6÷2(1+2)=??? 

No, the equation does not have two solutions SIMULTANEOUSLY. Rather it has one solution but we cannot decide which one it is. It may seem like it's the same thing but it's not... Edit: stop spamming "it's a joke", I didn't get it initially, it wasnt obvious to me. Regardless I shared useful information atleast to someone. About the argument "9 is the obvious answer", consider solving in using bodmas and pemdas.

​@@wetraccoonbetterthantrumpactually it's very simple, you just have to solve it in order and you get 9, it's not ambiguous and the guy above was just joking

​@@wetraccoonbetterthantrumpit's a joke

@@wetraccoonbetterthantrumpIt’s called a joke, ever heard of one?

​@@wetraccoonbetterthantrumpwell obviously the dude was joking

This will help you in your confusion ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-_HtJTPelgDo.html

wait for me brooo, I'm coming too!

"how to make isaac newton live again?" "Yes" Marvelous,very inspiring 100/100

I used BODMAS so I got 9. Anyone else ...

@@attaullahkhan4742 me too

it should be 'I saw a man through a telescope'

@@minorknight4491 No, Eddie is trying to explain to you the ambiguity of the statement. 6÷2(1+2) is a mathematical ambiguous expression and his example perfectly encapsulates that.

@@minorknight4491you are too dumb to understand it first try

​@@minorknight4491Yes, correct use of language (in this case Mathematical notation) removes ambiguity and clarifies the intended meaning. (6/2)(1+2)= I think everyone would agree on 9 6/(2(1+2))= I think everyone would agree on 1 Good notation writing is important. That's why most use two line fractions, they remove ambiguity also and reduce the number of required brackets. They are best practice.

@@minorknight4491you are an actual brick, crazy how ur still alive

Yes but those don't really make it clear how to interpret 6 ÷ 2(3) What does "brackets" mean? It's up to interpretation. In reality nobody writes this ambiguously so you don't even need to care

@@fahrenheit2101 it means multiplication. It has always meant multiplication.

​@@sergiobricenosIt can be juxtaposition, or implicit multiplication which is the same in algebra, in which 6÷2(1+2) where x= 1+2 would give 1

@@leohenderson2390 no 6÷2x, where x = (1+2) would still be 9. Maybe go to school

@@sergiobricenos No, it means priority multiplication - implicity multiplication by juxtaposition. In other words, distribute the factor through the parentheses. a(b+c) = ab+ac. Distributive Law.

The answer is 1

Super agree! You don't often see mathematicians or atleast a normal math teacher adressing the problems faced by normal peopleor students , due to an un-elaborated explanation, while dealing with problems they widely face across not just in their books, but also on daily basis. Nevertheless the answer is 9, under the rules of a sequential method called BODMAS or even often referred to as PEDMAS. Either ways it means: 1.Bracket open or Parenthesis 2.M multiplication 3.A addition 4.S subtraction I am a great admirer of eddie woo, especially from the calculus, permutations and combinations courses that I went through. It made me obssessed with a subject which I had hated in my earlydays! His reasoning behind math topics isn't often seen around, which makes it more accessible to the general public who has merely taken up with the fundamentals. Thanks for the videos, Eddie!

@@safenomore709Exactly! People don’t remember PEMDAS

Answer is one

This will help you in your confusion ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-_HtJTPelgDo.html

Agreed. This is how computers do it too. When in doubt plug it in a calculator

@@RazorM97 The problem with that is that calculators solve this in 2 different ways. Because this problem is a problem of a shorthand, linear, method of representing an equation and calculators sold inside us and outside will solve it in 2 different ways. It’s all got to do with if multiplication by juxtaposition has higher precidence.

@@JonGretarB i can't argue with that, you are correct, from what i've seen in most standard programming languages, the order will still give 9, but it's hard to even deny that these can't be arbitrary rules

@@RazorM97 I don’t know of any programming language where you can even write the formula with juxtaposition(implied multiplication). You always need to add the star sign in between, removing the ambiguity, and thus the answer would always be 9. But programming languages CAN differ in other things. Like what the modulo operator does like I learned the hard way.

@@JonGretarB Also in some countries ÷ sign has different meanings. That's why formulas should be written using ISO format and problem is gone ;)

what's the difference?

🤓

@@iceyspicey4802hey genuine advice… get off the internet 😊

​@@iceyspicey4802"wow tgis person is so much smarter than me, i should use this emoji to make them feel bad" You:

@@3000-DEN Aw but it made you feel bad despite literally nobody talking to you 😂

100%

genius

It really doesn’t matter, order of operations forces you to go from left to right.

With a telescope, I saw a man

@@zmoostofa Why would you saw that man??? AND WITH A TELESCOPE?????

He answered the question like a politician.

We have two equations 😂😂

​​@@Some1NamedPlays so again a question man is on moon or earth. 😂

Not anything! What he meant is that the 2 only POSSIBLE answers, 1 and 9 are both true because the way it was formulated can be interpreted in both way! Yes 6 divided by 2 is a division, but if the division symbol was the fraction symbol, we would all have understood that the division works as a single number rather than a division in itself to solve, and as now that Fraction Symbol is not there, even with PEDMAS, both answers are both true because the division symbol is not defined to us to have us know if it’s a fraction or not!

No, it *isn't* ambiguous. Solve it another way. 6 ÷ 2(x+1) = 1 6 ÷ 2(x+1) = 9 which one of those results in x being 3? A term attached to a parenthesis without an operator between them is *part of the parenthesis term*. Any such term must be distributed to the contents of the parenthesis before any other steps in the order of operation can be performed. Think of it as if it were a number attached to a variable. 6÷2x=n. If n is 1, then the value of x, the parenthesis phrase, must total to 3. If n is 9, then the value of x, the parenthesis, must equal 2/3.

No it's stupid to say either one. The convention is from left to right and precedence of operations with parathesis taking higher precedence so the answer is 9

@@walidyasin2039 but like in real life you wouldnt have this confusion. technically reading from left to right doesnt really matter. even if you wanted to solve it, it'd be in like an excel sheet where the confusion wouldnt happen.

​@@katheryne-bois A fraction bar acts as a grouping symbol. The only way to get a second answer to this problem would be to write it as a rational expression. But that changes the meaning of the expression, and therefore is a different problem resulting in a different answer.

100%

That's not the whole problem. It's generally ambiguous if non commutative operators are used with juxtaposition. Both is solved by using fractions but you could also always explicitly write out multiplication when used with a non commutative operator.

@@derblaue 100%

Why not prove the answer by using the golden rule of algebra. 6/2(1+2)=1 remove the explicit division be multiplying both sides by 2(1+2) 6=1*2(1+2) simplify 6=1*6 proven

well the answer is actually 9 cus explicit division has priority over implicit multiplication

So true

Well that's why some languages use comma. And then it's exact.

@Ashirwad Paswan well in my language it's simple. If you wanted to say that man has a telescope than you say: I see a man, with a telescope. If you omit the comma then you're looking at him with telescope. But in maths X(Y) is shorthand for X * (Y)

but language might be tricky, i dont understand how there is no logical correct answer of this?

@@manankjoshi981 it definitely depends on context. We could solve it using PEMDAS if it's deep & theoretical math. Solving it left to right is possible if we are computing in terms of accounting, economics, or finance. Both are right answers. Just depends on where the equation is being used

country name

@@user-zn1sc4bd4n Probably Britain, England. Their accent atleast is

@@wowzersfyi its australia

@@user-zn1sc4bd4nyou mean what country he’s from?

@@obbyistguywhodoessomeguides yes you are right

😂.. Might be. I told my wife I just manipulate or set up equations in an order that is faster and more efficient for humans to solve or where I would be more familiar making it easier to solve.

This will help you in your confusion ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-_HtJTPelgDo.html

fractions wouldn't help because it can be either 6/2(1+2) or 6/2 × (1+2)

​@@dooflegoof of course would help it's either Nominator: 6 Denominator: 2(1+2) 6 ----------- 2(1+2) Or Nominator: 6 Denominator: 2 Then multiply by (1+2) 6 × (1+2) -- 2 There is no ambiguity

@@NirousPlayers ohhhh, right, I didn't picture that in my head when I wrote the comment thanks for correcting my mistake

From the BODMAS(BRACKETS OF DIVISION MULTIPLICATION ADDITION AND SUBTRACTION) rule you must do the Division first then multiplication then addition and then subtraction so I guess the answer here must be 9

@@thegamingsuneo430the answer is 9 based on BEDMAS (brackets exponent division multiplication addition substraction), but it seems like you’re somewhat confused. BODMAS with your explanation doesn’t make sense, what does brackets of division mean? You also seem to think that you should operate in the strict order of division THEN multiplication THEN addition THEN subtraction, but that’s not the case. Division and multiplication have equal priority, same with addition and subtraction. What separates them in terms of order is which one comes first (left to right)

@@thegamingsuneo430 Neither division nor multiplication holds priority over the other. Once the expreession has been reduced to ONLY multiplication and division, and maybe addition and subtraction...you go left to right and resolve division or multiplication AS ENCOUNTERED. Then if addition or subtraction remains; you repeat. Left to right, and perform in the order encountered.

The entire reason for the confusion is the ÷ symbol. Higher level math doesn't even use it as it creates confusion, so they only use fractions as you can get answers with no confusion. The ÷ sign is for teaching division more than anything, as it looks more like the other math symbols like +, -, and x, making it easier to understand to a child.

Surely you mean brackets first...? The very first letter of BODMAS or BIDMAS or BEDMAS is for brackets... You do them first.

Not so fast, because juxtaposition exists, where the placement of operations implies priority, in such a way to remove unnecessary operators. 2*x means the same thing as 2x, but 2*x means the 2 is unrelated to x, but 2x is specifically telling you there is 2 of x without needing to write (2*x). So in that sense, countries with a syllabus that teach PEJMDAS will come to the answer of 1, and they are just as correct, since the order of operations is a collective agreement for communication, not a universal objective mathematical truth.

Grouping parenthetically is covered by PEMDAS. But there are also other ways of indicating grouping.

Academically, multiplication by juxtaposition implies grouping so writing 6÷2(1+2) explicitly before you simplify at all is 6÷(2×(1+2)) which gives 1. More literally/programming-wise, multiplication by juxtaposition implies only multiplication so writing it explicitly gives 6÷2×(1+2) which gives 9. Both are widely used so both are valid. That's why it's ambiguous.

Yes thats what i thought

2

Yeah the OG BODMAS rule (idk about O but i know the meaning) Brackets open->Division->Multiplication->addition->Subtraction (priority order)

6÷2(1+2). First bracket =6÷2×3. Then division =3×3. Then multiplication =9

Yes, words have an agreed meaning for a reason. Even acronyms! Without that established agreement, a conversation can not be constructive.

Correct

It also should be an exercise in rejecting improper or imprecise problems. As Eddie Woo pointed out, the problem is ambiguous, it can therefore be rejected as such. Any mathematician would reject this.

@John L I disagree. The agreed language of mathematics that allows for the same result every time dictates this problem be solved in a particular order. Ambiguity only comes when the language is not fully understood.

@@darkfieldcarnivore3928 So you understand what factorisation is?

100%

mhm yep

Isn't that just 49? -7 x -7 is 49, no? Why is there an issue? Thanks!

@@djkhemixit could be interpreted as (-7)^2 or -(7)^2, resulting in either 49 or -49

​@@baldabilityoh. Thank you

But then again there are are numerous questions like the math question presented in the video and no one would know what method to apply right?

@@SL_Beast well, the thing is, my peers also never learned bosmas or pemdas, so it never occurred to us to separate parentheses and use multiply. If I see 2(1+2), I've never separated it as 2 × (1+2), the first time I've seen this is exactly when this question first appeared. So for me, it has always been just 1 method. Same goes for my peers, I've never seen them separate the terms like that. But funny enough, we were never taught the phrase "multiplication by juxtaposition" either, took me a long time to even realise it's a thing. For us, the idea of multiplication by juxtaposition is more like a subconscious decision, or an unspoken rule.

@@yesno6360 That's 😎. For us I remember really well that in like 7th grade they added a whole unit dedicated to teach us BODMAS. And the thing is they didn't even teach us PEMDAS it was just BODMAS. And for the longest time I thought BODMAS was an Universal absolute math term and that there wasn't any other math terms besides it that is on the same topic/use as BODMAS. They really should teach these things in school to us.

@@SL_Beast The order of operations as it's taught, like BODMAS, is generally fine as it's handy to have a consistent way to simplify expressions that also reduces errors. (Give M and D equal priority and go L to R for equal priority, for example). It's great for people with a range of maths abilities. When you get older and more confident with it, you often stop using the literal BODMAS as there might be easier or alternative ways to simplify. For example, 4 + 3²×10/2 - 4 You can go in order: O: 4 + 9×10/2 - 4 M: 4 + 90/2 - 4 D: 4 + 45 - 4 A: 49 - 4 S: 45 You can alternatively also do: S: 0 + 3²×10/2 D: 3²×5 O: 9×5 M: 45 It's perfectly valid for that expression to do that and it's almost BODMAS backwards and it ended up being a step shorter. It's all about understanding grouping and the different strengths of the grouping of different symbols. Minutephysics did a great short video called "the order of operations is wrong" which talks a little about that. Worth a watch.

@@GanonTEK thanks! I'll look into this more this looks interesting. :)

Questions like this should never be given because it's terrible notation. Academically, multiplication by juxtaposition implies grouping so 6/2(1+2) means 6/(2×(1+2)) = 1 Programming-wise/more literally, multiplication by juxtaposition implies only multiplication so 6/2(1+2) means 6/2×(1+2) or (6/2)×(1+2) = 9 Both widely used, hence ambiguous notation. 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 here 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 (1). Just like Sharp does. TI who said implicit multiplication (1) 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 (1). TI later changed to the programming interpretation (9) 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 lesson here is use proper notation (6/2)(1+2) for 9 6/(2(1+2)) for 1 Those would be valid problems to test students. Better yet, two line fractions remove ambiguity and reduce the number of required brackets. They are best practice.

@@GanonTEK This is why I think fractions are way more useful than using ÷. ÷ is a great way to introduce the concept of division, but fractions are much more intentional.

@@Ninja0Pain Very true. Fractions on two lines are best practice. They remove ambiguity and reduce required brackets.

It depends on the syntax of the the calculator

These problems are dumb as hell, it doesn't test anything

well as a programmer, I see this as an absolute loss

@@beasthuntermohit567 I feel for you

@@KazamaKazuyoshi458 They aren't. 1 and 9 are both correct answers depend on how you write it or what is the context.

@@beasthuntermohit567 it dosent make sense to get 1 like we have always been taught to do these kind of questions by pemdas or Bodmas

@@OREO_____ It makes. Look it as a fraction. 6/2(2+1) =6/2*3 =6/6 =1

100% correct. The answer is 9.

They aren't equal the order is PEMDAS

Making up your own bullshit to solve the math

That's not the correct reason though, multiplication and division can happen in any order with the same result just like addition and subtraction. When you think of division as a fraction there isn't a way divide by 2 and (1+2) because one is a numerator and one is a denominator. It would have to be 1/(1+2) instead. Another way to write the problem is 6/2 * (1+2)/1, both are fractions and now the reason is more obvious. You can also rewrite everything as multiplication between fractions, so 6/1 * 1/2 * (1+2)/1, and the order that you multiply them makes no difference. The result should always be 9 unless (1+2) is explicitly part of the denominator, otherwise it is assumed to be on top. So it's not the order that matters, it's the assumption that multiplying 2 by (1+2) is possible in the first place. There would need to be parenthesis around the whole thing, like (2(1+2))

Actually lookup mathematical order of operations: M is before D... Mixed division and multiplication Edit In some of the academic literature, multiplication denoted by juxtaposition (also known as implied multiplication) is interpreted as having higher precedence than division, so that 1 ÷ 2n equals 1 ÷ (2n), not (1 ÷ 2)n.[1] For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division,[20] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[d]

1 is the answer

I have seen this in a comment section maybe it will add some light ( 6÷2(2+1) we can assume 2+1 to be x then we have 6÷2x by simplifying the fraction we have 3÷x recall x=2+1=3 we have 3÷3=1)

Depends if you follow the universally accepted rule of BODMAS or some random American rule. If you follow BODMAS it is 1.

​@@usmanbelloahmad6461no you are wrong it is 9 at least according to math we know of 6/2(2+1) is not 6/(2(2+1))

the fact that many people still debate over this is so ludicrous, instead of just accepting that the division notation is the heart of the whole matter, get over it, be thankful that fractional is more common instead, and search out for questions that aren't trivial (i.e. number theory, combinatorics/permutation/star&bars, trig, calc, ineq, ode/pde, group theory, abstract alg, etc, as long as it isn't at the level of this overrepeated problem)

Denominator :))

How about 6/2(3)?

Nope.6/2(1+2) is equal to 3*3 which is 9 Now why is like this. (1+2) is counted first Then you need to divide 6 by 2 And lastly multiply quotient with result of sum 2 numbers which was placed inside of the bracket.

@Zenix. Wrong way round. 6/2(1+2) = 6 over 2(1+2) = 6/2(3) = 6/6 = 1. For it to be 9 we need to change the _expression_ to (6/2)(1+2).

Facts

Yeah just like how it would have been written differently to be equal to 1

How about switching with a telescope with using one

The answer to the equation of the video is 12!

And.. Chat GPT was wrong. "Through" and "with" are both acceptable here, and have the same meaning.

@@stevejohn7459 no its 16

In the US we learn the same thing with the acronym PEMDAS, so yeah parenthesis are first and then after that is either multiplication or division whichever comes first left to right

they literally told you the order. You literally repeated it. From left to right ahaha

@@TheLifeLaVita no cause it’s more like P E MD (whichever comes first in the problem) AS (whichever comes first in the problem)

@@SilverPh3nix you wrote yes* wrong

@@TheLifeLaVita lmao I’m a dumbass whoops

Cringe

​@@jarredlucas4000ill mannered chap

But it's not ambiguous. Math has clear rules that give you the answer which is nine by the way. This is why Matt has rules to avoid this sort of ambiguity. This is like what middle school math was all about or remedial high school mat.

@@SuperWolfkin math is unambigous if you use the correct notations which this equation doesn't

@@signeCS I'd argue it's not ambiguous as it stands. You don't always need specific notations when it's covered by the conventions of the environment. Like you don't need to indicate which way to read the letters when you write a sentence because in English we have the convention of Left-to-Right. Likewise math has LTR/PEMDAS conventions that take any ambiguity out of this equation.

People on the internet with no qualifications when someone with all of the qualifications show up... and still call them wrong... are just so idiotic.

You can see this issue in their language. I've heard a bunch of people talking about how the other half can't do whatever grade maths. That's the problem! Most people are relying on elementary level maths without taking into account the other ways of doing things because their way is the right way, but it's only low level. Maths is unfortunately nuanced, despite us wanting to believe it's black and white.

100%

That's true for a lot of intentionally open-ended engineering/physics questions, but in this case, it's just a matter of remembering all the order of operations that we learned in grade school. In higher level math, we usually don't see things written this way because even though it's technically right, it's confusing, and some people can get the wrong answer (as made evident by this whole mess). This wasn't a trick question. It was a tricky one maybe. The answer is 9 btw.

​@@samshim3149 the writing is wrong. Without any context, this can be anything. Some calculators even said it 1. A fricking calculator! If this was a paragraph question, no doubt the answer will be wrong

@@youravghuman5231 Again, I'd have to disagree. I'd argue that the question isn't wrong. Poorly written maybe, but that was intentionally done to cause confusion. The only part that people are confused about is the implied multiplication. I guess some people were taught that if you see number touching parentheses, you need multiply it immediately, no questions asked to get rid of the parentheses. Back when they taught me math, they also taught us that you can only do that if it doesn't affect the rest of the expression. Maybe I only know this stuff because I used to get points taken off when I showed my work like that back then, but this is how I do math in engineering today, and it works great. If you wanna read my full explanation, I left a comment. TL;DR is the question is only "wrong" if they meant to say 6/(2(1+2)), but since they didn't say that, you gotta assume they're trying to trick you, so you should read it literally and solve it literally, without making assumptions as to what it may have meant. Just curious... What kind of context were you talking about? Just like extra brackets or something? Because my first thought was the context is it's an internet challenge.

@@samshim3149 im not good at English but it's poorly written. Intentional or not doesn't matter because it's a mistake in the question. Just read other comments they explain about juxtaposition. What i meant by context is like a question with a given scenario instead of giving an equation like this. That can be more understandable than this equation. If a student writes this equation like this in that question, no doubt he will be marked as wrong.

Division is division. Who tf would interpret it as one big fraction? That's not even a debate. The way you denote division doesn't change its precedence.

⁠@@paulblart7378just because u think it wouldnt be interpreted as a big fraction doesnt mean it would never be interpreted as such.

@@whojoue0000 the point is that it's still wrong. It's not a big fraction, and you can't just choose to interpret is as such, it would be wrong.

​@@whojoue0000Very true, and if you use the academic interpretation of multiplication by juxtaposition, which implies grouping, it is a big fraction and commonly interpreted as such, even by many modern scientific calculators

​@@paulblart7378literally in any level of math beyond grade 10. Ex. The quotient law of logs

_"So that means the extended way to write this equation is 6 / 2 * (1 + 2)."_ You can get away with 2 * (1+2) because it resolves to the same number in isolation, and where there is no preceding division in the expression. When you expand it the way you state, you are falling into a trap. 2 of (1+2) should become 6 over 2 of (1+2). but you are making it 6 over 2, of (1+2). See the difference? It should become 6 over (2 of (1+2)), not (6 over 2) of (1+2).

Writing 6/2(1+2) is just as ambiguous. Never write multiplication by juxtaposition after division on one line. 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. Follow that and there is no ambiguity. (6/2)(1+2) for 9 6/(2(1+2)) for 1

@@GanonTEK No it's not it's tough to put long / here. What I mean was try putting 6 as numerator and 2(1+2) in denominator. Now whatever way you solve it you'll get the right answer.

@@ScoRPy22 Ah, yes, using a viniculum, a horizontal fraction bar, does remove ambiguity. For example: 6 --- 2(1+2) / is not a representation of the horizontal fraction bar though. The horizontal fraction bar implies grouping. / does not. / is the unicode Solidus but isn't the real Solidus. ½ is the real Solidus which shows a clear two line fraction but on one line by using a steeper line and sub and super script. 1/2 is not as clear cut as the numbers are not offset so you get issues with 1/2(3). ½(3) is clear but 1/2(3) is not.

Bruh it’s 12!

Bro he didn't answer the question

I love watching cringe indian videos with cringe music, intro and voice

9

What else can it be

Bodmas

I can tell you are in 4th grade

This will help you in your confusion ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-_HtJTPelgDo.html

Underrated comment

he fye 🔥

9

@@_Nilu__ There is no multiplication in the question. It's juxtaposition. It functions the same as multiplication but it's undefined whether it has different priority or not.

It depends on which interpretation of multiplication by juxtaposition you follow. That's why it's ambiguous.

There is. Order of operations. The answer is 9. There is no ambiguity

@@seanspreckelsen3496 In math there's an order to solve operations. Copy this from a webpage. "First, we solve any operation inside of parentheses or brackets. Second, we solve any exponents. Third, we solve all multiplication and division from left to right. Fourth, we solve all addition and subtraction from left to right."

@@seanspreckelsen3496 If you follow the order is 1, but the calculator answer is 9. 🤷‍♀️🤷‍♀️

@@Dragonflies82 that's not correct, you're misusing order of operations. PEMDAS is the acronym, but the order should be P>E>MD>AS, it is NOT P>E>M>D>A>S. You multiply AND divide in the order the problem is written (left to right, top to bottom) then you add AND subtract in the same order. 6÷2(1+2)=x First the function inside the parentheses. 6÷2×3=x Then, being no exponents, you multiply and divide in order from left to right. The left most function is division so in this case you divide BEFORE you multiply. 3×3=x Then multiply SECOND. x=9 The problem isn't designed to be ambiguous, it's designed to point out a common misunderstanding of order of operations.

@@Dragonflies82 order says parantheses first, so you get 6:2(1+2)=6:2(3) which is 6:2*3. Now you need to follow the order from left to right. Youll get 9

Parentheses come first.

@@slamkam07 uh... Y...yeah? You're absolutely correct, but I just don't see how it connects to what I said

I know it as BODMAS Brakette Off Division Multiplication Addition Substraction🎉

I don't get it, i can just add parentheses in the equation if i want?

@@Luizedu If it follows the initial order, yes, it can help avoid confusions.

The order of operations doesn't interpret implicit notation and that's where the ambiguity occurs.

@@GanonTEK idk what your school taught, but my entire k-12, and even into College math courses, ALL flat out stated it's PEMDAS, sometimes written as PE(MD)(AS), Multiply/ Divide and Add/Subtract IN THE ORDER they appear. There is no ambiguity about it. The answer is 9.

@@Th1sUsernameIsNotTaken You've missed the point. The implicit notation is ambiguous here since there are two common interpretations in use. Academically, multiplication by juxtaposition implies grouping so 6/2(1+2) means 6/(2×(1+2)) written explicitly, which is 1. Literally/programming-wise, multiplication by juxtaposition implies only multiplication so 6/2(1+2) means 6/2×(1+2) which is 9. Nothing to do with the order of operations at all since the same order of operations gives 1 and 9 once the notation is written explicitly. Proper notation writing is the solution here. (6/2)(1+2) if you mean 9 6/(2(1+2)) if you mean 1 No ambiguity then.

@@GanonTEK It isn't ambiguous. You literally added parenthesis to the question when it wasn't what was given to you. You're FORCING ambiguity. And no, programming wise, if we're talking "best" practices, would be more along the line of (6/2)*(1+2), or whatever iteration you'd like it to be, to ensure there is NO ROOM for misinterpretation, even if it is obvious to you.

@@Th1sUsernameIsNotTaken They are implied if using the academic interpretation of multiplication by juxtaposition. Like how Sin2y means Sin(2×y), not Sin2×y or how ab/cd usually implies (a×b)/(c×d), not a×b/c×d. It's a common convention. Yes, explicit operators are pretty much essential for programming.

I dont really know what you mean with two kinds of math, but i would agree in the sense that this is why fractions are superior. These are not so ambiguous. The arithmetic division sign is honestly just an inferior operator sign because you can just get confused with the order of operations rather quickly if you are not used to using this sign. But if you do exactly as the order of operations tells you, you are fine. Math itself is not breaking. It's the misinterpretation that makes it look like there is no right answer. Maybe I misunderstood what you mean by "different types of math"

​@@DonPedro69 idk if this is what Talla was talking about, but the way i see it is that that expression mixes two types of notations. it's not exactly a rule, but if you think about it it's more likely to see "÷" and "×"together in one expression, the same way as seeing "/" and " ⋅ " in the same expression. so when you mix the two notations it gets confusing. so if you write "6÷2×(1+2)" it's clear, as well as writing "6/2(1+2)". mixing the symbols makes stuff weird

@@andreasibilla7855 maybe it was suppose to mean what you say, however that still makes no a big difference in the way you calculate. To make it clear. First of all, "÷" is never recommended to be used at all, but if you want to use it you can use it to show ratios between two numbers like 2÷3, even than it is prefered to use 2:3 or 2/3 (it's just a convention). If you have more than just a ratio between two numbers you should always use horizontal fraction bars like for example (5×6+2)/(5×3) (i cant write actual horizontal fraction bars but just imagine it) but also for algebra you should always use fraction bars. They are just much easier to understand and they also get rid of some parenthesis which makes it easier to calculate correctly. × and • have the same meaning tho and make no difference

​@@DonPedro69 the fun thing is that I believe this is mostly a mathematical debate because of the internet. I believe I watched a video once that explained that up ti a certain year people would prioritise the order from left to right, over the order of either ÷ or × having priority. This is a debate mostly because different generations and different countries get to look at this through the internet, and we can see that it really depends on how the rules of that country are, because in my case, if this was about a fraction, the second part would ned to be in brackets too. It is how I learned it, otherwise we just go from left to right. It's not that the math of me and other people is different, it's rhe way we have been taught to communicate it.

@@corneliahanimann2173 yeah that's the main issue i think, you are right it really comes down to how you learn it at school ig, but at a certain level like university things become more unified...atleast to my knowledge

But what determines which way to get the result?

​@@Rami-bi9xjAsk whoever wrote the question to rewrite it as a fraction instead, or if that's not available just give up

It’s not confusing, it’s not a special math problem, the answer is 9 and only 9.

It’s a straight forward term divided by a term. The term 6 divided by the term 2(1+2) gives the answer of 1 2(1+2) is one term containing 2 factors. The 2 and the (1+2). Factors multiplied, are a single term.

​@@yourmommydotcommy2650go back to school bro

Well tell whoever said 9 they are correct, and the other to go back to elementary school and learn order of operations

@@grapeman8612 order of operations is irrelevant. Multiplication and division have the same priority as they are the same operation (division is multiplication by the reciprocal). The only issue is that the problem is intentionally written poorly to cause arguments and generate engagement. Source: I have a math degree.

@@SappinYourSentries order of operations also state you go left to right, and this problem is incredibly simple when you follow that. Also there is no reason to do 2*3 first because then the actual problem is 6/2/3 and that is not what this problem is. There is no parenthesis around 2*3 so you don’t do it first Also I know you’re not arguing but it isn’t really written that poorly Source: I have a brain and am not 1 year old

​​@@grapeman8612 Number 1 is correct because of three reasons: 1) 1. 6/2(1+2) 2. 6/2(3) 3. 6/2(3) ≠ 6/2×3, so you solve 2(3) first 4. 6/6 5. =1 2) So you know in algebra if an equation is something like 2×(2a+2b), you multiply the two with both factors in he brackets, making it 4a+4b Well, the same thing happens 6/2(1+2) 1. 6/2(1+2) 2. 6/(2+4) 3. 6/6 4. =1 3) / is just the same as ÷, which means there is a fraction So that would mean that 6 is the dominator and 2(1+2) the nominator 1. 6 --- 2(1+2) 2. (Solve 2(1+2) which way you like) 6 --- 6 3. =1 UNLESS You interpret the 6 as the dominator and 2 as the nominator, which would then mean they both get multipled by (1+2). 1. 6 -- × (1+2) 2 2. 3 × (1+2) 3. =9 If that's how you solved the equation then that's fair. Anyways the answer is 1 and even any calculator says that And I realized I put way too much effort into my reply

@@hanmira you didn’t put too much effort in, it’s all fine lol, but anyways here’s why your wrong 1. Even if it is 6/2(3), it’s still be nine because you have to go left to right. (Parentheses rule does not matter because the 2 is not inside parentheses.) 2. Again you are ABLE to distribute, but distributing at that time would be going out of order. 3. And if you wrote it like an equation you would move the 2 to the denominator and the 6 and (1+2) (or 3) to the numerator. Still being 9 (18/2) I’ve done some more research since I’m confused as to how anybody gets this wrong, and apparently there is historical reasons. In the past the % (division symbol) would be differing from / (other division symbol) So a problem like this 8%2Y, would turn into 8/(2Y). That symbol used to mean “divide everything that comes after” but it doesn’t anymore” so now % and / mean the same thing, and there is no version where this equals 1, unless you’re living in the early 1900s Also I realize that is a percent symbol but I couldn’t for the life of me find a division one like the one in the vid

Nope. Multiplication and division always go from left to right if in the same consecutive order unless one of the two is in brackets

Exactly

It depends on how you interpret multiplication by juxtaposition. Academically, it implies grouping but the more literal/programming-wise interpretation is that it only implies multiplication. It's just a really poorly written expression. 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.

ok @@GanonTEK

Since when do they teach the material in the syllabus?

The problem here is the implicit multiplication (or juxtaposition) which makes it all ambigous and since implicit multiplication can only be used when it's not ambigous the whole expression is invalid.

BODMAS is just a mnemonic device. Multiplication and division are literally the same operation. Regardless, this moronic expression is ambiguously written. You never see this is in grown up math because no one writes anything ambiguously.

@@derblaue well maths has a serious problem if its being ambiguous rather than concrete

​​​@@kryptoncrescent that happens when people dont want to follow proper writing. If it was written properly it would either be 6/2*(1+2) Or 6/(2*(1+2)) in this case you dont need the * I think the biggest problem is that people dont even know how PEMDAS and simmilar stuff works. They think since addition is before subtraction it makes it a higher priority, luckly in my country we dont have such acronyms and we are just thought as it is.

@@kryptoncrescent Math doesn't have a problem, people who don't know how to write math do. There is a difference. This is a poorly written problem plain and simple. The fact that it is poorly written does not affect math itself only this problem and those who read it.

Exactly. So many people are trying to pass themselves off as superior and so smart because they think it's a simple question... all they're doing is showing their failure to understand the topic.

6/2(1➕️3)🟰❌️ 6➗️2(1➕️3)🟰✅️

6/2×(1+9)=9 tho

LIIIEEES! I heard the same all the time.

I mean to be fair it is very often the case

There’s a right answer: 9.

@@DonPedro69 Yeah, but as any mathematician will be quick to point out:" Well, it's not ALWAYS though, is it?!" x'D

@Monkey Business good point...I say it the way I say because I'm really not sure xD

The order of operations doesn't prove one answer over the other, unfortunately. It can't, because it's the *notation* that is ambiguous. If you interpret the implict multiplication literally, you convert 6÷2(1+2) to 6÷2×(1+2) which is 9. If you interpret the implict multiplication academically, you convert: 6÷2(1+2) to 6÷(2×(1+2)) which is 1. That's where the ambiguity is. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. Both are widely used and nothing to do with the order of operations used at all. It's also why anyone using the order of operations to prove one answer over the other is just making a circular argument and proving nothing. All they are doing is assuming a notation interpretation and saying the one they picked is the right one, when it's not the only right one. Mr. Woo is correct and multiple institutions and professors agree.

⁠@@GanonTEKThere is no ambiguity, you either come up with some parenthesis out of nowhere or just do math

@@413XUIFC With ab/cd you get (a×b)/(c×d), so parentheses out of nowhere. With Sin²x you get (Sin x)², so parentheses out of nowhere. So, it's quite common and an integral part of maths. The issue here is the notation, which isn't maths. Notation is language. Language is used to describe the maths, but the language isn't the maths itself. Language is arbitrary, like our base 10 number system. You can use any number system you like, and all the rules and properties of maths still hold. Pythagoras theorem in base 10 a² = b² + c² Pythagoras theorem in base 2 a¹⁰ = b¹⁰ + c¹⁰ Even indices notation is arbitrary. aa = bb + cc is valid. Notation and rules are completely separate from each other.

@@GanonTEK Dude, you are just trying to mix the little knowledge you have about basic math just to try to show you know, when you don’t Having as an argument that you can invent parenthesis out of nowhere because of a trigonometry “shortcut” is the most ridiculous thing I’ve read in a while Copied comment: You are inventing a parenthesis out of nowhere, leading to this new creation: 6/(2(1+2)) Which would be 1 BUT THAT IS NOT THE CASE So now read carefully because I will explain something you should have learnt at the age of five: There is something called “commutative property”, which only applies (in basic math) to sums and multiplications, but NOT DIVISIONS (turn /6 into 6^(-1) and see how you can now freely multiply), and that’s why you can’t do right side first. There are many different ways of demonstrating why 1 is simply NOT correct, but that is the most basic example I could come up with. You clearly have no idea mate, I don’t know what you are studying, but it is just being a waste of time

@@413XUIFC All implicit notation conventions are shortcuts. That's the entire point of them. Don't you know that? 2(1+2) is another form of implicit notation, like ab, or Sin²x. You just don't believe it is, with no evidence. Here is something you should know since you were 5: Notation is interpreted before any rules are used. Why? Notation is language and is arbitrary. 6/2(1+2) using the academic interpretation of multiplication by juxtaposition convention means 6/(2×(1+2)) which is 1 Commutative property: 6/(2×(2+1)) = 1 6/((2+1)×2) = 1 6/((1+2)×2) = 1 Commutative property holds. Replacing division with multiplication of the inverse: 6×½×1/(1+2) = 1 For ease I'll write 6×½×⅓ = 1 Commutative property: 6×⅓×½ = 1 The multiplicitive inverse, commutative, distributive properties etc. all hold for 1 and 9. Why? 1 and 9 are not determined by the rules here. They are determined by the notation convention applied. If you use tte more literal/programming-wise interpretation of multiplication by juxtaposition you get 6/2×(1+2) instead and then, after you have removed all the implicit notation you can apply the rules which all give 9 in this case instead. The rules do not describe the language. The language describes the rules. So, you haven't demonstrated any way 1 isn't valid yet. Every way you can get 9, there is a way to get 1, without fail. It's impossible not to, since it comes down to notation conventions abs not rules at all. I've a lot more knowledge than you presume. You're the one with no idea, which is obvious when looking at the facts. Modern international standards like ISO-80000-1, the American Mathematical Society, calculator manufacturers, multiple professors and mathematicians all agree it's ambiguous. They are people and institutions with authority on the subject. If you think you know better than them, go ahead and take it up with them yourself.

It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. That's the issue. You can't prove either answer since it comes from notation conventions, not any rules of maths. 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

There is no multiplication in the question. It's juxtaposition. It functions the same as multiplication but it's undefined whether it has different priority or not.

Is Canada we use BEDMAS which means division comes first. To be honest I always thought they were interchangeable

@@dglogan4647 they both are same but the important part is the order of evaluation which is left to right. So division comes first. 6/2 is evaluated first. Answer is 3. Then 3(3) the answer is 9. Similarly if it was (1+2)2 / 6 then the (3)2 / 6. the answer would still be 9

@@dglogan4647multiplication and division are interchangeable. In cases where you have a problem with multiplication and division you would work the problem LEFT to RIGHT.

We have BODMAS here

In India we got "Bodmas" i.e: Bracket of Division Multiplication Addition and the last but not the least Subtraction.

the thing is, if we are already using brackets to make it clear that 1+2 must be done first, then it is obvious that if we leave the second brackets out of the divider 2 and the factor (1+2) that they must be seen seperately, otherwise i could doubt the means of the first brackets too

Oh calculators disagree depending on what convention they follow. Even calculators by the same company.

​@@enraqusbail6314 yeah, 6 ÷ 2 (2 + 1) is 6 ---- (2 + 1) = 9 2 While, 6 ÷ (2 (2 + 1)) is 6 ----------------- = 1 2 (2 + 1)

Naaah. I still go with 1 for following logic in a sentence 🤣

what are you talking about? the order of operations say that you solve the paranthesis first? so it's 2(3) = 6 and then 6/6 = 1. You have a multiplication between the 2(3) since you have a paranthesis and a multiplication it's going to be 6/6 = 1. The multiplication and division in the order of operations you find in PE(MD)AS while you find partanthesis in (P)EMDAS. Obviously since you have both M and P first before divsion the more dominant answer is 1. When you have a multiplication with a paranthesis it's like it's glued together in a sence, you can't just neglect that fact and take away the glue and divided 6 with 2 before you resolve the glue with 2(2+3)

Very smart very clear, math isnt ambiguous, its clear, its truth

Algebra has a very specific grammar, which is designed to be unambiguous. That is to say, there is a clear answer: 9. Algebra is not like natural language

@@Supreme_Lobster point me to this problem in the algebra text book

100%

I would agree with you, but you can just use order of operations to solve and get 9. (1 + 2)=3, 6 divided by 2 = 3. Both threes are next to each other so you multiply them (3x3) to get 9

It’s already in fractional form. 6 divided by 2(1+2) Answer is 1 6/2 cannot be the factor of the parentheses as factors must be whole numbers. Not fractions. And the term 6 is separated from the factor 2 by explicit division. So the factor 2 being juxtaposed with the parenthetical expression containing the factors 1 and 2, having a higher priority over explicit division and multiplication, must be simplified first.

The debate is where juxtaposition falls on the order of operations. If it fell before multiplication and division, it'd cause 2(1+2) to become 6 before the division sign is even considered.

Funnily enough, there is a version of PEMDAS that does take multiplication by juxtaposition into account called PEJMDAS. Where J is for juxtaposition. It's above regular multiplication or division. Maybe BOJDMAS is the BODMAS variant? Hard to say it though! I agree with your comment completely.

@@GanonTEK That's cool! I have never heard of variants with a J in them! Thanks for being so cool!

@@jacckkaboii3528 Thank you for being so kind!

@@GanonTEK I never called it multiplication by juxtaposition but I always used it and thought it was part of the P or B in both version like, 2(5 + x), I always thought regardless of whatever is done to the 2, it will first become 10 + 2x first before anything else is done to it. So this 6/2(2+1) has always been 6 all divided by 2 into 2+1. It's the same as 5 x 8^2, because there is no bracket covering the 5 x 8 part of the equation, I would think that the thing being squared is the 8 and not 5 x 8.

@@tjossai9302 For the last example, exponents have higher priority than addition so that's why it's only the 8 being squared. Outside brackets are not part of the B/P step, only inside parentheses are. It's a notation convention, though, that puts implied brackets around expressions like that, so it's a perfectly valid interpretation but it's not the only one in use. That's the problem. It should be written properly as (6/2)(1+2) or 6/(2(1+2)) to remove all ambiguity.

No, no.......so 6/2 = 3 is just wrong. Where does the = come from? Sure, 2+1 = 3, but you don't put a = into the calculation. The way you wrote it, it would be 3=3 which doesn't make that much sense in that context. Even though you got the right answer. First, you solve the stuff inside the brackets, then you do the "point before line calculation" don't know if it is called that in English, what I mean is, you calculate multiplication and division before + and -. 6/2(2+1) ---> 6/2(3) ----> 3*3 = 9

but wut if u live in An Asian country where they write right to left?

@@jujucasar2003 it doesnt matter. just use bodmas

@@jujucasar2003 Math, is a language, if they speak their own language, they can write their own direction, but like english, math is ALWAYS written from left to right.

ayo bro , here the question is not about using BODMAS, it is about whether the (2+1) is in numerator (answer = 9) or denominator (answer =1)

While I completely agree with you regarding the distributive property part, I think the problem is the horizontal notation, rather than the ÷ symbol, because even with the / symbol, it's quite ambiguous whether the (1+2) belongs to the numerator or the denominator.

nerd

It's a notation problem yes, replace ÷ by a fraction / . It is (6/2)*(1+2) = 6/2 + 12/2 = 18/2=9

true , writing mathematical equation in one line is biggest crime in mathematics

Bro i was searching for this 😂 it's disappointing to see that we have forgotten the basic fundamental rules of mathematics

Thank you!!!!! Simple and straight forward Use BODMAS

Well there is a piece missing on this, because there is also an agreement that if two mathematical operations have the same priority, you have to solve the exercise from left to right, which is the case in this exercise after you solve the parentheses: 6 ÷ 2 * 3

@@jethropumbwe4515 The thing is that school level maths don't show the more detailed problems that arise from careless use of juxtaposition (multiplication by just writing two things next to each orher: a*b = ab).

BODMAS is just a mnemonic device to help children memorize. There are only TWO binary operations in the ring of real numbers, NOT four _(all other operations are built off of those, for example, a^2 is just a times a; the square root of a is just the inverse of a^2; a^3 is just a times a times a; a^b is just a multiplied b times; a! is just a times (a - 1) times (a - 2) ... times 1. etc)._ Those TWO binary operations are addition and multiplication. Division is simply a type of multiplication. Specifically, division such as a/c is just a times b where b = c^(-1). In other words, division is just multiplication by the multiplicative inverse. The only note of caution here is that the additive identity does not have a multiplicative inverse (that is, you can't divide by 0, because there is no a such that 0 * a = 1).

There is no multiplication in the question. It's juxtaposition. It functions the same as multiplication but it's undefined whether it has different priority or not.

BEDMAS? Brackets before division, though? I'm confused

Ok so if A/x(y+z) and x(y+z) is also (x*y+x*z) then A/x(y+z) can also be a/(x*y+x*z) so now if you calculate it your number if unambiguasly wrong

there is no left to right rule. what in pemdas stands for left to right? nothing does.

@@signeCS This is what people don't understand. Left to Right is not a law, its a suggestion designed to reduce the amount of thinking you have to do. You literally never ever use left to right in anything higher than grade 9 math

@@kato2048 thank you, someone who actually understands math