This will help you in your confusion kzfaq.info/get/bejne/la6kfbeBytLLdKM.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

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

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

what are they teaching ppl these days 😂

​@@bdat6321??

How about? 6÷2(1+3) =6/2(3) =2/2 or 3/3 =1

didn't solve the brackets fully

I used pemdas

I am pretty sure it is

same

Nah it’s 1

But wouldn't that be 9 if 6÷2 was also in brackets... I mean (6÷2)(1+2)... But when the equation has no separation that clearly means that (1+2) is only multiplied with 2 before that and not the solution of 6÷2

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

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 kzfaq.info/get/bejne/la6kfbeBytLLdKM.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 ;)

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 😂😂

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

the P applies to these parentheses you yourself have around (3) order of operation say P first, not M, this will cause2(3)=6....6/6=1

​@@bonitawilliams-anderson6820 At first, I thought of using PEMDAS this way and got 1 initially. Looking at it now, since the "2" is not inside the parenthesis with the "3" which would be (2 • 3) and is bound by the ÷ sign in 6÷2 then the multiplication isn't prioritized over the division which would lead to 9 if looking at it from the left-to-right angle. Calculators are telling me it's 9 anyways.

@@brandonhiguchi9623 "1" is correct (apparently "9" is an acceptable answer, however not understanding why)

​@@bonitawilliams-anderson6820 the P in PEMDAS means to solve what's in the parenthesis, not to multiply them. Multiplication and division are interchangeable in PEMDAS and work from left to right, so you need to divide 6/2 first to get 3, then you multiply that to 3, which is 9

@@thecubingsquare101 So the answer is 1 and 9 is an acceptable answer.

That is a way to do it. Just like saying "I saw a man with a telescope" can be interpreted as "I saw a man by using a telescope". It's poorly written, and math nerds will just avoid using the ÷.

After you add 1 + 3, you still have the 3 in brackets. The rule is to remove the brackets first, correct? If so, then the next step is to multiply 2(3). Final answer is 1

I agree. I have a degree in math. However I remember back in the basics way way back during junior high algebra, you go from left to right as you said. Thus the answer is indeed 9.

​@@jpjp9111nah its not that simple its pemdas Parentheses Exponents Multiply and divide left to right Addition and subtraction left to right. The reason this cant be answered in its current state is because its written bad and can be solved 2 different ways

@@markpage5567No you do the division before the multiplication. Every schoolboy knows this....BODMAS,

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

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.

The obvious answer, what fools we all were for not seeing it sooner

Lmao

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

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.

😂.. 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 kzfaq.info/get/bejne/la6kfbeBytLLdKM.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

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

He just said it can be 1 aswell

Only one answer and it's 9@@Gh05tt

@@Youness324 it can be 1 your wrong I take advanced mathematics

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

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?

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. :)

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.

​@@placeholderfornow4766you are an idiot or just bad joking?

​@@unbreakablebedrock2313They are correct. There is no juxtaposition in the regular order of operations since it is a notation convention. There are variants that include it like PEJMDAS and around half of scientific calculators effectively use that concept (you can see it in their manuals). Implicit notation needs to be interpreted before you use any rules unless you use a variant that includes it.

Yes, using appropriate brackets like (6/2)(1+2) for 9 or 6/(2(1+2)) for 1 removes ambiguity and is what modern international standards like ISO-80000-1 says to do in this case. Two line fractions are best practice and also remove ambiguity, and also reduce the number of required brackets. 6 ---(1+2) for 9 2 6 --- for 1 2(1+2) Correct notation is the solution here.

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.

Exactly what I teach my students. Do away with the × and ÷

Right on. All this nonsense about "ambiguity. There are well defined rules.

The issue is with the notation used, not about the order of operations or priority. Academically, multiplication by juxtaposition implies grouping so 6/2(1+2) implies 6/(2×(1+2)) Using whatever order of operations you like gives 1. Literally/programming-wise, multiplication by juxtaposition implies only multiplication so 6/2(1+2) means 6/2×(1+2) which is 9 instead. The order of operations doesn't interpret implicit notation so it doesn't resolve the ambiguity.

@@GanonTEK I am a computer science teacher in highschool and I barely remember what you 're saying (my fault), from University. Since then I can not recalI another time. I believe in practice (e.g. programming languages, Excel, calculator) there is (almost ?) never a group implied. EDIT: I just tried it in my Android calculator: 6÷2×(1+2) and it gives 9. Not that this proves anything, just an example. If anyone can find some programming language which gives a 1, I am interested. Happy New Year!

@@dimitristripakis7364 Programming languages and spreadsheets do not tolerate missing operators in a math or arithmetic expression (as you of course know). Thus the expression 6/2(1+2) will give a syntax error in every programming language that I'm familiar with. In order to fix the error, one must insert the omitted multiplication sign. If inserted the way you did, they will all give 9. Some people interpret 6/2(1+2) such that the whole of 2(2+1) is in the denominator. They will have to also add brackets, i.e, 6/(2*(1+2)). The result from programming languages is then 1.

@@donmacqueen Yes of course I added the *, it does not work otherwise.

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)

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

Yeah, maths classes don't even use that stupid ÷ sign once you get get to high school. Fractions are just objectively better

@Condred Idk why we ever used that because the divide symbol is literally just a blank fraction

Yeah completely agree!

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]

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

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.

thank you like shut up i used to argue with people but once I heard from an expert aka today that thy could coexist I know that both are right

A better example is "I saw a man with a stick" No issues there, the man has a stick. "I hit a man with a stick" Hang on, who has the stick now? Are you using a stick to hit him with (verb attachment) or are you hitting the man from the first sentence who had a stick (noun attachment)? Both are valid interpretations of the sentence. So it's ambiguous. With 6÷2(1+2) It depends on which interpretation of multiplication by juxtaposition you use. Academically, juxtaposition implies grouping and multiplication (1). Literally, juxtaposition implies multiplication only (9). Both use the same order of operations after the implicit notation is interpreted. So, it's also ambiguous.

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 do PEMDAS and I get one. P(Parentheses) E(Exponents) M(Multiplication) D(Division) A(Addition) S(Subtraction)

That is a big misconception. Multiplication and division as well as addition and subtraction are done from left to right. Division comes before multiplication if it is to the left. I think that a lot of schools fail to teach that.@@user-dy4fx9lc9w

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

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.

No, both 6÷2(1+2) and 6•1/2(1+2) are ambiguous for the same reason: they both have implicit multiplication after division on one line. That's the main problem. You need to write 6•(1/2)•(1+2) or (6/2)•(1+2), for example, to be unambiguous there and something like 6•(1/2)•1/(1+2) for 1

Yep

Yes

No. If you see 2*3÷7+2(5) The solution to that is, 76/7, because you always do x(y) first

please visit 4th grade

I agree this is correct. PEMDAS / BODMAS / BIDMAS are the fundamentals for a question involving multiple types of operations. But also, once the numbers inside the brackets is added up, there is however a rule when we end up with only + and/or - in an equation or division and/or multiplication, you must go left to right. There can only be one right answer, because saying 'yes' basically implies 1 = 9 which is completely false.

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.

Pemdas makes it 9, md is multiplication OR division (done left to right). “Left to right” on its own is not a valid rule for order of operations.

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

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

It depends on which interpretation of multiplication by juxtaposition you use. Academically, juxtaposition implies grouping and multiplication (1). Literally, juxtaposition implies multiplication only (9). Both use the same order of operations after the implicit notation is interpreted.

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.

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

@@JustaTechSavvyGuy.Multiplication and Division have the same weight

country name

@wowzersfyi its australia

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

@@obbyistguywhodoessomeguides yes you are right

@@user-zn1sc4bd4n hes french i think

Correct answer, but for the wrong reason. Multiplication and division have the same order precedence, so you operate from left to right.

If you treat the division as a fraction, you get (1/6) * 2 * (1+2), which is 1. Eddie says it's ambiguous because of this, because you can interpret the division sign as a fraction sign.

@@aguywhouploadvideos693 If you treat the ÷ sign as a fraction, you actually get 6 * (1/2) * (1/(1+2)), as the right side of the ÷ sign would be the denominator, resulting in 1. I simplified it to 1/6 to avoid the extra parenthesis and to make it a bit clearer. (Because in this case A == B)

Yeah that's my answer and I entered this vid to make sure but there was non except for this comment

actually, why did people thought of using the division in this equation as a fraction

This will help you in your confusion kzfaq.info/get/bejne/la6kfbeBytLLdKM.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.

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.

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

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).

@@allenchang61856/2(3) means 6 divided by two times three. Thus, we do it from left to right, because multiplication and division have no precedence. We then get 3*3=9, our final answer. The bracket means nothing when it has already been dealt with.

It depends on which interpretation of multiplication by juxtaposition you use. Academically, juxtaposition implies grouping and multiplication (1). Literally, juxtaposition implies multiplication only (9). Both use the same order of operations after the implicit notation is interpreted.

It's not ambigous. The rule is precise. The answer is 9.

@@I_am_RazielIt depends on a notation convention, which are completely separate to rules or laws or axioms of maths. Academically, juxtaposition implies grouping and multiplication (1). Literally/programming-wise, juxtaposition implies multiplication only (9). Both use the same order of operations after the implicit notation is interpreted.

Yep and I guess we can keep doing that

well your opinion is wrong lol

​@@lembarkii8669Exactly, the brackets are first and foremost, but we calculate merely what's inside, then once we've done that we look at +-÷x, since that it's ÷ and x priority simply goes left to right, therefore 6÷2x3 is 3x3 which equals 9, that simple

@@BigNasouyi so the guy in the vid is wrong?

⁠​⁠@@BigNasouyiLogic doesn't work with variables. n(x + y) = nx + ny. Dent this and maths broke. Furthermore, conversion of ÷n to ·(1/n) is very valid as well. And there isn't a strict left > right order past primary school, everything could be moved around in a clear notation. The above interpretation is valid.

It's not. You don't multiply before you divide as you need to solve from left to right.

no, the ‘brackets’ means whatever is INSIDE the brackets. the 2(3) is the EXACT same as 2x3 so its multiplication, not brackets. 6÷2x3, you go left to right because division and multiplication are paired in the order of bimdas, so its 3x3, then 9

Bruh it's 9 you're supposed to do the division part before you multiply

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.

I read allat 🔥🔥that was fire

I like how you disproved the clarity of the order of operations immediately after stating it is very clear.

@@bramvanduijn8086 my thermodynamics homework the other day provided a gas constant in units “kPa·m^3/kg·K” which is wrong according to the literal rules outlined by order of operations. It technically should be kPa·m^3/(kg·K). You can never trust division symbols in place of fractions lol

In my country, your method would be considered unacceptable

YOU ARE INCORRECT. First of all you added algebra II and calc which are not very helpful for this problem. Brackets are first so we do (2+1). But here's everyone's mistake. Instead of multiplying (it's that because the number outside of the brackets is multiplied by the corresponding one.) And since it is also considered to be a part of the brackets expression because it's multiplying it by a constant then it becomes 6. Then you divide 6 by 6 and get 1 to be your answer. Try it on a calculator if you don't understand :]

It depends on which interpretation of multiplication by juxtaposition you use. Academically, juxtaposition implies grouping and multiplication (1). Literally, juxtaposition implies multiplication only (9). Both use the same order of operations after the implicit notation is interpreted.

Exactly

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

The issue is BIMDAS doesn't include implicit notation conventions. Those are separate, interpreted first, and are the the cause of the ambiguity.

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.

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

There kind of is. 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.

@@GanonTEK I mean, the rule I stated in the original post should literally be taught as a part of PEMDAS. “Undefined” should be the official answer for an ambiguous equation that can have multiple answers depending on the grammar rule. A student should not be penalized for using one set of grammar rules over another.

@@jakemccoy 100%

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.

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).