6÷2(1+2) = ? | Correct Answer Inside Finally Solved!! | PEMDAS/BIDMAS is Wrong?!

[@michaelkrailo5725]

As a programmer, this is simple. The order of operations are programmed in and you have to know how it is computed but I never leave it up to the conventions as shown. My first thought about this is that the initial question is completely ambiguous due to the missing parentheses around the right side of the division symbol (if that is what was intended). The bottom line is never leave out brackets which leaves you at the mercy of the order of operation rules. It's just a cruel ambiguity otherwise which causes the robot to explode "That does not compute, that does not compute...". I don't have a problem with multiple levels of parenthesis, that's the very thing that takes the ambiguity in our thinking out of the equation. Instead, simply break the equation down into smaller parts using variables. That's what we do in the programming world and it is much easier to read and understand. Do not rely on ambiguous conventions or induce ambiguity. Remove all ambiguity using variables or parenthesis and we will all get the same exact result. Edit: for instance, a=6, b=2, c=a/b, answer=$c*3, and 9 is the answer but this has nothing to do with any rules of operation as it is clearly defined what the intension is by using in smaller sized chunks instead of stuffing it all together and having to rely on conventions. It's almost as if, this was thrown out there just to cause controversy for the sake of stirring the pot on who knows the conventions and who doesn't. To be honest, I don't care. All that really matters is that whatever was indented is accurately interpreted by the person needing to get the correct answer.

[@TheFlax33]

No doubt. My Pascal teacher in college would of threw my printout at me for ambiguous coding .

[@earthstick]

a la LISP, fully parenthesised prefix notation.

[@insoft_uk]

As a programmer • is always implied, one thing I like about TI 83 calculators they use * / and * is implied, I hate my CASIO it’s childish looking symbols the multiplication is x wtf use • it’s a graphical calculator and it uses what I call baby’s symbols

[@KathrynLiz1]

Yep learned to do it that way... used to be an Assembler, Cobol, Fortran and RPG programmer back in the 60s....(Assembler mostly,,, IBM).

[@snagswolf]

It's only ambiguous if you don't know the rules of math.

[@marcosmercado5648]

In other words, if you want an A grade in the class, you must do it the way the teacher explained it, otherwise you are screwed! 🙂

[@utoothheartyeight]

Ah, You got it!

[@highlanderthegreat]

@@utoothheartyeight or you can prove the teacher WRONG...LOLOLOL

[@78tag]

@@highlanderthegreat ....that would get you kicked out of school with an "F" to boot. That is how the leftist school hierarchy works now. Scientific truth is no longer allowed - only "their truth".

[@78tag]

This is always the case when "the teacher" is part of the current education system in the obiden regime.

[@lolguy-x9n]

😂 Do whatever you can to find out what your teacher prefers.

[@timsmith4089]

A month or so ago, I went around and around with people on a similar Facebook post where I argued for the ambiguity of such an expression. In some cases I was accused of seeing ambiguity where none existed because of PEMDAS (or its variations) after all! But the fact that many on the post responded with differing answers demonstrated the ambiguity despite protests to the contrary. I argued, and still do, for clarity over convension as this video discusses. As a retired engineer, clear communication was important throughout my career. Such ambiguity would never be tolerated in the real world where the consequences can be catastrophic. I'm all for convension where its limits are understood and adhered to.

[@RS-fg5mf]

You can argue for better clarity BUT failure to understand and apply the Order of Operations and the various properties and axioms of math correctly as intended doesn't make the expression ambiguous and isn't a valid argument against the expression... On average 70% of adults incorrectly believe that 5+2×10=70 and swear they were taught that way. Does that mean 5+2×10 is ambiguous?? Absolutely NOT... Ignorance of the basic rules and principles of math does not equal ambiguity...

[@timsmith4089]

@@RS-fg5mf I stand by my statement that the limits of conventions need to be understood. Even if, in your example, 25 is obvious to me, and even if only 30% (or less) of adults incorrectly calculate 5+2x10, even may be too much risk in the real world where clear communication regarding equations far more complex trumps the consequences of adhering to mere convention. Unfortunately, people cling to calculators too much these days yet even in the video the guy's calculators yielded differing answers. That was a surprise to me, and in my view, supports the point I'm trying to convey.

[@RS-fg5mf]

@@timsmith4089 I understand your point but your point is irrelevant as a majority of the people who get this wrong or get 5+2×10 wrong do not work in professional fields where Math is a major priority. In fields where Math is a major priority a vinculum would be used in almost every situation... Now understand my point... Failure to understand the basic rules and principles of math does not make the expression ambiguous it means too many people are ignorant (lack the knowledge) of the basic rules and principles and can't evaluate a basic 4th grade arithmetic expression correctly... You suggest giving a person a fish I suggest teaching a person how to fish... Give them a fish they eat for a day. Teach them to fish they eat for life. I teach math and I am an advocate of promoting the Order of Operations and the various properties and axioms of math correctly as intended...

[@snagswolf]

The only ambiguity being created is from those who insist that juxtapositional multiplication takes precedence when you're dealing with non-monomial expressions. Order of Operations was put in place to remove ambiguity. You all are working against those efforts.

[@RS-fg5mf]

@@snagswolf EXACTLY 💯

[@xnjgirl99]

If I have 6 pizzas to be divided between 2 tables, where each table seats 3, how many pizzas would each person get? Certainly not 9.

[@DoseofScienceDoS]

You did math wrong lol. It would be 18 pizzas divided by two tables. How sad 😢

[@xnjgirl99]

@@DoseofScienceDoS - what’s really sad is you truly believe it’s 9. Clearly, the educational system has FAILED you. 🥲

[@Of_UnCommon_Sense]

@@xnjgirl99 Dose doesn't really believe that. He is just trolling.

[@gbhxu]

Exactly!

[@gregorybeekman1081]

I know I'm late to this party but that's a great answer - love it!!

[@scottnassler]

Excel says: "We found a typo in your formula and tried to correct it to: =6/2*(1+2)" It gives the answer as 9. It doesn't like the input.

[@smartmanapps5588]

Yeah, there's a lot of bad programs out there which have been written by programmers who haven't checked the rules of Maths first and they won't allow this perfectly valid way to write an expression. 6/2x(1+2) is indeed equal to 9, but 6/2(1+2) is equal to 1, but some programs won't let you even enter that.

[@avibhagan]

it's in agreement 6/2*(1+2) = 9 6/2(1+2) = 1 it ain't difficult 6÷2(1+2) ≠ 6÷2*(1+2) this is the mistake , that makes people get 9 as the solution. THIS : 6÷2(1+2) = 6÷2*(1+2)

[@berndmayer3984]

in the distribution, the parenthesis must be retained for the time being because the parenthesis has top priority. 2(1+2)=(2+4)

[@MichelSLAGMULDER]

The problem is very simple. The symbol ÷ is illegal. That all. I 've made maths studies in france and during this period I have never used this symbol. No teacher used it. We have other solutions.

[@MuffinsAPlenty]

Eliminating the obelus (÷) wouldn't resolve the ambiguity. Because then people would write 6/2(1+2) and still have a debate over whether 1 or 9 is the correct value. And I know this because lots of people talk about the problem with the solidus (/) too, and all the same confusion/disagreements pop up. The issue is inline notation where implicit multiplication directly follows division. And this problem is unlikely to ever go away, because inline notation will likely always be used by people somewhere, and the differences of opinion on whether implicit multiplication gets special precedence or not are "natural" on both sides of the debate.

[@doughendrie5468]

This is a simple term divided by a term consisting of 2 factors. The 2 and the (1+2).

[@antonharmacinski276]

5:20 - PEDMAS isn't wrong. At least as I was taught it. The issue is that people are REMOVING the parenthesis instead of RESOLVING the multiplication they infer first. The way I was taught PEDMAS was to RESOLVE all parenthesis before moving to the Mult/Div Left to Right portion. The people who get 9 for an answer are skipping this step and removing the parenthesis instead of resolving them. The way I was taught was that if the result of the parenthesis were to be done in the MDL2R phase, a multiplication symbol would be added before parenthesis to notate that it should be done in that order. As this problem is written, ALL PARENTHESIS should be RESOLVED first. This includes the multiplication implied by PARENTHESIS as there is no notation to resolve it with eth MDL2R phase. I've recently learned that its because people are being taught to do this in US High Schools. So I can't fault younger generations for getting the wrong answer when the older generations choose to fail them and skimp out on their education. IDK if it was to lessen the mental burden on the teachers or students but either way it's only going to lead to a disaster over time when we run out of people who know how to do math properly. I'm sorry if it's confusing or complicated, but life is often confusing and complicated. Math is but a reflection of that fact. You really can't take shortcuts in math, and this problem illustrates why. The people responsible for the space program in the 1960's, and professional engineers all get the answer of 1 and people have backed it up and continue to back it up with their lives on a daily basis. Or maybe, just maybe, hear me out now, they are cat people. We humans only get 1 life so we have to be more careful and precise. Cat people get 9 lives so they can afford to take a shortcut and see what happens. :P

[@Of_UnCommon_Sense]

Actually the do not understand that the integer preceding the parenthesis is the common factor for the parenthesis.

[@DoseofScienceDoS]

Lol, clearly nobody taught you to check your work. Let’s try the answer as one!!! 6/2(3)=1 6/2=1/3 3=.33334 Clearly one isn’t the answer. Please don’t be the type of person who argues even when the proof is in front of you

[@antonharmacinski276]

@@DoseofScienceDoS The proof is in front of me every time I dive a car, or my house doesn't collapse on top of me. All of the people who use math in a way that our lives depend on it, also get 1. I'm gonna stick with them. BTW you didn't resolve the actual problem, you changed it. IDK where you got 1/3 from but, 2(3) is processed first because of the order of operations. To get 9 the equation would have to be 6/2x(1+2). The problem is 6/2(1+2). There would be a multiplication symbol added if the result of the parenthesis were to not be treated as it were still in parenthesis. This is the real issue. There are a whole lot of people out there with incomplete instructions and that is why they get 9. I'm sorry our education system let so many people down, but my evidence put satellites into orbit and provides transportation and housing to millions if not billions of people. No one was getting 9 in the 60's, 70's, 80's or 90's because they weren't being taught to skip steps or change the equation. I think we are discovering the real reason our space program died. We are running out of people who were taught math correctly and the government doesn't want to deal with all of the disasters that will happen as a result of using "new math" for anything other than balancing a personal check book.

[@DoseofScienceDoS]

@@antonharmacinski276 lol, you are so wrong. I was showing that one can’t be a solution, clearly you didn’t understand so how about this. Two is a divisor not a multiplier, this makes it (3)/2 not 2(3). (3)/2=1.5 6*1.5=9 I’m really sorry if you don’t understand basic math but I assure you that you are very confused. Yes, we are running out of people who know how to do math… including you.

[@Of_UnCommon_Sense]

I think it is intentional part of dumbing down America. If they can't understand math they can not excel.

[@photon-9551]

I think this is where scientists score of mathematicians. Mathematicians prioritise convention where scientists prioritise principle. Principle equations are used to represent relationships, Principle theses relationships should be expressed in their simplest form Principle the simplest form is achieved by the reduction of operators thus 3+4 is simply represented by 7 where e remove the’+’ operator To help with simplifying equation operators follow a pedmas convention and then a left to right convention Convention must not break simpler conventions thus 2(1+2)= (2+4) by factorisation = 2(3) by principle of simplification Applying the principle of simplification by removal of operators and the convention of pedmas than to remove the brackets implied multiplication should be carried out so 2(3) simply becomes 6 with the removal of ‘()’. Thus your equation 6 divided by 2(1+2) simplifies to 6 divided by 6 = 1. No ambiguity. Just simple principles of arithmetic.

[@donmacqueen]

If "6 divided by 2(1+2)" means that 6 is in the numerator and 2(1+2) is in the denominator, then indeed it simplifies to 1 without ambiguity. But it is neither obvious nor universally agreed upon that "6/2(1+2)" means that 6 is in the numerator and 2(1+2) is in the denominator. Hence, the presenter's conclusion that it is poor notation is correct. We do not have a widely accepted agreement on how to interpret this particular form of notation.

[@doughendrie5468]

6/2(1+2) is the term 6 divided by the term 2(1+2). And yes it means the 6 is the numerator and the 2(1+2) is the denominator.

[@donmacqueen]

@@doughendrie5468 To some people it means that, but certainly not to everyone. As best I can tell, there are highly educated mathematicians, scientists, engineers, etc. on both side. So absolutist claims of which view is right don't fly, at least not with me.

[@doughendrie5468]

⁠@@donmacqueenAs far as I’m aware, 6 is still a term, 6/2 is still 2 terms, 2*3 is factors multiplied, and is a single term. So 6/2*3 is the term 6 divided by the term 2. Show me valid references that back up your statement about terms and factors. Educational maths websites or algebra textbooks would be acceptable.

[@pegasisilver6249]

Having 5/2x coming out as 5x/2 is pure idiocy, who came up with that?

[@klm2558]

Nobody except the interpretation of poorly written expressions

[@wadabid6165]

Well, when we commonly use / we probably see it as the fraction bar --- but tilted over, so normally we would think its 5 / (2x) since we are used to fraction bars. But the rules are the rules, and it is true that it may be misleading for thay exact example of 5/2x, but in an operation like 17/283•1939/848•27 it is helpful to just follow bodmas and forget about everything else, solving each sign (/ or ×) in the order they appear.

[@perjohanaxell9862]

​@@wadabid6165the thing is I didn't know of the left to right rule even tho I teach math. Her in Sweden it's not in the math books because we don't write expressions in a single line. That's simply not how it's done at any level of our skolsystem. If you do you therefore have to use paranteses to make it clear.

[@lynnrathbun]

elementary arithmatic teachers.

[@williamjackson2089]

If 6/2(1+2)=9 if calculating from left to right and 6/(1+2)2=4 calculating from left to right and 6/2(1+2)=1 if using juxtapostion multiplication convention and 6/(1+2)2=1 again using juxtaposition multiplication. Since 2(1+2) and (1+2)2 both =6.Also 6/(2(1+2)) and 6/((1+2)2) both =1

[@harrymatabal8448]

6÷2×3 = 3×3=9. 3÷2=3/2 ×6=9 6×3=18÷2=9. Simple grade 5 maths

[@harrymatabal8448]

Sir I respect your way of thinking. But take note of the following . The question is. 6÷2(1+2) and not 6/2(1+2). Don't be tripped on you own feet by talking about left to right and juxtaposition

[@williamjackson2089]

@harrymatabal8448 What is the difference according to you, is the difference between 6 divided by 2(1+2) and 6 over 2(1+2)?

[@ejrupp9555]

x = (1)x ... you could do that to every number to change 2 ÷ 2 = 1, to 2 ÷ (1)2 = 4, if you do the division before the parenthetic ... therefore, it is obvious that the parenthesis when used without the multiplication sign takes precedent.

[@UncleJunior52]

You solve what needs to be solved first then move to the rules. The first 2 is pre solved so you do the right sided problem first while following standard math rules.

[@j7ndominica051]

We don't even use an inline division sign in high-school algebra. If you want to input the equation into a computer, you usually have to put all the parentheses in. A basic chinese scientific calculator won't even understand the implied multiplication without an operator.

[@avibhagan]

6÷a(1+2) = 9 , solve for a I bet you won't get a=2.

[@smanzoli]

There IS an occult multiplication operator before "(". It could be a dot, an X or an * but it IS there by common used mathematical notation, when it's hidden, the 2 parts ate ONE single part... 2(3) is one thing... a hidden parentheses is there 2(3) = (2*3) That's how mathematicians, physicists and engineers use it in books and papers. According to professionals, PEMDAS is just a simplification, for kids. So: 6/2(3) = 1 {a/b=1} 6/2*3 = 9 {a/b*c=9} It's NOT ambiguous for scholars, mathematicians, physicists, engineers. That's exactly how it's used by 100% of them myself included), 100% of time. Use say an HP10s to calculate this, and get what any mathematician would answer: 1 ANOTHER thing is the notation a calculator or excel demands... and it's simply because some were not programmed to assume that. It must be user friendly for everyone, including kids, so professional/book notation is not allowed. Texas do use PEMDAS. But HP, justaposition multiplication comes first than division.

[@MrGreensweightHist]

"a hidden parentheses is there" No such thing as a hidden () You are adding () where there are none The correct answer is 9 and only 9 "justaposition multiplication comes first than division." There is no juxtaposition multiplication in this problem

[@smanzoli]

⁠​⁠@@MrGreensweightHistyou are free to be as wrong as you want. Have just tried that in my HP10S and the answer is 1, as ANY mathematician would say, as ANY book related to math, physics or engineering uses all the time. Check how ALL formulas and expressions ate written in books and papers.

[@smanzoli]

⁠​⁠@@MrGreensweightHistyou are free to be as wrong as you want. Have just tried that in my HP10S and the answer is 1, as ANY mathematician would say, as ANY book related to math, physics or engineering uses all the time. Check how ALL formulas and expressions ate written in books and papers.

[@MrGreensweightHist]

@@smanzoli Except I am not wrong and your HP10S is. Or to be more precise, GIGO, Garbage In Garbage Out. "ANY mathematician would say," I taught math. Mathematicians do not agree with you.

[@craigward4868]

The equation isn't mathematics. The question is a description of mathematics, i.e. a language. As with any language, if it is used poorly, it can be ambiguous. This equation is ambiguous. Some of that ambiguity can be removed by observing that division is a form of multiplication. Rewrite the statement as 6 x 0.5(1 + 2) process the parentheses first and get 6 x 0.5 x 3 No matter which way you process the operators, the answer is 9. If the author of the statement wanted the divisor to be the product of everything after the division operator, she should have added more parentheses or used other symbols available to arithmetic as a language. The two calculators in the video were given different equations so it is not unexpected that they would give different results. It just illustrates how ambiguity can be written in any language.

[@avibhagan]

That is wrong sir. While you are 100% Correct to say : 6÷2*(1+2) = 6 x 0.5*(3) = 9 Let me show you where you went wrong. If we let y = (1+2) Then 6÷2(1+2) = 6÷2y AND 6÷2y = 6/(2y) while 6÷2*y = 6* (0.5) * y This is how math is done. There is a REAL LAW in math, that governs this (juxtaposition). PEDMAS is not a law or rule. It's a notation for little kids to use. That is all. I hope this explains the problem. In essence 6÷2y ≠ 6÷2*y Which means 6÷2(1+2) ≠ 6÷2*(1+2) So 6÷2(1+2) = 1 ≠ 6÷2*(1+2) = 9

[@kychoi2653]

Implied multiplication or ratio - has implied parentheses around it, so 'ab' means (aXb). If we code it in a program, we should add ( ...) to it. 48÷2(9+3) is 48÷(2(9+3) ).

[@MrGreensweightHist]

No

[@J-kd2qc]

No it doesn't. Implied multiplication is only where the multiplication is implied. Adding parentheses changes the expression. Look at this: 10 - 5 + 2 should be 7 right? Adding parentheses to 5 and 2, 10 - (5 + 2), this now equals 3. Implicit multiplication is so that you can write letters next to each other without having to explicitly have a multiplication symbol. A * B * C * D = ABCD. That's all implicit multiplication means. In this case, 6/2(1+2) is the same as 6/2 * (1+2). All you need to do is multiply what's outside the parentheses with what's inside the parentheses. 6/2(1+2) 6/2 * 1 + 6/2 * 2 3*1 + 3*2 3 + 6 9. Think of it this way, one half of 20 is 1/2(20) = 10, not 1/40th.

[@lawrenceandrews4367]

@@J-kd2qcwrong

[@thierrypauwels]

When I was at secondary school, we were taught that multiplication had a higher precedence than devision, not only when it is juxtaposition, but even when explicitly written with 'x'. Then I started to programme in Fortran, and suddenly multiplication and division had the same precedence. I really had to be very careful when programming a formula like (a+b)(c+d)/(e+f)(g+h).

[@culwin]

You were taught wrong then.

[@thierrypauwels]

@@culwin It is not wrong. It is just a different convention. Conventions are not part of the mathematical concept and are not dogms. There is not such a thing as a right convention or a wrong convention. Same with driving on the left or on the right. It is just a convention. And Britain has chosen a different convention then the US. That does not mean that one of them is wrong. And if you go to another country, you should know which convention they use before hiring a car...

[@culwin]

@@thierrypauwels You're wrong.

[@DoseofScienceDoS]

I agree, you weren’t taught right and you are still wrong. There is only one answer and solving your example equation is ridiculously easy

[@BrianG61UK]

@@thierrypauwels But conventions are supposed to be ideas that most people use. Does your country only use calculators made locally and only program computers in locally designed programming languages?

[@RichM3000]

I'm an engineer. The solution is 1 99% of the time, but the real answer to how to solve the equation is to go back to the underlying formulas to verify what's really going on. I've never seen something so ambiguous in real life but, if someone handed that to me written in that form, I would refuse to sign my name to 1 or 9 without knowing the background. So, I agree that, in the real world, the answer is "undefined", "Insufficient data", etc. Or, if the boss demands an answer: 1 ≤ y ≤ 9 :)

[@thierrypauwels]

Be careful. Maybe your boss is a dogmatic adherent of one of the conventions...

[@timsmith4089]

As a retired engineer, I couldn’t agree more!!!

[@Of_UnCommon_Sense]

I am also an engineer, and there is nothing ambiguous here at all. The integer leading the parenthesis is a common factor of the parenthesis. 2(1+2) = 2(3) = (2x3) = 6 So 6/2(1+2)=1 See my comment above for more clarification.

[@thierrypauwels]

@@Of_UnCommon_Sense It may not be ambiguous to you, because you have a convention about how to interpret such a thing. But when someone else writes it down, you do not know whether he/she adheres to the same convention as you do. And that is why it is ambiguous

[@RichM3000]

@@Of_UnCommon_Sense It's not ambiguous to me either..not in the slightest. It's 1. But, knowing it's ambiguous to others means it's ambiguous by definition, as there's a lack of clarity. Sad, I know. I share your pain.

[@MisterTutor2010]

The only problem is ambiguous notation. This is why I don't use ÷ when writing an equation. The horizontal version of / makes the intended order of operations clearer.

[@herbie_the_hillbillie_goat]

No, it's pretty clear. Follow the established order of operations and you unambiguously get 9.

[@MrGreensweightHist]

÷ and / mean the same thing, and if you write it with a /, the people getting the wrong answer or 1 are still going to tell you the wrong answer of 1. The sign is not the problem

[@syncradar]

I think that ÷ and / should be differentiated. 6÷2(1+2)=9 While 6/2(1+2)=1

[@MrGreensweightHist]

@@syncradar ÷ and / are the same thing. There is no difference and no reason there should be.

[@syncradar]

@@MrGreensweightHist Its for my own convenience.

[@highlanderthegreat]

another way it should be written is 6 divided by (2(1+2 ))= you do what is inside the parentheses first. then you divide

[@joenarbaiz1640]

So, you agree with me that it should be written that way if you want to emulate multiplication by juxtaposition. Your distributive property example has been discredited.. Thank you. 😊

[@highlanderthegreat]

@@joenarbaiz1640 nope not the way i was taught back in late 50s and early 60s..i still say the answer is 1

[@joenarbaiz1640]

@@highlanderthegreat And I was taught that the answer is 9 for that expression. I don't give a damn what YOU were taught. You did exactly what I said that you have to do to emulate the function of a vinculum. 6÷2(1+2)=9. 6÷(2(1+2))=1. It is blatantly obvious that you don't know the difference between an obelus and a vinculum. An obelus and a solidus have the same functionality as opposed to a vinculum. Have a good one. 👍 This conversation is over. Adios! 😆🤣😂😹

[@highlanderthegreat]

@@joenarbaiz1640 in the vids i sent you my juxtapositions of the problem have been affirmed as the correct way...

[@joenarbaiz1640]

@@highlanderthegreat What videos? Why I said you agree with me is because you added another set of parentheses to the expression. 6÷(2(1+2)) is how you represent a vinculum in inline text. You do what is inside each set of parentheses first before you divide and obtain an answer of 1. You just illustrated the proper way to indicate juxtaposition but it doesn't match the original expression. 😕 The original expression is merely multiplication. The obelus (÷) and the solidus (/) are functionally different from a vinculum or horizontal fraction bar. 😆🤣😂😹

[@detroittigersandotherbaseb7220]

The only rules involved here is the commutative and distributive principles. Why are people doing 6/2*3? Multiplication and addition are commutative so the numbers can be swapped. And using distributive principle you have the answer of 1 It should be 6 6 ----- --------- 2(2+1) or 4+2

[@doughendrie5468]

Or it can be 6/(1+2)2=1 Or 2(1+2)/6=1

[@samael338]

So the solution to this dilemma is to stop writing, in lower level maths, ambiguous math equations if you honestly expect an unambiguous solution. It is pure laziness or incompetence to complain about using extra brackets or parentheses. It is like complaining about using vowels in words.

[@GanonTEK]

100%

[@MrGreensweightHist]

9 is the only correct answer. the problem is not ambiguous. People who think it is ambiguous simply don't understand how to read a math problem correctly.

[@samael338]

At least we know how to read English.@@MrGreensweightHist

[@MrGreensweightHist]

@@samael338 I doubt it. but by all means, continue with the insults instead of addressing the facts. 🙄 This problem is NOT ambiguous. 6÷2(1+2) 6÷2(3) 6÷2X3 3(3) 9 Is the ONLY way to read it

[@samael338]

Hey, it will be ok. I know words can be hard sometimes. Stay strong brother!

[@mybluemars]

The takeaway I get from this is: Understand how your calculator (dedicated or called) interprets mathematical expressions and use parenthesis whenever possible.

[@donaldnelsonbarger2978]

Do the job on paper and let the calculator do the multiplications

[@zakelwe]

The problem with calculators is that they don't have two lines to work with a a ------- or ------ * c bc b so don't get the notational clue on which to do first. Hence why ( ) are important to reduce ambiguity going from classical two line notation down to one.

[@rizzwan-42069]

​@@donaldnelsonbarger2978or use memory keys

[@marksolum1794]

Use enough parenthesis (grouping) to remove all ambiguity (like the way this equation was written). (6/2)*(2+1)=9 or 6/(2*(2+1))=1. Work your way from the inside out no need to work from left to right. The obelus, ÷, because it is ambiguous is not used anymore. If working on paper then use a horizontal line, vinculum, for division and things correctly fall into place. For inline equations the vinculum is replaced by the solidus, /, and proper parenthesis. If one has the ideal gas law PV=nRT and wants to solve for T by dividing both sides by nR one has T=(PV)/(nR). adding an explicit multiplication sign T=(P*V)/(n*R)and everything is fine. If one writes using an obelus and uses PEDMAS and left to right convention one has T=P*V÷n*R that will give an incorrect result.

[@MrGreensweightHist]

There is no ambiguity People getting 1 are just wrong

[@marksolum1794]

@@MrGreensweightHist There is the real world (ISO 80000-1,2 and scientific journal notation standards) and this jr. high math that is teaching bad habits.

[@KyleTremblayTitularKtrey]

​@@MrGreensweightHist1 is correct. If you attempted to follow your same logic through other standard equations they break as the OP already showed you. Whether you see it or not you do actually swap between bedmas and juxtaposition having priority.

[@nfpnone8248]

My problem with this viral problem, is why is it viral in the first place, because if you know what constitutes a parenthetical statement, and how to expand parenthetical statements, then the only problem is how to input it into the calculator or spreadsheet of your choice to get the correct answer. The answer is 1 , today, yesterday, and tomorrow! You cannot change an implied operator from multiplication to division, which is what you are doing when you change the problem from 2(2 + 1) to (2 + 1)/2, which violates every mathematical principle there is!

[@CEntertainArt]

The reason many calculators return 9 is not because they calculate (2 +1)/2. The reason they return 9 is because they evaluate the equation in the following way: 6/2(1 + 2) = 6/2(3) = 6/2*3 = 3*3 = 9 Whether you believe it is truly 9 or if it's 1 is yours. I am simply explaining how they solve it. The only way a calculator would see it as a fraction is if there are two parenthesis calculations with a division symbol in between. Otherwise, it is handled as division. I hope you see I'm not trying to argue and simply explaining the computational aspect.

[@DoseofScienceDoS]

Uh oh, someone doesn’t know how to do math. Let’s check your answer of one. 6/2(3)=1 6/2=1/3 3=.333334 Clearly one isn’t the answer. Please don’t be the type of person to argue when the proof is right in front of them

[@H00H]

​@@DoseofScienceDoSwtf are you doing ???

[@DoseofScienceDoS]

@@H00H simple math. What are you doing?

[@DoseofScienceDoS]

@@H00H I figured out where you are doing your math wrong. The two is a divisor not a multiplier. So it would be (1+2)/2 not 2(1+2) Now you can see 3/2=1.5 6*1.5=9 Sorry you don’t get math good job deleting your non sense comment lol.

[@royg123]

Great video. I've always considered these sorts of questions less ambiguous, and more malformed. Division via the quotient symbol is part of one notational system, whereas implied multiplications part of a different notational system. If you are using implied multiplication, you would always use fraction notation for division, and you would never have ambiguities as a result. PEMDAS really applies specifically to the notational system that uses explicit multiplication and quotient symbol for division.

[@perjohanaxell9862]

Yes it's all that one line notation that makes it ambiguous. It's all so much clearer when using two lines so you can write the divisions properly.

[@pompejio]

@@perjohanaxell9862 As a tutor I teach my students right from the beginning to use a fraction line instead of the division sign. At first they don't like it, then they love it. Not only because you can see how to reduce fractions way earlier... It's the same with negative powers, just put them down the escalator (denominator).

[@MrGreensweightHist]

Uh, no. This is a straight forward problem. the only possible answer is 9 You are misapplying the term implicit multiplication

[@zakelwe]

@@perjohanaxell9862 Spot on, techically you should even use the parenthesis on the two line notation but as it is so obvious what is intended there is no need. In the old days of course this would never have even popped up, as all handwritten on two lines, but as we move to one line notation it is causing issues. People are starting throwing MD or DM or left to right at it depending on their fancy, none of which actually apply to two line notation very well ! Instead if on line use parenthesis to indicate which of the two line ones you are representing. Simple. Hence why that is the suggested convention and not MD or DM or left to right .

[@seanclark6438]

The obelus does cause ambiguity, the obelus is indicative of a fraction and a fraction means divide, so BODMAS is easily misinterpreted

[@Vinraymi]

in school, I leaned that multiplications and divisions, as well as additions and substractions don't have a specific order, as long as you do P E (M/D) (A/S) Edit: forgot to mention that you have to do it from left to right in cases with multiple M/D or A/S

[@danquaylesitsspeltpotatoe8307]

LEARN TO MATHS its grade 6 you derps!

[@dorothymacritchie]

if pemdas to my knowledge the anwser is 9

[@tobesgamer3976]

@@dorothymacritchie any version of pemdas like bemdas and bodmas always get 9 because multiplication and division is equal to you do from left to right

[@zakelwe]

There is a left to right rule for addition and subtraction but not one for M and D, this is a common misconception. The reason why none for D and M , even though it has been suggested in the past, is because originally this format of a/b*c was written over 2 lines and the notation indicated what was required to give the two possible answers. Doing left to right always means you get (a/b) * c whereas the person writing it might have wanted you to actually do a/ (b*c). The two line version are obvious without the brackets. For one line you need the brackets to tell you which way to pick, D first or M.

[@natetheg379]

3:30 What he was talking about here, I say that if there is an invisible 1 before variables without a coefficient, then there's invisible parenthesis around the coefficient and its variable or two multiplied variables before divided ones that just causes an exception to the order of operations.

[@Of_UnCommon_Sense]

There is no exception needed. Just do the parenthesis first and remember the integer in front of the parenthesis is the common factor of the parenthesis and is part of the parenthesis. It is not a separate term. 2(1+2) = (2+4) = (6) = 6

[@MrGreensweightHist]

@@Of_UnCommon_Sense Wrong, as usual

[@MrGreensweightHist]

"then there's invisible parenthesis around the coefficient and its variable or two multiplied variables" That is true, but in... 6/2(1+2) There is no variable. There is no coefficient

[@J-kd2qc]

@@Of_UnCommon_Sense You can't just separate the 2 from the 6, you multiply what's outside the parentheses with what's inside the parentheses. 6/2(1+2) 6/2 * 1 + 6/2 * 2 3*1 + 3*2 3 + 6 9. Think of it this way, one half of 20 is 1/2(20) = 10, not 1/40th.

[@Of_UnCommon_Sense]

@@J-kd2qc The 2 is separated from the 6 by a division sign. Numbers do not attach to divisions. They do however attach to parenthesis. Go ask your high school math teacher if you do not believe me.

[@steveross8364]

What's to debate? It's 1. Because you complete the calculations on each side of the inital operator first, then solve the equation.

[@TomTerrific-vm3qg]

Where in your order of operations does it allow you to rewrite an equation? Your third step is invalid. That parenthetical has an integer directly correlated to lt That integer indicates that that parentheses denotes the multiplication operation. That doesn't mean you can rewrite the equation. Follow the order of operations solve the parenthetic and the answer is {1} not ambiguous.

[@Of_UnCommon_Sense]

Exactly

[@donaldnelsonbarger2978]

We also should look back to the description of "÷", the page said {expression1} "divided by" {expression2} so, for this case we don't have {expression2} until the second parentheses.

[@F-14_Jockey]

The American Institute of Physics - "never write 1/3x unless you mean 1/(3x) " [5]. This style is extremely common in many professional contexts, for instance " 1/2π " is a very common factor and it is universally understood to mean " 1/(2π) ". [S. Hargreaves] Hence the answer is '1', using an HP calculator with RPN you will always get one. Also... “Implied multiplication is where we have our next conflict. The American Mathematical Society has previously followed the convention that "multiplication indicated by juxtaposition is carried out before division" [3], and this is still implicit in the examples in its current style guide [4].

[@rizzwan-42069]

​@@F-14_Jockeymy scientific calculator shows me 1.57 for 1/2π and 1.59 for 1/(2π) weird.

[@MartinBeerbom]

@@F-14_Jockey The Author's Guide for Journals published by the APS states clearly that multiplication (ANY multiplication) has higher priority than division, period, for a few years now.

[@nomis4913]

I just tried this using a Casio fx-991MS calculator. If I enter 6/2(1+2)= it gives 1 If I enter 6/2x(1+2)= it gives 9 (where / is actually the division symbol in both cases) Two different answers from the same calculator depending on whether the multiplication is implicit or explicit. Go figure. For what it's worth, if I were presented with this in an exam I would have written down 1.

[@donmacqueen]

There are two ways to interpret the expression 6/2(1+2). When you enter 6/2(1+2) the calculator chooses one of the two ways. (some other calculators choose differently) When you enter 6/2*(1+2) you are telling it which of the two ways you want. There is lots of disagreement over which of the two ways is correct, with adherents on both sides adamantly insisting they are right and the other is wrong. As best I can tell, there is no universal authority to settle the matter. The ISO has two relevant documents, ISO 80000-1 and ISO 80000-2, and they do not settle the matter.

[@abbyz5388]

Yeah when I see a number next to the parenthesis I carry it over I don’t just add a multiplication symbol between them.. it’s so weird my brain hurts 🗿

[@utoothheartyeight]

Does a cow give milk? NO, you have to take it.

[@p.thomas7843]

Great calculator uses BODMAS!!

[@bartsky1945]

and 1 is only correct answer to equation that is written this way

[@Mike80528]

HP RPN is the way to go if using a calculator. Anything else is second-tier. Shortcuts are ALWAYS simplifications. If you think they are the end-all and be-all, you know less than you think. Advanced math would never use the ÷ symbol. When properly written as a fraction the equation is much more explicit and proper.

[@larryhutchens7593]

After much number crunching with my pocket calculator and using a flat earth based geometric system which negates the forces of gravity I came up with 42 as the correct answer.

[@confusious7433]

😜

[@donmacqueen]

Congratulations! You have now been granted membership in the Secret Society of Special Symbols.

[@larryhutchens7593]

@@donmacqueen Oooooh, I like secret things.

[@drziggyabdelmalak1439]

😂Brilliant!

[@blokkadeleider]

🤣 Gold!

[@sander_bouwhuis]

Very useful video! It goes to show I either never encountered improper notation, or I simply didn't realize it was ambiguous.

[@Of_UnCommon_Sense]

There is absolutely no ambiguity see my explanation above.

[@skilz8098]

It's not ambiguous! The only correct answer is 1. What's ambiguous is the lack of ability to properly interpret or evaluate said expression not just based on the order or precedence of said operators, but also based on upholding the basic properties of arithmetic or elementary algebra being, the distributive, the associative, and the commutative properties. The convention is only part of the story and most who try to argue about convention, always seem to forget about the basic properties and axioms of said mathematics.

[@MrGreensweightHist]

@@skilz8098 It is not ambiguous,. but the correct answer is 9

[@Joe_Narbaiz]

@@Of_UnCommon_Sense You seem to think that 2(1+2) is juxtaposition and therefore you modified the original expression to include additional parentheses. 2a is juxtaposition but 2(a) definitely is not. 6÷2(1+2)=9 and 6÷(2(1+2))=1.

[@Of_UnCommon_Sense]

@@Joe_Narbaiz If 2b is juxtapositional and b=(a), then 2b = 2(a) and either is juxtaposed

[@davidparker5345]

With known numbers PEMDAS is followed is how I was taught. At 2:25 the problem now only has 2 numbers. (1) and (2x) the "number" 2x is a single number which is found by the formula 2 times x. Working with known and unkown is not done the same.

[@MrGreensweightHist]

"2x is a single number" True, But 2(x) is not. And this problem is a 2(x) format, not a 2x format.

[@doughendrie5468]

When replacing variables with constants, the following shows how to do it. In finding the numerical value of any literal expression the student should observe the following RULE, First, put in place of each letter its numerical value; second, simplify the result thus obtained. In any but the simplest expressions the student should always observe the two steps of the above rule separately in the order in which they are stated. To mix the two in an attempt to perform mentally both processes at once, is sure to result in many errors and consequent loss of time. EXERCISES In Exercises 1-16, let a = 3, b = 1, c = 5, d = 7, and f = 2. Substitute for each letter its numerical value, and then simplify the results according to the rule of page 13 : 1. 4a²-7b Solution. 4a²-7b = 4*3²-7*1 = 36-7 = 29 Hawkes, page 15, First Course in Algebra

[@MrGreensweightHist]

@@doughendrie5468 yeah still doesn't support your claims no matter how many times you cite it, because there are no variables in the original expression Let a = 3 2a = (2*3) because you replace and simplify. 2(a) = 2*3 because the a is not attached and there is nothing to simplify. Is there a reason you keep citing off topic aspects from that book instead of addressing the actual topic? 4a²-7b = 4*3²-7*1 = (4*3²)-(7*1) Why didn't the book put parentheses? Because 4*3²-7*1

[@hurktang]

No! Multiplication by juxtaposition have priority! That's it. You know it, you saw it clearly in your research. People who use PEMDAS in grade 2 are not told about juxtaposition so they will not write that formula. Clearly PEMDAS cannot solve juxtapositions right. So it's okay if you say "error I don't know", but I'll tell you. This answer is 1.

[@daviestj]

with your 2 calculator example, one calc was processing an extra pair of parentheses, forcing the wrong answer. Some calcs are inserting the extra parentheses when you hit the equals button.

[@colleenpatino3156]

The problem is, they could both be correct, 9 or 1 could be the correct answer. The question wasn't written properly. It should have been written with two sets of parentheses. It needed to be written like this... 6÷(2×(1+2)) or (6÷2)×(1+2) Without using both of those parentheses, you're unable to see what the second movement needs to be. With only one of those in parentheses, after getting the answer of 3 from the first set of parentheses, you're unable to see if the 6÷2, then multiplying by 3 is the correct way to solve, or if the 2 needs to be multiplied by the 3 first, then 6÷6. Without both of those parentheses, either answer could be correct. If someone doesn't truly understand it, they get 9. If they understand that the 2 needed to be multiplied by the 3 first, they would get the correct answer of 1.

[@doughendrie5468]

The ONLY way to get 9, is by (6/2)(1+2). The parentheses round the 6/2 forces the multiplication to take place. 6/2 without parentheses cannot multiply the (1+2) as factors must be whole numbers. They cannot be fractions. 2(1+2) is a distribute. The common factor 2 must be distributed back into the parentheses. The expression is the factorisation of 6, and must be simplified back to 6. The term 6 is separated from the factor 2 by explicit division. By the rules of algebra, what you do to one side of an equation you must do to the other side. Solve for X. (6/2)(2+1) cannot be anything other than 9 so the 6/2(1+2) must have a different result as it’s a different expression 6/2(1+2)=X one explicit division, so multiply both sides by 2(1+2) to give 6=X*2(1+2) simplify both sides to give 6=X*6 divide both sides by 6 to give 1=X proved

[@doughendrie5468]

As a follow up using distribution 6/2(1+2) distribute the 2 6/(2+4) simplify 6/6 parentheses removed In accordance with the distributive law. PARENTHESES () Any parts of an equation that are written inside a set of parentheses are done first from the inside out. MAtHNASIUM 6/2(1+2) inside done first 6/2(3) from the inside out 6/6 parentheses removed. If you use the rules of algebra correctly the correct answer is the result. No requirement for extra parentheses.

[@colleenpatino3156]

@@doughendrie5468 I know, the problem is, they don't know juxtaposition yet until they are probably in high school. Without that, they believe it goes back to left, then right. They don't understand that you still need to go with the parentheses first, making it 6, then dividing 6 by 6, getting the correct answer of 1. It needs to be "Please excuse *just* my dear Aunt Sally". PEJMDAS.

[@rockymarciano6750]

If you set this problem, you could use brackets to prevent ambiguity.

[@MrGreensweightHist]

The fact a second set of brackets is not used means it is not ambiguous. The answer is 9

[@zakelwe]

@@MrGreensweightHist Or 1 depending if you do the multiplication first. With format a/ b * c there is ambiguity without brackets on whether to do the division or multiplication first, there being no convention actually for DM or MD or left to right .

[@MrGreensweightHist]

@@zakelwe " there being no convention actually for DM or MD or left to right ." Yes, there is. Multiplication and Division share priority. When operations share priority they are done left to right. That is the convention Basic orders of operations. 6th grade math

[@DrDaveW]

This is a great analysis - yes you have to be explicit when writing equations, but also use the minimum amount of notation that removes ambiguity. However, I would say that this is not ambiguous. Ther problem arises when 2(3) is expanded to 2 x 3. It shouldn't be. It should take precedence and be calculated as 6. By expanding it, you are removing the precedence that the parenthesis imply. Don't convert a non-ambiguous expression into an ambiguous one.

[@MrGreensweightHist]

"Ther problem arises when 2(3) is expanded to 2 x 3. It shouldn't be" Yes, it should. " It should take precedence and be calculated as 6." There is no rule in math stating any such thing. " By expanding it, you are removing the precedence that the parenthesis imply." Parentheses only give precedence to what is INSIDE the parentheses. Never to what is outside them

[@J-kd2qc]

You only need to multiply what's in the parentheses with what's outside the parentheses. 6/2(1+2) 6/2 * 1 + 6/2 * 2 3*1 + 3*2 3 + 6 9. Think of it this way, one half of 20 is 1/2(20) = 10, not 1/40th.

[@KyleTremblayTitularKtrey]

​@@MrGreensweightHist2(3) should not be expanded to 2x3 if they wanted to expand it to 2x3 he would have written 2x3. Its literally less symbols. 2(3) is 4 symbols, 2×3 is 3 symbols. Why are we using an extra symbol? Because we are actually writing (2x3) with LESS symbols. 4 instead of 5. You dont make a shorthand thats longer than the original notation, you'd just use the original.

[@MrGreensweightHist]

@@KyleTremblayTitularKtrey "2(3) should not be expanded to 2x3" All existing rules of mathematics disagree with you

[@edmondgautier8301]

As the expression is ill formed ( non compliant with ISO 80000-2), there is no answer possible.

[@GanonTEK]

Non-compliant with ISO-80000-1 also.

[@wizrom3046]

The divide symbol used in the example is freely interchangable with the / (divide) symbol. So the question is 6 / 2(1+2) And the answer is, and has always been, 1

[@dookoonu2741]

I'm no Math genius but I disagree ...My understanding was that the parenthesis should be cleared first. well it seems mathematics is not an exact science

[@fabmystery6433]

Yea man I do agree as math is a variable which is utilised to find out the laws and behaviour of the universe.. actually we have created maths, it does not exist, just as we use variable to find out unknown value..and yea math is inaccurate 😊

[@MrMousley]

I would do it like this .. 6 / 2(1 + 2) = 6 / 2(3) = 6 / 6 = 1 I'd do the 2(3) before the divide because it's 2(3) and not 2 x 3 If it was 2 x 3 I'd do the divide first .. working left to right

[@syncradar]

6/2•(1+2) 6/2•3 6/6 1 6÷2•(1+2) 3•(1+2) 3•3 9 I used ÷ and / differently.

[@J-kd2qc]

All you need to do is multiply what's outside the parentheses with what's inside the parentheses. 6/2(1+2) 6/2 * 1 + 6/2 * 2 3*1 + 3*2 3 + 6 9. Think of it this way, one half of 20 is 1/2(20) = 10, not 1/40th.

[@tigershark94]

@@J-kd2qcnope. One half of 20 would be expressed as 1 --(20) 2 Or, better yet, as 20 -- 2 What you wrote was 1 1 --- = --- 2(20) 40

[@J-kd2qc]

@@tigershark94 HAHAHAHA, wow, you're so bad at math dude. Go back to second grade and learn the order of operations.

[@mktwatcher]

Only left me more confused. Thank you

[@Steve-re9md]

I had a lot of problems with maths at school. Basically I did not know my tables and being asked a question relating to them frightened me to death! I often found myself thinking 'why am I learning this rubbish' In any event I decided on a career at sea and found out that I needed virtually everything I had studiously rejected! Somehow I bodged my way through all the exams and ended up as Captain of a High Speed Passenger Catamaran running from Gibraltar to North Africa. The moral of my story? well I had to turbo charge my self confidence when I went for a job, I became very good at first impressions and gaining the upper hand in interviews but oh how much easier if I had been told why I needed all those maths things in the first place.

[@dannygjk]

Sounds like you cheated your way thru.

[@pulsar22]

Even a street vendor need math. Even a farmer in some 3rd world country needs math. It really is the onus on teachers to make practical use of math apparent right from the start so that students will be more motivated.

[@skilz8098]

@@dannygjk I wouldn't say cheated, but I could joke at the point of saying: faking his way through with a big charming smile lol...

[@Of_UnCommon_Sense]

If you had gone to my school this would not have been a problem. When we learned our times tables, until you could recite your times tables to 12 x 12 you could not go out for recess. A good incentive to learn.

[@charlesdarwin5185]

You should have done koomon😂

[@johnyeates2156]

As a long retired engineer that's done a lot of maths, I can't recall using this ÷ sign to denote division. In my era, we always used a horizontal line with the numerator above the line and the denominator below the line.

[@chutalotr]

Obviously from the UK where we study Maths rather than Math.

[@nicholasjh1]

But the slash gives precedence to the numerator and denominator so it is incorrect to use in a line unless you intend a fraction

[@helenamcginty4920]

Well im 75 and English. im sure we used a division sign before we started algebra in senior school. But can only visualise doing long divisions which we wrote differently. Nb in Spain where I now live the 1st set of accounts I read had the total at the top. And kids at least write long division backwards compared with how I was taught. Plus full stops are used where we use commas and the decimal point is a comma. Eg. 550.234,00 not 550,234.00. What is the system in other countries?

[@josephpoley4805]

The division sign is just a placeholder for correct notation which would be 6 / 2(1+2) = 1. 6 numerator; 2(1+2) denominator. Left to right is BS. Also the brackets/ parentheses aren’t cleared until the multiplication is completed and =6.

[@MrGreensweightHist]

@@josephpoley4805 that is incorrect.

[@jannetteberends8730]

Can’t we just say that writing a problem like that is a writing error. As soon as there is some ambiguity, it’s not math anymore. The problem was written down incorrectly.

[@GanonTEK]

100%. 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.

[@jannetteberends8730]

@@GanonTEK exactly

[@is7728]

Problems of Dividing by a(b) always causes arguments 😮‍💨

[@Billicyhot]

This is some calculators will return an invalid expression error until the implied multiplication is explicitly written and replaced with the "boatload of brackets"

[@jmlongdg]

@00:35 the equation is different on the two calculators. The right calculator show 6/(2(1+2)) when written out like that even by order of operations the answer is 1. Why are people trying to use rules of geometry, and trig to disprove basic order of operations?

[@GanonTEK]

Many scientific calculators put extra brackets after you press equals. They have not been manually typed. They are to show you how the calculator is interpreting the implict notation. A very good feature. The same expression was typed on both. It depends on the scientific calculator but here are some that give one or the other: These give 1: Casio FX 83GTX, Casio FX 85GT Plus, Casio 991ES Plus, Casio 991MS, Casio FX 570MS, Casio 9860GII, Sharp EL-546X, Sharp EL-520X, TI 82, TI 85 These give 9: Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES, Casio 570ES, TI 86, TI 83 Plus, TI 84 Plus, TI 30X, TI 89. 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 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.

[@doughendrie5468]

The Casio fx-ES Plus calculator range also shows the calculation. As follows- • If vou execute a calculation that includes both division and multiplication operations in which a multiplication sign has been omitted, parentheses will be inserted automatically as shown in the examples below. - When a multiplication sign is omitted immediately before an open parenthesis or after a closed parenthesis. Example: 6 ÷ 2(1 + 2) - 6 ÷ (2(1 + 2))

[@turtlerockfire3664]

To help sole the problem, write it as a fraction. 6/ 2(1+2). In math, you can write this as a fraction. The answer is 1.

[@joanseymour4382]

@@turtlerockfire3664 Indeed. And, to stop all the arguments and make it simple, it SHOULD be written in fraction form. The answer will still be 1, but without all the conflict!

[@highlanderthegreat]

you do what is inside the () parentheses first and since the 2 is outside of the parentheses you do that next so that is 2 x 3 =6 then you divide so its 6 divided by 6 the answer is 1...at least from the way i was taught back in the 50s

[@joenarbaiz1640]

You don't understand that implied multiplication and explicit multiplication are still multiplication and fall under the rules of multiplication with respect to the Order of Operations. 😆🤣😂😹

[@highlanderthegreat]

@@joenarbaiz1640 i will go so far as to say this good sir,,, both answers can be correct and both answers can be wrong, all depends on how and when you learned, i learned in the late 50s early 60s, my answer is 1, with the so called calculator the answer is both 1 and 9....so in my old fashioned way the answer is 1 in the modern day after the year 2000 i guess the answer is 9 so can we just flip a coin,,, heads i win tails you lose...lololol aint that fair??????

[@joenarbaiz1640]

@@highlanderthegreat It is interesting that CASIO calculator models can provide different answers to the same expression. Actually, the historical interpretation ended in 1917. Again; you are ignoring the multiplicative inverse property and the proper interpretation of the distributive property. Thanks for admitting that you live in the past. Your comment about coin flipping is at the least ludicrous. 😆🤣😂😹

[@highlanderthegreat]

@@joenarbaiz1640 awwww joe, i thought the coin flip was sorta funny,,, no way i could lose that way...lololol yes i am using shall i say, somewhat old method,,, and no i dont live in the past, although the way the problem is written does lead to the 2 different answers from the way i was taught so you could also say you can not solve the problem the way it is written, thats why i say coin flip to solve it...

[@YallDotBiz]

The problem is poorly stated. What I see is 6 ------- 2×3

[@一本のうんち]

no. ÷ sign is not the same as a fraction or / sign. ÷ sign is mostly used in simple linear arithmetic taught in early prealgebra and follows regular PEMDAS order. unlike a fraction it does not separate the equation in clear 2 sides.

[@davetooes6179]

I'm 75 this is the first time my head exploded!! My maths learnt in the UK and Australia was BODMAS similar to the two you show. I ALWAYS thought people who got those math problems you see being posted WRONG were just not properly schooled in the way of doing those problems. Similarly seeing two electronic calculators getting differing answers just made it worse.

[@Patsoor]

@davetooes6179 His 2 calculators are showing a different formula with an extra ( so no wonder it shows a different result.

[@davetooes6179]

@@Patsoor mate I missed that. bastard played with the data to get the results he wanted to show. Interestingly neither of those two formulas is acceptable in Excel without additional operands. I see previous replies esp. from engineers have the answer as 1. I'm the wrong one out my answer would have been 9 due to the maths I had been taught. But I never went passed year 10 Maths so there is that.

[@MsDani899]

My Math in Canada utilized BEMDAS for order of operation: brackets, exponents, multiplication, division, addition, subtraction. It has been years ago since learning this and have never forgotten it. lol

[@davetooes6179]

@@MsDani899 and using that your answer was 1 ??? My BODMAS gave it as 9

[@cliffordschaffer5289]

@@davetooes6179 As a computer programmer I would fire anyone who didn't put parentheses in to make it clear. Nobody should be guessing what someone meant. Cheaper and easier to put a parentheses in than to deal with mistaken results later one.

[@BAMozzy69]

Its intentionally ambiguous and isn't written the 'best' way to communicate the intended Answer. To me, the way its laid out, it could be written as 6 / 2(1+2) more like a Fraction - therefore the Answer is 1 If the person setting this problem intended us to get the answer 9, It should have been written (6÷2)(2+1) - but that's not how this is written so its the fault of the person setting this question and not communicating clearly.

[@TeddyOG]

Thats how I thought it was, but twitter is convinced it's the other way. I'm so confused now

[@DavidLee-qe3rd]

How calculators respond to implied multiplication varies depending where they are being sold and is based on canvassing the opinions of local maths teachers and examination boards (not professional mathematicians in general) for marketing purposes. The order of operations is always stated in the calculator manual - but of course almost nobody ever reads these! Likewise in publishers' style guides for printed mathematics in books and journals. Interestingly international Sharp calculators have always correctly and consistently interpreted implied multiplication and return different values with and without the "×" operator. eg my Sharp EL-531returns: 6 ÷ 2 × (1+ 2) = 9 but: 6 ÷ 2(1+ 2) = 1 Also interesting to note that, whilst log x² is ambiguous, the ambiguity is eliminated by convention in trigonometric functions. eg sin θ² = sin (θ²) whereas sin² θ = (sin (θ))²

[@brybb3775]

Best to just follow the ISO 80000-2 standard for mathematical notation and do away with the ÷ sign altogether. I don't recall seeing that symbol being used beyond lower elementary arithmetic anyway. In cases where teachers choose to use it, they should clearly define what it means... whether or not ÷ is equal to / or if ÷ means that the upper dot represents the value to the left and the lower dot the values to its right. Still, I prefer they do away with it from the start as it may cause confusion for their students later on.

[@cryofpaine]

The problem isn't the ÷. You would have the same problem if you used / as well. In fact, it would be more ambiguous because it wouldn't be clear whether you meant 6 - (1+2) 2 or 6 ----------- 2(1+2) But with the ÷, it's clear that the six is divided by the rest of the term, not that there is a fraction coefficient on the term in the parenthesis. The only ambiguity is people not understanding that multiplication by juxtaposition is not the same as multiplication, and comes higher in order of operations.

[@MrGreensweightHist]

@@cryofpaine "But with the ÷, it's clear that the six is divided by the rest of the term," That is wrong.

[@MrGreensweightHist]

÷ and / mean the exact same thing. Either way, you get... 6÷2(1+2) 6÷2(3) 3(3) 9

[@brybb3775]

​@@MrGreensweightHist, when no distinction is made between X over Y (X/Y in this case as I could not express the formula s intended in this thread) and X ÷ Y, one can get confused over the order of operations. In higher level maths (e.g. Physics), I don't recall seeing the ÷ symbol being used... which I think is due to this ambiguity (hence the ISO standard referenced). I am inclined to think that symbol was used due to the limitations in printing technology but nowadays, anyone can print out the actual unambiguous expressions nowadays making this symbol unnecessary for anyone past the lower elementary grades.

[@MrGreensweightHist]

@@brybb3775 "one can get confused over the order of operations" Only if one does not understand the orders of operations. "I don't recall seeing the ÷ symbol being used." Understandable as it is not commonly used, but it means the exact same thing as / so once you know that, it should not matter. "which I think is due to this ambiguity " There is no ambiguity. To think the problem is written ambiguous, due to use of a seldom used symbol, even though the symbol has clear meaning, is akin to claiming a book is ambiguous because it uses a word you haven't learned yet. It is not a fault in the writing of the book. It is merely a gap in your knowledge. A gap that, once filled, makes the book crystal clear. "making this symbol unnecessary for anyone past the lower elementary grades." Until it pops up somewhere you didn't expect, like an internet brain teaser, and you find yourself lacking the knowledge to solve it. Not knowing something is never a fault. Not learning when the opportunity arises is. Whether it is written 6÷2(1+2) or 6/2(1+2) or 6 --(1+2) 2 It remains the same problem

[@grumpyolddude439]

ab/xy is different than a*b divided by x * y. ab written as such, is a singular value. Like 13. 13x7, despite 13 = 10+3 and 7 = 5+2; does not work when written 10+3 x 5 + 2. Likewise ab while when isolated IS the same as a*b; when in an expression, may NOT be as simple as a*b.

[@thierrypauwels]

We could swith to RPN (reversed Polish notation). No more ambiguities ! ;-) 6 2 1 2 + x : or 6 2 : 1 2 + x

[@mengjohnny7228]

2(3) =(2x3), not =2x3. unless there is no other expression or multiplication or divide etc.... I have seen lot of videos lately, people just simply explain 2(3) = 2x3. when you do math step by step, you should 6÷2(1+2) =6÷2(3)=6÷(2x3)=6÷(6)=6÷6=1, you must understand how brackets work. 2(3) is a group, 6÷2(3) =(6÷2)÷3 NOT equal to 6÷2x3. I believe people just misunderstand 6÷2(3) =6÷2x3. Casio fx-991MS gives a correct answer 1. （x² +xy)=x(x+y), means x taken from the left side of the equation to outside the bracket. 6÷a(a+b) NOT = 6÷a x (a+b), THIS IS ABSOLUTELY NOT CORRECT.

[@dgkcpa1]

Bravo! Glad to see someone who knows real math.

[@eonshade6297]

2(3) is just a different way of writing 2*3 What is 4a÷4a in your opinion? Is it: 4a÷4a=1 Or 4a÷4a=4a*(1/4)*a=(4a^2)/4=a^2 Multiplication and division have the same priority, so you go left to right. 6÷2(1+2) 6*(1/2)*(1+2) (6*(1+2))/2 (6*(3))/2 18/2 9 You cannot go wrong by using lots of brackets (where they can be used). 6÷(2(1+2)) 6÷(2(3)) 6÷(6) 1 If the problem were written like 6÷(2(1+2)) Then the result would have been 1, but it isn't. Brackets really tell everything. Or even like this with distribution 6÷(2(1+2)) 6÷(2*1+2*2) 6÷(2+4) 6÷(6) 1

[@RealMesaMike]

Yup, It is argued (convincingly, I'd say) that multiplication by juxtaposition constitutes a grouping which is part of the P (or B) in the mnemonic.

[@isaacogunmuko3947]

6÷2(1+2)=9 PEMDAS: Parentheses Exponents Multiplication Division Addition Subtraction BIDMAS: Brackets Indicates Division Multiplication Addition Subtractions Note: Brackets/Parentheses first means to find the value of what's inside the parentheses. (1+2)=3 is solving parentheses first. 2(3) means to multiply 2 *3 = 6, notes solving the parentheses first. In both Multiplication & Division are done left to right, Addition & Subtractions are done left to right P: (1+2)=3 B: (2+1)=3 E: None I: None M&D: 6÷2(3)=>6÷2*3=3*3=9 D&M: 6÷2(3)=>6÷2*3=3*3=9 A&S: None A&S: None

[@plektosgaming]

@@isaacogunmuko3947 You DO know that 2(1+2) is (2+4), right? This is literally basic algebra. As in 2(1x + 2y) - except x and y are both 1 in this case. You must resolve the parenthetical terms/group first. Before you even get to anything basic. This may appear to be "simple" math, but it has algebra in it and the order of operations changes as a result. ÷ and / are not identical. One implies that a fraction must be made and the other is showing an existing fraction. PEMDAS is too simple to catch this error as it assumes the symbols are interchangeable. EG: C = 5/9(F-32) is NOT C = 5÷9(F-32) The first is read as: 5/9ths of (F-32) The second is read as: 5 divided by 9(F-32) And you'll never see it written with a ÷ symbol in any book for that reason. You can back this "extended order of operations" up in any number of physics and calculus formulas as well.

[@dbtest117]

Great we need more of these accurate videos on this issue. Thanks!

[@MrGreensweightHist]

Except this video is inaccurate

[@dbtest117]

@@MrGreensweightHist Nope it's accurate. The only other acceptable answer would be that it's ambiguous. And that is only because people have been taught PEDMAS and similar stupid conventions that goes against previous conventions that made sense.

[@degrass5408]

​@@dbtest117nope it's not accurate because he wrote in the calculator 6÷2(1+2) vs 6÷(2(1+2)) which is not equal to each other because the second one has (2(1+2)). This is done first because it's in parentheses. Thus, according to PEMDAS, we get 6÷2(3)=6÷2×3=9 vs 6÷(2(3))=6÷6=1. PEMDAS may be an arbitrary convention that is accepted but his example doesn't demonstrate the ambiguity of PEMDAS

[@dbtest117]

@@degrass5408 Did you watch the video even? You are making a fool out of yourself.

[@enomiellanidrac9137]

If it's written manually: learn how to write unambiguous formula is the answer. If it's an input into a calculator or math program: RTFM is the answer.

[@virtualDon]

Your calculators are correct: The one on the left had 6/2(1+2)=9 and the one on the right had 6/(2(1+2))=0. The extra brackets on the right calculator do in fact make it equal to 1. Also, your analogy of 1/2x is not the same as 1/2*x

[@teknul89]

That’s because the one that give 1 was an outdated way to calculate it from 1917 but it’s outdated Historically the symbol ÷ was used to mean you should divide by the entire product on the right of the symbol. But the correct answer is 9 since we do order of operation aka pemdas Since there is a hidden multiplication before parenthesis It’s easier to understand it if you say (6/2) * (1+2) = 9

[@GanonTEK]

Depends on the scientific calculator but here are some that give one or the other if you type it exactly as in the video: These give 1: Casio FX 83GTX, Casio FX 85GT Plus, Casio 991ES Plus, Casio 991MS, Casio FX 570MS, Casio 9860GII, Sharp EL-546X, Sharp EL-520X, TI 82, TI 85 These give 9: Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES, Casio 570ES, TI 86, TI 83 Plus, TI 84 Plus, TI 30X, TI 89. Online calculators don't agree either. Microsoft Math gives both answers on screen. The notation is ambiguous. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. Half of calculators use the academic way (Sharps, most Casios and some TIs) and half use the more programming/literal way (Some Casios and most TIs). Both equally valid. The expression itself is the problem and is not valid.

[@romerypb13]

@@GanonTEK nice speech

[@etzjunior_]

So explain why 0:26 , The calculator on the far right and the one in the middle still say 1? No extra brackets

[@tonitalas1757]

The one on the right gives 1, not 0. There is no correct simplified answer to this expressio!

[@fcey337]

Thank you so much for your excellent explanation.

[@Dethneko]

I consider the distribution property of multiplication, which states: 2(1 + 3) = ((2*1) + (2*2)) = ((2(1)) + (2(2))) = (2(1) + 2(2)) Therefore 6 / 2(1 + 3) = 6 / (2(1) + 2(2)) and the operations should be clear from there: = 6 / (2 + 4) = 6 / 6 = 1 If you argue that the distribution should be (6/2(1) + 6/2(2)) then there's already a clear problem, because to distribute the entirety of "6/2" then the problem should be written as: (6 / 2)(1 + 3) = ((6 / 2)(1) + (6 / 2)(2)) = 3(1) + 3(2) = 3 + 6 = 9 My 8th grade math teacher taught me the cumulative property of addition. A college professor taught me that subtraction is adding the opposite. Therefore: 2 - 1 = 2 + (-1) = (-1) + 2

[@DoseofScienceDoS]

Well you forgot that your seventh grade teacher taught you to check your work because you got a big fat F.

[@davidbrattain1446]

This is right and what I have been arguing in comments on these posts. This is a problem of syntax that plagues mathematics as much as it does the English language where the use of homonyms and synonyms as well as various punctuations can produce sentences with very different meanings. We must provide all at our disposal to clarify problems and not leave it to the student to make assumptions. This is the kind of thing that creates unnecessary noise in a subject that the majority of people already find challenging. Good work! ☺

[@DoseofScienceDoS]

What?? This is an easy equation to answer. There is no confusion, the answer is nine

[@AliKhan-jt6zj]

Let me tell you why 1 is the correct answer for this. Consider it this way: 6÷(2+4). Is it equal to 6? Now factor 2 out of the bracket. 6÷2(1+2)=6 As a matter of fact multiplication with '×' is not the same as direct multiplication with parentheses.

[@indrasubagja7870]

Now change 6÷(2+4) as a fraction, what u'll get? 6 as numerator and 2+4 as denominator, right? In fraction form u omit the bracket. Then factor the 2+4, u got 2(1+2) in the denominator. Now, change it again to the inline division form, u'll get 6÷(2(1+2)). If we omit the bracket during inline to fraction, we should do the reverse during fraction to inline which is add the bracket. And that's different from the original question. Understand?

[@gbhxu]

Exactly. You cannot divide 6 cakes between 2 tables of 3 people and get 9

[@grumpyolddude439]

Let me tell you why you are wrong.....Order of Operations defines the process. After () and exponents; go left to right and execute in order, as encountered.

[@AliKhan-jt6zj]

@@grumpyolddude439 what will be the denominator if you express this as a fraction? That “Let me tell you why you are wrong” was too confident though.

[@gbhxu]

@@grumpyolddude439 You have to get rid of the brackets before you can do anything. Hence 6÷2(1+2) goes to 6÷6=1

[@lutzfilor8253]

This is why you should never use a calculator if you don’t know what you are doing in mathematics. This problem only exists in America not following internal conventions.

[@TheYaddayadda]

"Who would want to write out a boat-load of brackets like that every time?" Programmers: >.>

[@Mesa_Mike]

LISP programmers!

[@J-kd2qc]

Set a = 6/2; set b = 1+2; set c = a*b; Done, no brackets at all. c = 9.

[@Robert_McGarry_Poems]

As written in the thumbnail... That reads: 6 over [2*(1+2)] If the multiplication of the 2 and the sum of parenthesis, is implied because of standard notation rules, then there is automatically an implied set of brackets around the whole piece, they don't need to be written in. The division sign is a sign the the equation will be in two parts. The numerator and denominator. Since there can only be two parts in a division, brackets and parenthesis are automatically implied, again you don't have to write them, they exist by definition. Unless you otherwise explain notation changes, or expressly bracket everything... (6 over 2) * (1+2), is not what is written because that is not how notation conventions work.

[@ivankrushensky]

Absolutely not true..... explain K=1/2m(v^2) with your statement

[@Robert_McGarry_Poems]

@@ivankrushensky half mass multiply parenthesis. What is there to explain...? You still need context to understand it. Meaning, they are implied parenthesis. But you know best...

[@ivankrushensky]

@Bob456969 according to what you're saying (and the math problem) it means: K=1/(2mv^2)......which it does not.

[@Robert_McGarry_Poems]

@@ivankrushensky no. I didn't, and no I am not. You obviously are just being obtuse... Look in the dictionary for the word obtuse, before you come at me again. Implied doesn't mean, I'm telling you were they need to go. But as written the implication is in the actual convention.

[@Robert_McGarry_Poems]

@@ivankrushensky 1/2 without parenthesis, is still (.5). the location and context of the notation matters for the implication. You are just assuming that I meant every division sign. But I know you are smart enough to figure it out. Possibly... Maybe, actually. I have no idea if you can figure it out.

[@tomtke7351]

when one writes an equation that can only be solved by PMEDASS rather than to employ correct parentheses then I conclude that an unnecessary math element is being relied upon. Restated: the equation was INTENTIONALLY written to be confusing without PMED ASS.

[@doughendrie5468]

Casio fx-570EX explains exactly how to carry out this problem. Input the original expression and the calculator automatically inserts an extra parentheses to emphasise how the expression does not include the 6, so the 2(1+2) is fully simplified to 6, then the 6 is divided by 6. Answer is 1. “If you execute a calculation that includes both division and multiplication operations in which a multiplication sign has been omitted, parentheses will be inserted automatically as shown in the examples below. - When a multiplication sign is omitted immediately before an open parenthesis or after a closed parenthesis. Example: 6 ÷ 2(1 + 2) -> 6 ÷ (2(1 + 2)) So 6/2(1+2) 6/2(3) 6/6 1

[@detroittigersandotherbaseb7220]

The only rules involved here is the commutative and distributive principles. Why are people doing 6/2*3? Multiplication and addition are commutative so the numbers can be swapped. And using distributive principle you have the answer of 1 It should be 6 6 ----- --------- 2(2+1) or 4+2

[@degrass5408]

Not true because (2(1+2)) is done first because this entire operation is in parentheses, which leads to 6÷6 while in 2(1+2), the 1+2 is done first because it's in parentheses. This leads to 6÷2×3, which leads to doing the operation from left to right. Thus you get 6÷2×3=9 and 6÷(2(1+2))=6÷(6)=1. It's like people (and the uploader) doesn't even realize this. I'm saying this because that's how he wrote it in the calculator, that's why both calculators gave different results.

[@steveklemetti8035]

@@degrass5408 I know that 2(1+2) is done first. That's what I said. 2(1+2) or 2(2+1) is the same and that results in (4+2) or (2+4) and that resolves to 6. No, 2(1+2) does not resolve to 2*3. Because it is one unit, one expression. The 2(1+2) must be completed as one. It is not to be split to 2*3. That is where everyone is going wrong here. They think it is 2 expressions when it is not. We don't care about what calculators say. Calculators are programmed by people. We use math rules. The rules are Associative, Commutative and distributive. Both the Commutative and distribute rules are used here. 2(1+2) can be written as 2(2+1). That is commutative. And distributive says the 2 in front of the ( gets distributed to the operands inside to make it (2*1+ 2*2). That gets resolved to (2+4). The problem is that people are relying a a children's rule of PEDMAS, which is not even right. It is more properly PEMDAS. When people should be relying on adult rules.

[@doughendrie5468]

2(1+2) is a single term 2(3) is a single term 2*3 is a single term. 6/2 is two terms 6/2(3) is two terms 6/2(1+2) is two terms

[@steveklemetti8035]

@@doughendrie5468 Well, that's what I'm saying. When the two terms are computed, we end up with 1.

[@nats50]

In short, the equation must be written in the way the final answer is intended. If the intended answer is really 9, it should be written like this: (6/2) (1 + 2) = 9 or (6 ÷ 2) (1 + 2) = 9 And if the intended answer is really 1, it should be written like this: 6 / [2(1 + 2)] = 1 6 / [2×3] = 1 6/6 = 1 Making use of parenthesis and bracket remove the ambiguity?

[@GanonTEK]

100%

[@plektosgaming]

It's actually written correctly. Lol. That division symbol is literal shorthand for "insert fraction line here" or "this is a fraction". As such, you read it as 6 (fraction line) (all the rest below it). Not remotely ambiguous. Yet somehow people.. well.. they make things difficult. lol.

[@sqratlas4097]

6:2 (1 + 2 ) = 6/2 * (1 + 2 ) = 3 * ( 3 ) = 9

[@grumpysanta6318]

Calculators are more and more using PEJMDAS, where multiplication by juxtaposition takes precedence over other multiplication. That's where the answer of 1 comes in. ... Which is exactly what he covers. Sigh... :)

[@plektosgaming]

Which is exactly how my father and I were taught it - before pocket calculators were common. The division sign is always shorthand for a horizontal line, or was for hundreds of years. 300 years ago the answer would have been 1. Or 100 year ago. In fact, it's literally a fraction on its side - with the x and y just reduced to dots. As such, the "new" methods are simply wrong. There is no ambiguity except in that people are being taught it wrong.

[@KathrynLiz1]

I was taught (back in the 1950s) to resolve what is in the brackets first then using the usual priorities go left to right.. which yields 9. Nested bracket can remove all ambiguities... having been a programmer, nested brackets pose no issues for me, so when writing such expressions I use them to remove all possibility of misunderstandings.

[@werffjvander]

1

[@arnoldca8377]

resolve what is in the brackets first. -- Yes you keep working until the brackets are out of the equation, at 2(3) the brackets still exists therefore must go first....which yields 1

[@plektosgaming]

The singular thing that people who teach this forget is that the sign is literal shorthand (as per the creator's intentions in 1659) for a horizontal division line. Which was created due to how typesetting worked at the time. This saved several lines on each page and being two dots and a horizontal line, caught on quickly as it was very clear what it meant. All they need to do is remember this fact and it is clear - none of the idiocy above. 6 ----- 2(3) Which there is only one answer to no matter how you try to solve it. But computers and lazy teachers...

[@peterpan408]

The question is just malformed.. Rewrite it into an explicit form (either version), and it becomes easy.

[@DoseofScienceDoS]

So you get the answer nine right?

[@colleenpatino3156]

​@@DoseofScienceDoSThe problem is, they could both be correct, 9 or 1 could be the correct answer. The question wasn't written properly. It should have been written with two sets of parentheses. It needed to be written like this... 6÷(2×(1+2)) or (6÷2)×(1+2) Without using both of those parentheses, you're unable to see what the second movement needs to be. With only one of those in parentheses, after getting the answer of 3 from the first set of parentheses, you're unable to see if the 6÷2, then multiplying by 3 is the correct way to solve, or if the 2 needs to be multiplied by the 3 first, then 6÷6. Without both of those parentheses, either answer could be correct.

[@cakepigs]

I never knew about implied multiplication, I always treated implied multiplication as bracket multiplication making it B in BIDMAS therefore making it higher priority.

[@alext8828]

Yes, exactly. I see the number just outside the parenthesis as part of the parenthesis. You have to be a maniac to disregard that number. Or just a robot.

[@joenarbaiz1640]

@@alext8828 The only process in the brackets step is evaluation of the contents WITHIN those brackets. The remainder of that bracketed expression is evaluated in the multiplication step. If you want to interpret 2(1+2) as a bracketed expression, include another set of brackets.

[@alext8828]

@@joenarbaiz1640 No, that's not true. Real math is not done that way. Complicated equations are not processed that way. Casio calculators were all changed to reflect that error. That was silly math to teach 3rd graders.

[@joenarbaiz1640]

@@alext8828 You do know that CASIO calculators can give either answer dependent on the model, right? 😆🤣😂😹

[@Of_UnCommon_Sense]

@@joenarbaiz1640 As 2(1+2) = ( 2+4) = (6) = 6 Solve parenthesis first.

[@edvsilas8281]

Finally ,somone recognizes that there is ambiguity in the answer . Finally ,someone recognizes that conventions change but the mathematical answer can not change simply because a particular convention ( or fashion) says so. Finally,common sense has won the argument .

[@drziggyabdelmalak1439]

It hasn't though, really, has it? Look at my comment above.

[@Of_UnCommon_Sense]

Mathematics describe the real world. If we used this formula in 1916 6/2(1+2)= #gunpowder and we needed 1 lb of powder in the cannon. Now you say we changed convention so it says we need 9lbs now. Will you touch off the cannon with 9lbs now? After all we changed math. NOT Some professor wrote a book. But nobody noticed.

[@edvsilas8281]

@@Of_UnCommon_Sense The quantity of powder charge to a cannon being changed is simply a recipe change that will change the magnitude of the blast ;however, changing the order of mathematical operation is akin to not only changing the quantity of powder but changing the powder composition itself .

[@xenuno]

No ambiguity at all. The fact there is no sign between the 2 and the parenthesis indicates a special case where it is multiplied immediately on the value calculated within the parenthesis. It's that simple ..

[@edvsilas8281]

​@@xenunoAmbiguity if you use pemdas or bidmas convention since juxtaposition prioritizing does not apply but in algebraic math it does . I would completely ignore any series of operations having an obelis and parentheses in same series .

[@jpopaldana]

It's quite simple. Multiplication with a parenthesis has a higher priority over multiplication with a multiplication sign. The answer is 1.

[@Neptunianist]

It’s a great video and absolutely understands the ambiguity in the cases where a mathematical question is written in this way. However I think there is an approach, an extension of the one taken in the video (which inspired this idea, actually). The approach will lead to only one answer: 1. My thinking on the solution to this is that we need more information about the question. What do the values represent? What real-world problem or situation is the question trying to portray? Are we, say, dividing six pieces between 2 teams where each team originally had 2 people and then one more joined each team? Are we saying that we have six pieces, or six of anything actually, which are divided equally between 2. But then multiplied by the sum of the numbers in the brackets; 3. For this we have to see that the things can be scaled up, increased, and therefore they can’t be physical things unless we’re talking in abstraction. This could be units of heat or force, instead. It could be physical but abstracted, yes, like eggs. We could say we need so many for a recipe so start with 6, divide by 2 and then multiply by the number in the brackets, which might be people, to give the number of eggs required. We cannot multiply something that we have already broken and divided into pieces. Where is the extra material, the extra pieces, coming from to allow this multiplication, this expansion? So the last one has to be abstract. If we had not divided the six first, it would be fine because we would not be combining two different types; the pieces and the people in the team. The same applies to any object, where division is used. It’s always one type of thing being divided into quantities, a different paradigm, quality and thing entirely.

[@bepohal]

in the real world, outside the US, the answer is always 9. always...

[@Neptunianist]

@@bepohal Yes, because we use BODMAS. I get that rule, completely. But I do think that it’s a rule which shouldn’t be blindly followed. In my view we should try and understand what the values in any equation are and what the equation is trying to represent.

[@bepohal]

@@Neptunianist generally spoken this equation will only lead to 9. pemdas is what the whole world uses. especially the stem field

[@MrGreensweightHist]

@@bepohal This isn't a U.S. thing. The answer is 9 everywhere. And there are people getting it wrong, everywhere as well.

[@jmm1233]

the 2(1+2) is iteration of a function of n(x) so it should be wrapped up in brackets like a parcel as in 6/{2(1+2)} , which works well in python and c++ interpretation

[@MrGreensweightHist]

"the 2(1+2) is iteration of a function of n(x)" no it isn't "so it should be wrapped up in brackets like a parcel as in 6/{2(1+2)} " Except you just changed the problem because you assumed 2(+1+2) was grouped when it clearly isn't. If they had meant 6/{2(1+2)}, they would have typed 6/{2(1+2)}, and not 6/2(1+2) Work the problem as it is, not as you want it to be

[@KyleTremblayTitularKtrey]

​@@MrGreensweightHist Except people write 2x and know its (2x) If I write 4/2x it means 4/(2x). If I write 4^xy every single math textbook interprets 4^(xy). Juxtaposition combines without needing brackets.

[@typicalhguy9290]

But when you did it on your own two calculators (at 0:40), you didn't enter it the same way. You entered 6/2(1+2) in one case, and 6/(2(1+2)) in the other case. Since multiplication/division associates left to right, in the first case the 6/2 is executed first, giving 6/2 = 3, 3(1+3) = 9. In the second case, with the extra layer of parentheses, the (2(1+2)) is executed first, giving 2(3) = 6, 6/6 = 1. What do you get when you actually enter it the same in both, without forcing the precedence with the extra parentheses? Since both calculators are the same brand, I would expect they're both implementing the same rules and will give the same result.

[@GanonTEK]

No, the same expression was typed on both. Some calculators after you press = put brackets in to clarify how it interprets implicit multiplication. It's a great feature. Depends on the scientific calculator but here are some that give one or the other if you type exactly 6÷2(1+2) into all of them: These give 1: Casio FX 83GTX, Casio FX 85GT Plus, Casio 991ES Plus, Casio 991MS, Casio FX 570MS, Casio 9860GII, Sharp EL-546X, Sharp EL-520X, TI 82, TI 85 These give 9: Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES, Casio 570ES, TI 86, TI 83 Plus, TI 84 Plus, TI 30X, TI 89.

[@lucientjinasjoe1578]

Let's materialize it, I have 6 divisions of armed vehicles and divided them to two which contains blue and yellow and sub divided one with black armband and two with white armband and the outcome is 9 on paper and 1 on the ground , funny isn't it 😊

[@LeeDukes-f6s]

Thanks for posting this need for clarity of presentation in mathematics. It also reminded me that some of the calculators that give different answers when the data was entered with the division sign can give the same answer of 1 when the data is entered as a fraction where the 2(1+2) is completely in the denominator.

[@detroittigersandotherbaseb7220]

The only rules involved here is the commutative and distributive principles. Why are people doing 6/2*3? Multiplication and addition are commutative so the numbers can be swapped. And using distributive principle you have the answer of 1 It should be 6 6 ----- --------- 2(2+1) or 4+2

[@emresahin94]

question is not 6/(2(2+1) so you cannot convert the question to a fractional number.@@detroittigersandotherbaseb7220

[@MrGreensweightHist]

"2(1+2) is completely in the denominator." And that is a fault of the calculator because if you change to division to a fraction, the denominator is the 2 alone, not the (1+2)

[@detroittigersandotherbaseb7220]

@@MrGreensweightHist One does not use calculators in this. All division is a fraction. In division there is the dividend / divisor = quotient. The dividend is the numerator and the divisor is the denominator.

[@degrass5408]

The input into the second calculator is completely different than the first calculator since he put in 6+(2(1+2)), which means that 2(1+2) is done first because it's in parentheses. While the first calculator doesn't have parentheses around 2(1+2), thus 6+2(1+2) doesn't equal 6+(2(1+2)).

[@kripalsinghsaini1821]

How can we write the statement : Six is divided by twice the sum of one and two mathematicaly.

[@runfayalife]

You know that video of the world class violinist playing in a NYC subway and getting zero attention? That's this comment.

[@TheMathManProfundities]

6/2(1+2) would work fine for established mathematicians and engineers but it we wish to include children and others who still rely on operation order acronyms we would have to use 6/{2×(1+2)}

[@soilomasbello1156]

That's why I use RPN calculators so I'm in control of operations order if the order is incorrect is because of me not inadvertently the calculator.

[@RichM3000]

Same. Even on my iPhone I use an RPN calculator app.

[@Eewec]

Sad that North America removed juxtaposition from the order of operations in their educational system. It's in their early text books teaching PEMDAS but it seems that somewhere along the way, lazy students who never went passed the first page that listed the rules, never went through the examples that show things like (a/c)*(b/d)=ab/cd. Then those same students became the teachers and never taught what they never bothered learning. Even the calculator companies put it back in after it was pointed out that only North America removed juxtaposition. Now it seems that NA is trying to foist their dumbed down ruleset on the rest of the world. I believe Casio had a period where they removed implied multiplication, then to my understanding they re-introduced it.

[@doughendrie5468]

Casio fx-570EX calculator. • If you execute a calculation that includes both division and multiplication operations in which a multiplication sign has been omitted, parentheses will be inserted automatically as shown in the examples below. - When a multiplication sign is omitted immediately before an open parenthesis or after a closed parenthesis. Example: 6 ÷ 2(1 + 2) - 6 ÷ (2(1 + 2)) Calculation Priority Sequence 7 Multiplication where the multiplication sign is omitted 8 Permutation (nPr), combination (Cr), complex number polar coordinate symbol (2) 9 Dot product (•) 10 Multiplication (x), division (+)

[@rebeccaabraham8652]

We were taught BODMAS back in the 60’s and 70’s and implicit multiplication was quite acceptable then. These ‘modern ideas’ are wonderful things - but we started unlocking the universe with the old methods and principles - and anything that conflicts with what we know works can’t be acceptable. Anyway - we didn’t rely on calculators to parse equations - that’s just sheer laziness and puts your results at the mercy of a programmer who may/probably doesn’t know what they’re doing!

[@jeremyhorne5252]

In logic, we call this a "well-formed formula problem". This is an interesting problem you present here. I am not sure if there are any parallels in logic, where precedence of operators always is critical.

[@alext8828]

Why is the 2, just outside the parentheses not included in the parentheses operation. That would clear up the entire issue.

[@joenarbaiz1640]

@@alext8828 Parentheses only include the contents WITHIN the parentheses. 2(1+2) is implicit multiplication. 2*(1+2) is explicit multiplication (2(1+2)) is juxtaposition.

[@doughendrie5468]

2(1+2) is a parenthetical expression. The factor 2 is connected to the parentheses by juxtaposition. The expression is the factorisation of 6, and to simplify the 2(1+2) back to 6, the factor to must be distributed back into the parentheses, either by the Distributive Law or Order of Operations. 2(1+2) distribute the 2 (2*1+2*2) simplify (2+4) simplify 6 parentheses removed. Or by O of Ops 2(1+2) inside parentheses first 2(3) inside to outside next 6 parentheses removed The term 6 that is to be divided has no connection to the factor 2, due to 6/2 is a fraction, and factors must be whole numbers. “p22 Elements of Algebra Chapter IV - Of the Nature of whole Numbers, or Integers, with respect to their Factors p38. If, therefore, we consider all whole numbers as products of two or more numbers multiplied together, “ NOTE - Whole Numbers

[@degrass5408]

​@@joenarbaiz1640this is true if there is no other × or ÷, but the way it's written in the calculator it's clear that 6÷2(1+2) doesn't equal 6÷(2(1+2)) because (2(1+2)) is done first since it's all in parentheses: 6÷(2(1+2))=6÷(2(3))=6÷(6)=1. While the quantity 2(1+2) is not in parentheses, thus we get 6÷2(3)=6÷3×3=9. He may have a point, but his examples (or at least this example) doesn't say anything beyond that PEMDAS is an arbitrary convention that is accepted.

[@MrGreensweightHist]

@@joenarbaiz1640 "2(1+2) is implicit multiplication. 2*(1+2) is explicit multiplication" Wrong. 2(1+2) is explicit multiplication. 2*(1+2) is explicit multiplication 2x is implicit multiplication. Implicit multiplication ONLY applies to a coefficient's relationship to its attached variable. It does not apply to parentheses, which are, in their own right, and explicit sign

[@youngone1985]

PEMDAS is NOT math. It's a memory aid. If its programmed in calculators, that's so the calculator gets the same answer each time. Even if the problem is so poorly written, there's no answer.

[@daddoo5268]

At 0:42 when showing the two Casio calculators side by side to demonstrate how they get "different" answers, the equations are significantly different in that the right side calculator has a second set of parenthesis.

[@GanonTEK]

No, the same expression was typed on both. Some calculators after you press = put brackets in to clarify how it interprets implicit multiplication. It's a great feature. Depends on the scientific calculator but here are some that give one or the other if you type exactly 6÷2(1+2) into all of them: These give 1: Casio FX 83GTX, Casio FX 85GT Plus, Casio 991ES Plus, Casio 991MS, Casio FX 570MS, Casio 9860GII, Sharp EL-546X, Sharp EL-520X, TI 82, TI 85 These give 9: Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES, Casio 570ES, TI 86, TI 83 Plus, TI 84 Plus, TI 30X, TI 89.

[@powdercowboy90]

So this is the way I see it....wouldn't you see the problem like this.....the 6 is bieng divided by the 2(1+2). So 2(1+2) is its own separate expression that must be simplified first? Am i Incorrect? To simplify you use the distributive property to get 6.

[@joncrichton7670]

my 9 year old thought this same thing but I guess not at higher level math

[@srikrishna2561]

Yes. That's the correct one indeed.

[@eonshade6297]

Multiplication and division have the same priority, so you go left to right. 6÷2(1+2) 6*(1/2)*(1+2) (6*(1+2))/2 (6*(3))/2 18/2 9 You cannot go wrong by using lots of brackets (where they can be used). 6÷(2(1+2)) 6÷(2(3)) 6÷(6) 1 If the problem were written like 6÷(2(1+2)) Then the result would have been 1, but it isn't. Brackets really tell everything. Or even like this with distribution 6÷(2(1+2)) 6÷(2*1+2*2) 6÷(2+4) 6÷(6) 1

[@muffaloaf]

@@eonshade6297 the problem doesn’t have to be written like that to equal 1. Until the 2 is multiplied with what’s inside the parentheses , then the parentheses remains. And getting rid of the parentheses is the 1st step

[@natenatenate10]

@@muffaloaf Once an expression inside of a set of parenthesis has been resolved, the parenthesis are removed. Afterwards, the expression is solved from left to right. Parenthesis don't remain and emit residual effects on the order of operations.

[@yasdnilknarf1885]

Applause. Clear and precise explanation.

[@cliffordschaffer5289]

Author should be fired for not having parentheses to make it clear to everyone reading it.

[@doughendrie5468]

@cliffordschaffer5289 The term 6 divided by the term 2(1+2) answer is 1

[@cliffordschaffer5289]

@@doughendrie5468 Different calculators and computer systems will give different answers. The person who wrote that mess should be sent back to include parentheses so it works the same in any system.

[@TootlinGeoff]

Excellent video that explains clearly why the formulation of the problem is ambiguous. A vital counter to those dogmatic "mathematicians" that insist the answer is 9.

[@justuseodysee7348]

The answer is 9. There's nothing wrong with PEMDAS - it's perfectly logical and rational. It's the mathematicians who don't adhere to the rules due to lazyness.

[@DoseofScienceDoS]

Ummm… I hate to break it to you but the answer is nine.

[@TootlinGeoff]

@@DoseofScienceDoS 👇👇

[@TootlinGeoff]

@@justuseodysee7348 👇👇

[@TootlinGeoff]

@@DoseofScienceDoS 👇👇

[@1paultv22]

Someone pointed out that if a teacher used this problem on a test, they would have to accept both answers because of the different ways math is taught To elementary/primary school children before they get into higher maths like algebra and higher where the ÷ sign is no longer used.

[@totoitekelcha7628]

Why higher match don't used ÷ sign?

[@quantumtacos]

​@@totoitekelcha7628 On youtube there's a video by Syed Institute that explains why that symbol is avoided. TLDW: because it requires more brackets to achieve the requisite precision and it doesn't read as well anyway.

[@Lord_Volkner]

@@totoitekelcha7628 I minored in math in college and never once saw that symbol throughout all four years. I never thought about it at the time, but I think the OP is correct.

[@drziggyabdelmalak1439]

Not in the UK's GCSE level papers. They'd expect '9' as the answer. Anything else and they'd give you zero marks.

[@juliavixen176]

@@totoitekelcha7628 In Germany ÷ is used for subtraction.

[@len2524]

In higher level math you will always have units associated with the numbers and the units will dictate the order of operations. The archaic division symbol is never used in Engineering.

[@MrGreensweightHist]

That is a lie

[@RPSchonherr]

There is a rule for these expressions, it's distributive process. When you have 2(3) you can't just get rid of the parentheses and go left to right. The parentheses means there is still an operation to be done and that is to multiply what is in the parentheses by the adjacent number. With the original expression of 2(1+2) the better way to process it is ((2*1)+(2*2)) = (2+4) = 6 . BTW * means multiply for any who don't understand. Doing it correctly you get 6/6=1. My character map doesn't even have that division sign, so when I write it, it always comes out 6/2(1+2), a fraction.

[@DoseofScienceDoS]

The correct answer is never one, you have made a very simple math mistake and I can show you where you went wrong if you would like

[@RPSchonherr]

@@DoseofScienceDoS If you get anything other than 1 you are wrong and ignoring rules of math. Go away.

[@DoseofScienceDoS]

@@RPSchonherr nope, you have made a simple mistake. I would have taught you but you are rude so enjoy being arrogant and ignorant!! You 100% don’t know what you are talking about and seem 😜

[@Of_UnCommon_Sense]

@@DoseofScienceDoS yes show me where this educated person is wrong. Learn about removing a common factor and the distributive property of multiplication.

[@detroittigersandotherbaseb7220]

The only rules involved here is the commutative and distributive principles. Why are people doing 6/2*3? Multiplication and addition are commutative so the numbers can be swapped. And using distributive principle you have the answer of 1 It should be 6 6 ----- --------- 2(2+1) or 4+2

[@DustinSilva]

Maybe I need to watch it again, I dont quite understand why implied multiplication by juxtaposition isnt a thing, ie, if its used, the answer is one...Isnt it supposed to be used? Is the reason its ambiguous is because calculators differ on the answer? Seems like poorly coded calculators are the issue then. When I attended college, there were a list of acceptable calculators for my Algebra/2, Trig, Pre-Calc/Calculus classes. I HAD to have a calculator that was on the list, and the one I still have 17 years later still says this is 1.

[@juliavixen176]

Computers are dumb, and to use as little memory and cpu time as possible, they only look at the two tokens immediately to the left and right of any operator symbol that they find. Ultimately, what's going on is that the composition of multiple binary operators forms a binary tree, and the algebraic infix notation that we're using because of historical accident is really bad at representing a tree structure as a linear sequence of discrete symbols read from left to right. Basically a/bc means something to a human depending on which context it appears, but machines don't know the context. It could mean (a/b)*c if they evaluate in the most lazy way possible, or it could mean a/(b*c) which is usually what the human meant. Math notation was created by humans for humans to read, not electronic devices with only two general purpose registers.

[@plektosgaming]

Your response explains it, actually. :) [ Algebra/2 ]. This is a convention used by early typesetters and typewriters to use this [ / ] symbol for multiple things to save space. The proper division sign is with a horizontal line, and the meaning is crystal clear when looked as what it is, which is shorthand for a fraction. Fractions written properly are more complex to write but absolutely clear on the order of operations. Shorthand for the same thing changes nothing. It's what it was originally used for as well, over 350 years ago. There is no "debate" - just students who were unfortunately taught incorrectly.