towards other people opinions 😌

[@mirkotorresani9615]

The problem is that if you ask to any professional mathematician about the problem in the video, the answer would be something like "I refuse to answer, let's talk about topological algebra instead".

[@jamesrobbins26]

I never thought of this problem this way but you are right. The problem was thrown out to create a little controversy because the originator understood people could and would come up with 2 different answers and both would be correct because enough info was not given.

[@MGmirkin]

It's more insidious than that. It was created not to edify, but to explicitly be ambiguous and to drive "interactions" on a given FB page or Tweet. The idea is not to arrive at a "correct answer" [none is given, and no winners declared]. The idea is simply to create drama and dissent, which leads to more clicks, more page views, more comments, and arguably more reputation for the page, and thus possibly more monetization, etc., in some form or other. They're not here altruistically to teach people anything, but to sow discord and make money off of it, whether driving clicks to other pages / sites / videos, or growing some subscriber base and then selling the page to some new chump willing actually pay something for it for some unknown reason, with a built-in subscriber/liker/follower base that can then be advertised to or whatever.

[@mikestuart7674]

@@MGmirkin Exactly right, the authors of the videos saying the answer is 9 are doing it for money, despite the harm that they do to society. It is shameful.

[@kimf.wendel9113]

No some people just forgot what they learned i school and got confused. As such they turned to social medias to verify they weren't the only ones to forget how math works. Then more fot confused becuase they were in doubt aswell, and then a confusion spread.

[@Andrew-it7fb]

​@@kimf.wendel9113sometimes that's the case, but different people have been taught differently as well. Some people have been taught that multiplication by juxtaposition has priority over other multiplication and division and some were taught that it's bo different than any other multiplication.

[@shaunpatrick8345]

@@Andrew-it7fb that doesn't mean the latter group is right. If they were taught that + was "divide by" there would not be an additional right answer, they would just be wrong.

[@InsaneDark]

omg finally found someone who is saying the same thing as I am and if you follow that then there is no confusion as to the correct order of operation it goes from left to right (which we learn in elementary school) if you as a creator of equation want it differently then it's your job to emphasise it with any of the means you mentioned in video and I always say the same thing

[@tonitalas1757]

Thank you for explaining clearly the answer to this expression 😊

[@johnsciara9418]

First of all, I agree with you. 6 ÷ 2(1+2) is poorly written. Besides the better way to write the problem that you included, there is another example of what this could have meant. That has to do with factoring. For example (2+4) If we uses variables first to put it into a format that is recognizable such as ab + ac how would you write this? You could write it as a(b+c) so to factor (2+4) to simplify it to the lowest prime numbers you could write it as 2(1+2) Using the distributive law, when you "solve" this expression you could follow PEMDAS and add the values in the parentheses together first (1+2) =(3) and then multiply the 2 outside the parentheses to get 6 or you could distribute the 2x1 + 2x2 and still get 6. If you had an example of a factor a(b+c) and expanded the problem to include a division operation such as 6 ÷ a(b+c) what is the denominator? is it a(b+c)? If this is a factor, do you separate the variables a from the (b+c) before you obtain the value for the factor? Is 5(7-5) actually the factor expression for (35-25)? If you had (35-25) how would you write it as a factor? in 60 ÷ 5(7-5) what is the denominator? If 5(7-5) a factor of (35-25) do you separate the 5 from the (7-5)? Why is there an implied multiplication operation between 5(7-5) if it was a factor? If you write a(b+c) can you call that a factor some of the time and not a factor other times? Would I have to read your mind to know when you consider a(b+c) a factor and when you don't consider a(b+c) to be a factor? If you didn't want a(b+c) to be considered a factor why not write it as a x (b+c) then there would be no confusion.

[@devkird6069]

thata numbers right there

[@axelmac7856]

Im in 8th grade and that’s the exact same thing I thought but with other examples, I finally found someone that knows his stuffq

[@axelmac7856]

On this operation

[@georgearnold841]

That last sentence is exactly my argument against the answer 9. a(b+c) implicates the entirety as a factor that needs to be resolved first. Otherwise order it as a×(b+c) to separate the functions to 6/2 × 2+1.

[@bambajoe1721]

Too much wordas for 1 math problem my friend

[@xoxoxoxoxoxoxo6921]

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

[@DrTrefor]

Thank you!!

[@chrisgriffith1573]

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

[@rotten6253]

Its an ambiguous problem. It doesn’t have a proper answer because it’s improperly made.

[@mushtaqkasba8702]

Mathematics is all about presenting ideas in a symbolic form to make abstract and complex ideas simple.but those symbols should be clear. If the symbols used are ambiguous then you are presenting it in a wrong way.

[@IncredibleFlyinSquid]

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

[@SoraRaida]

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

[@user-cd3bu8uj8i]

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

[@phoenix2634]

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

[@habacue713]

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

[@trwent]

Yuck.

[@mirkotorresani9615]

You are not the only one. It's sad that most of the people don't have any clue about the wonderful mathematical universes that unravel, once these stupid problems disappear.

[@DrTrefor]

Ok which you all just sent this viral again:D

[@isovideo7497]

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

[@robertnewell5057]

I think this is very well put. Another poster has suggested that a similar problem to this one EITHER has a correct answer based on PEMDAS applied L-R OR is inherently ambiguous. Logically, not both these statements can be true. I think you demonstrate that neither is true. The fallacy in the original argument is that the ambiguity cannot be removed. It always can, but the way(s) in which it is removed yeild different answers because reveal that there are different questions being asked in each case. It is the 'many masquerading as one fallacy' in logic.

[@vaginalarthritis1753]

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

[@DrTrefor]

We are going to agree a lot then!

[@notahotshot]

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

[@angelmendez-rivera351]

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

[@Zephyr-tg9hu]

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

[@RealMesaMike]

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

[@namewithheld8115]

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

[@trwent]

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

[@Matt-qq8dh]

Thank you very much for this explanation. I was starting to get really confused. Removing the ambiguity by the person writing the math formulas makes really good sense. Parenthesis clarifies a lot of things like this per your explanation. I use the enter key on an RPN calculator kind of like a parenthesis and do the series of operations afterward when solving formulas, like resonant frequency. Again, thank you for explaining this and letting me know that a good mathematician would not write a problem like this.

[@PWingert1966]

Is there any indication in the Principa mathematical on order of operations and its interpretation?

[@josephvenegas5655]

good explanation, I never disliked math I was always challenged I loved it. I always found out a different way to solve equations and I was always told "no that is not how we learn it" was never taught away from a systematic perspective. However I did not care I always went against teachings, the schools I attended definitely does not show this side of math.

[@RS-fg5mf]

You can evaluate this expression at least 6 different ways but you still get the only correct answer 9

[@jjh7611]

RS can get his head dropped 1 or 9 times as a baby and he’d never graduate elementary school

[@mirkotorresani9615]

A curious person like you would be amazed from the mathematical universes that unravel, after these stupid problems, equation, and expression disappear

[@sclark223]

Interesting ... you make excellent points that notation is meant to clarify and simplify, and sometimes it doesn't do that well. I don't see this problem as ambiguous, though. Just highlighting a weakness in how PEMDAS is taught. I think 6 / 2 * 3 means (6 / 2) * 3. This is consistent with how in-line calculators process the problem too. If you want 6 / (2 * 3), then you have to put the grouping symbols in. If you use a horizontal fraction bar, then the bar acts as a grouping symbol for the denominator, as you say (much like the roof of a radical sign groups everything in the radicand). These problems are appearing in college algebra textbooks to encourage better understanding of how PEMDAS works, especially the MD stage, done in 1 left-to-right pass. If they are included in books, then I think the point is to encourage teaching the rule better, not to try to confuse people.

[@andrewboyer7544]

Well, if this trend is to be considered it seems to just be confusing people.

[@RS-fg5mf]

@@andrewboyer7544 many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...

[@RS-fg5mf]

It is a lack of understanding of the Order of Operations and the various properties and axioms of math... 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 an inline infix format extra parentheses 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 = 0.5 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....

[@MGmirkin]

It is confusion over the lack of explicit parentheses unambiguously stating which things are grouped together in what way **AND** conflation with the manner in which "distribution" or "factoring" is typically written **in short-hand* when factoring out polynomials and such. For instance x^3+x^2y+xz is often factored / simplified to x(x^2+xy+z). So, one can be **somewhat forgiven** for seeing an **implicit** distributive grouping in 2(1+2) analogous to something like x(x^2+xy+z) and thinking there **should be** an extra **implied** parenthesis (2(1+2)) whereby the 2 is distributed over the contents of the parenthesis, just as one would reconstitute the original x^3+x^2y+xz by distributing that x(...) over the parenthesis' contents. It is this ambiguity that these click-bait @r$3h0l3$ use to create these intentionally ambiguous memes.Not for any clarity or edification but explicitly to drive misunderstandings, arguments, and inevitable more clicks / views / follows / likes and hopefully more $$ through whatever monetization system the page / video / etc. uses. It's not about edifying anyone, it's about driving interactions to make the page money. It's insidious, and should pretty much be banned, IMO. It's a waste of our collective time and attention as a society / species. Just saying. ;)

[@01ant100]

brilliant, loved the explanation ... and the implicit correction, don't be ambiguous in your choice of symbols, what could be clearer?

[@MrJoosebawkz]

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

[@Technium]

The best possible take

[@TenTonNuke]

The best I've heard it explained is that even after reducing 2(1+2) to 2(3), you still haven't dealt with the parenthetical expression. In other words, the P of PEMDAS still isn't finished. And by restructuring the equation as (6/2) * 3, you've changed the equation entirely. Instead of distributing the 2 throughout the parentheses to satisfy the P, you've just kind of removed it. Instead of turning a(b+c) into ab + ac like you're supposed to, you've changed the equation to (1/a) * b + c. TLDR: The multiplier of the parentheses must be distributed to satisfy the P in PEMDAS.

[@GanonTEK]

Except that P is for inside parentheses only. Juxtaposition is either a separate step after Exponents, like in PEJMDAS, or it's a notation convention that needs to be interpreted and written explicitly before you start to simplify at all. Easy to show with 3²(4) If the P step is still present, how can you do P before E here? What's the next step? It's bad teaching to say outside parentheses is part of the parentheses step.

[@Pajo25ify]

@@GanonTEK this might actually be hard to understand because the answer to 3²×4 and 3²(4) are the same but the way they are calculated is different. 3²×4 = (3×3)×4 = 9×4 = 36 3²(4) = ((3²)(4)) = ((3×3)(4)) = ((9)(4)) = (9×4) = (36) = 36 This becomes more obvious if you begin with 3²(2+2) instead of 3²(4). 3²(2+2) = (((3²)(2))+((3²)(2))) = (((3×3)(2))+((3x3)(2))) = (((9)(2))+((9)(2))) = ((9×2)+(9×2)) ((18)+(18)) = (18+18) = (36) = 36 The thing is 3²(4) can be calculated as 3²×4 = 9×4=36 but if it were to be part of a bigger equation 3²(4) doesn't become 3²×4 but (3²×4).

[@simongpunkt]

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

[@ninjaslash52_98]

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

[@balthazarbeutelwolf9097]

I encounter this sometimes when I teach programming. Typically, I then say that it works like that in programming language X, but do not rely on that as you will work with other programming languages with different conventions. I once reviewed the code of an MSc where the student tried to model cell movement in a computer simulation by dividing the speed a cell was moving by the square of the mass. But, they had written it as x/y*y, and in that PL that was just dividing by y and then multiplying by y. So, once you code (or: write machine-checked proofs) these distinctions matter, not on a deep level, but still.

[@felipedosso8086]

Hey Dr. Trefor Bazett Where can I buy a shirt like yours? I really liked it

[@DrTrefor]

I couldn’t find the exact one, but check out the description I put one similar in my Amazon store

[@felipedosso8086]

@@DrTrefor Thank you

[@DrR0BERT]

As a fellow PhD, I have been presented this problem a number of times. Initially I was in the hard lined order of operations, but the more I revisited the topic, I started noticing the number of examples of when PEMDAS is overridden without confusion. (e.g., cos2x and 1/2x) Now when presented, I go into the ambiguity of the expression should have been addressed by the author and not the reader. A good analogy is the importance of being aware of removing any potential ambiguity when writing a sentence involving a list and not using the Oxford comma.

[@txheadshots]

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

[@keekwai2]

PHD in WHAT, you clown?

[@ibarskiy]

There is a slight argument that scalar multiples may be interpreted that way but even then it's ambiguous. But when all symbols are in the same general realm (being variables or numbers, but all the same) - that argument goes away. And even then, it's just bad form to create ambiguity and virtually all math people... scratch that, people that use math to communicate e.g. +physicists etc. - would write it in an unambiguous way

[@txheadshots]

@@ibarskiy Exactly. I have a Bachelor's degree in Mathematics and what I usually tell people is that if I had written a formula like that on a test paper where I was showing my work, I would have gotten points off for writing something so ambiguous

[@keekwai2]

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

[@leonardriceiii934]

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

[@GanonTEK]

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

[@119Agent]

As a life long NASA/DOD contract engineer, this notation would never be accepted. Ambiguity is a problem and should be avoided.

[@theedspage]

I agree, this problem needs clear notation.

[@mokooh3280]

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

[@kenevans233]

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

[@jacplanespotting314]

where did you get the fun blue t-shirt you are wearing?

[@BernardGreenberg]

This is exactly right. Don't forget another use of parentheses: f(5) = 25. Is f equal to 5, or any number of an infinite set of functions whose value is 25 at 5? These are notation jokes. Bravo for giving the "right answer"!

[@CiscoWes]

I’ve been caught up in this debate every time it pops up on Facebook. My argument was that a college level math teacher wouldn’t write a problem on the board like 6 / 2(1+2). Instead it would be written like 6 with a line under and then 2(1+2). We would instinctively tackle the 2(1+2) first to simplify and then end up with an answer of 1. But the angry comments yelling at us about PEMDAS strongly disagreed.

[@kimf.wendel9113]

Pemdas says it is 1, P stands for Parenthesis. To solve a a(b+c) parenthesis you end up with ab+ac. So 2(3) is not solved, it is shortened, 2x1+2x2 is the solved state which is to be reduced to a 6.

[@roscius6204]

It seems logical that the use/or not of symbols has implications. I know no-one can be definitive about implications To me, a number hard up against a bracket implies connected as against the 'partition' that a symbol would imply.

[@papatomsthoughts]

I was taught, many years ago, that when you have a number associated to brackets, 2(2+1), that you solve that first as one equation, so that is 6 so 6÷6 =1. Now if it is shown as 6÷ 2 x (2+1) then go from left to right, equals 9. That was 50 plus years ago.

[@ZeHoSmusician]

Removal of the multiplication sign is just shorthand, so 6÷2x(2+1) = 6÷2(2+1) [= 9]

[@AndyCole-nc6dn]

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

[@GanonTEK]

100%

[@tcmxiyw]

From another Ph.D. in mathematics: Thanks for doing this video. These types of problems are pointless. Those who have memorized orders of operation rules get an answer consistent with those rules. Those who haven’t memorized those rules get an arguably plausible answer. If you are entering an expression into a poorly designed calculator interface or writing an exceptionally complicated expression for a program, then order of operations rules must be clearly understood, but these situations should be avoided as they are error prone. Break the expression up into two or three lines. Get a calculator with a postfix user interface (6 2 1 2 + * /). I learned the order of operations rules in high school and have rarely used them since. Mathematicians have a knack for writing expressions so that they will be clearly understood without even thinking of rules for order of operations. There is beauty in a well crafted expression. Programmers will improve the clarity of a computation by expressing it in two or three lines.

[@billgreen576]

Great answer. I would suggest that the use of incorrect expressions is one the reasons we hear the excuses that the software does not perform as required.

[@GrieverSSBU]

I guess the best question, is which answer would you write down that won't get it marked wrong by your teacher if they hypothetically were to pose this question to you and your grade depended on it? I like your video a lot, but I've been marked wrong on questions like this and it always messed with me a lot. My daughter got marked down for an answer that with the two different arguments could be both right and wrong. I guess if Mathematics can be interpreted differently that somewhere along the way, they must have found a common answer that they could identify as the correct answer without ambiguity. Guess I'm just salty for getting a worse grade on a test for a correct answer.

[@klaymen0]

I think part of the confusion is the fact there are several symbols for division (/, :, and the unusual and huge ➗ symbol), while the multiplication symbol (x, *, or a dot) can even be left out. While addition and subtraction have only one symbol. By using the huge division symbol with a dash that looks like a spatial fraction, and at the same time skip the multiplication out at all, the reader is fooled into seeing a spatial fraction. „6/2*(1+2)“ would probably give less controversy, even if it’s the same thing.

[@GanonTEK]

Funnily enough, ÷ means subtract in some countries. Anyways, academically, multiplication by juxtaposition implies grouping so the division symbol used wouldn't matter if using that convention. You would get 1 using it. If using the more literal or programming interpretation then multiplication by juxtaposition implies only multiplication. 6/2×(1+2) is far more clear than 6/2(1+2). (6/2)(1+2) is the proper way for 9 and 6/(2(1+2)) is the proper way for 1. No issues then.

[@roger7341]

I would never write down an ambiguous expression like this and expect my readers to figure out what the hell I meant. Don't be a fool, keep your cool, use parentheses.

[@GanonTEK]

100%

[@johnli2761]

My take on evaluating 6 / 2(2 + 1) is this: 1) Argument 1 using the so called order of operation - answer is 9 6 / 2 (2 + 1) = 6 / 2 (3) = 6 / 2 x 3 = 3 x 3 = 9 2) Argument 2 using simple mathematical convention - answer is 1 This is based on the fact that when expanding A / B(C +D) = A / (BC + BD) since B is the common factor In other words, B is closely associated with what's in the brackets, almost like a function such as 2a or cos(a) which are taken as one entity, so 6 / 2(2 + 1) = 6 / (2x2 + 2x1) = 6 / (4 + 2) = 6 / 6 = 1 3) Argument 3 - solution is undefined as question is ambiguous (This is the correct answer) The problem with writing inline equation is that if we're not careful we'll be causing all sorts of problems as the above arguments demonstrated where you can have two answers to a simple arithmetic question which in fact should have only one answer. So the moral of the story is that: A) If you expect the answer to be 9 and you try to represent 6 ---- (1 + 2) 2 then you should write it as (6 / 2) x (1 + 2) = 3 x 3 = 9 or better still write it as 6(1 + 2) / 2 = (6 x 3 ) / 2 = 9 B) If you expect the answer to be 1 and you try to represent 6 --------------- 2 (1 + 2) then you should write it as 6 / ( 2(1 + 2) ) = 6 / (2 x 3) = 6 / 6 = 1 The fact that people, even mathematicians have to debate whether the answer should be 1 or 9 on this simple arithmetic primary school question is already a prove in itself that the problem lies with the question where you can interpret it in different ways and come up with different answers. The beauty of mathematics is that you can use different ways or methods to arrive at the same answer. Maths also teaches us to have an open mind and to accept that different people have a a different approach to solving problems and yours is NOT the only method or the only right method that works or that Bodmas is the ONLY rule on earth and to be ignorant of general sound mathematical principles and techniques. For example: Evaluate 875 x 99 + 875 x 1 Now Bodmas will say you have to do the two multiplications before the addition but not so because if you factorise 875 x 99 + 875 x 1 = 875 (99 + 1) = 875 x 100 = 87500 And you don't need me to tell you which is quicker especially if you don't have access to a calculator. Now, using brackets is cheap and free, so may as well use it if they can overcome any ambiguity to explicitly represent what you're trying to convey, whether we're trying to write inline equations on paper or to code an equation or formula in a programming language such as C++, Ada or Python. Imagine a software engineer writing a piece of flight critical software in Ada to compute the coordinates of latitude and longitude of a moving target you're trying to intercept with a missile and he meant to represent: A Target Lat = --------------- B(C + D) Instead of writing Target_Lat := A / (B*(C + D)); he mistakenly writes Target_Lat := (A / B)*(C + D); or Target_Lat := A / B*(C + D); And the result would be catastrophic because the first one is coded wrong and the second one he's relying on the compiler to decide what should be calculated first rather than telling the compiler explicitly what to do with his calculations! Of course in real life, to calculate lat and long would involve much more complex functions such as sine, cosine, velocity, direction, time, distance, etc.

[@garrettpyscher7852]

You are wrong removing parentheses. 6÷2(1+2) 6÷2(3) 6÷6 Changing ( ) to a x to simplify it is changing the order of operations. It is not 6÷2x3. The statement is 6÷2(3). Adding inside the parentheses does not remove the parentheses when there is a number adjacent to the parentheses.

[@TCMusic-iv4nd]

It's only a viral math problem because people are treating it as a *math* problem whereas, as pointed out here, it's actually a *language* problem. It's the purple people eater in math form.

[@sleethmitchell]

the introduction of calculators necessitated a linear method of feeding the problem into the calculator. when a person does math regularly, one sees groupings of terms rather than a linear parade of operations.

[@7rich79]

A good video, I think especially the image at 3:09 clarifies for many how they are applying either one or the other interpretation. I think it doesn't quite resolve the issue of how you deal with ambiguity. You can of course say that you shouldn't have ambiguity, and should seek to resolve that. But with maths as with language, you need to have a method of dealing with ambiguity when no other information is available. For example, imagine someone delivering a speech where they want to acknowledge the people who helped them: I want to thank my parents, Taylor Swift and Eddie Murphy. You can interpret this two ways: 1) They're thanking their parents as well as Taylor Swift and Eddie Murphy 2) They're thanking Taylor Swift and Eddie Murphy who are their parents You can of course (correctly) say that the sentence is ambiguous, and the person should make use of the Oxford comma or rewrite the entire sentence. In the absence of that information, should we have no way of making a default interpretation?

[@RS-fg5mf]

The answer is in the basic rules and principles of math when understood and applied correctly as intended... When you actually understand that GROUPING SYMBOLS only group and give priority to operations WITHIN the symbol of INCLUSION as a priority and that the 2 is not WITHIN the symbol of INCLUSION and there is no math book that states "with the exception of " then you will understand the only correct answer is 9 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 an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... ________ 2(1+2) and (2(1+2)) both have two grouping symbols ________ __________ 2(1+2) = 2×1+2×2. Distributive Property. Parentheses REMOVED. One grouping symbol... (2(1+2))= (2×1+2×2) Distributive Property. Inner parentheses REMOVED. One grouping symbol 6÷2(1+2) does not equal 6÷(2×1+2×2) as you have not REMOVED any parentheses and you still have the same number of grouping symbols.... 6. 6. 6 -------(1+2) = ------- ×1 + ----------×2 2. 2. 2 The same as 6÷2(1+2)= 6÷2×1+6÷2×2

[@danieljohn6106]

great take, math is a language with its own symbols

[@bobh6728]

The answer is what the writer intended it to mean. He just didn’t communicate it clearly.

[@Acme633]

This is the best video on the topic so far that really addressed the problem. Recently there had just been too many nonsensical videos on this "viral math problem" where the mathematical expressions were not written correctly in a professional manner and the uploader then proceeded to tell viewers what the "right" answer was. In fact, as you said, greater care in writing the expression with the proper use of parentheses, brackets and braces would have eliminated any ambiguities easily.

[@TARASTItheloser]

The way i was taught i use bimdas and i substitute the 1 with x and since i know the value of x already i know its 3 but i put the x there bc theres an invisible multiplication sign there which makes it easier for me then i solve both of them (both 3) then multiply them bc of that invisible multiplication sign. Idk if this sounds crazy to people but i got 9

[@teknul89]

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