It has millions of views because the problem initially looks too simple to have a video (and it is an excellent video). I wondered if I missed something and chose to watch. So, the order of operations rules were revised and both 9 and 1 are correct answers. I thought it was 1 (the algebraic grouping of terms as you noted). Great to know the rules changed. Thanks for making the video.
Great video, still slightly confused because I am taught that x(y) is one term and should be treated as 1 number but glad to learn that there are 2 different systems
MaxMisterC Both of them state that multiplication and division have the same importance, and are some left to right. Put it in a calculator if you disagree.
@@MaxMisterC Heck it wasn't even that for me. I always just ASSUMED (don't remember if it was actually in the education I got) that anything next to a bracket was in itself inside "invisible" brackets. So if you had 2(1+2), it would simply be read as (2(1+2)) = 6 regardless of what was put in front of it. I guess I never really bothered searching up if this was wrong, OR that my teacher might have been in the same group that still insists on this sort of thinking. Either way the new method (answer of 9) is correct and there just isn't much you can do about it. That's the rule and that's how maths works I guess.
as an engineer who has done advanced university level maths for about 7 years now, I would get 1. its the convention usually followed in physics/engineering textbooks to solve as terms and let implicit multiplication (brackets esp) go first
I'm in a similar position and I 100% agree. It's disingenuous by the video to imply there's one correct order, when so many physics and engineering books do operations in the 2nd way. The video is also wrong in stating that calculators all calculate in the same way. Mine doesn't. I guarantee that if any engineers I know saw something like "6÷2x" they'd calculate the 2x first. It has nothingo to do with the division symbol. Implied multiplication (for example, 2x rather than 2*x) in all the engineering I've learned always takes priority over normal multiplication. If you write it as 2(1+2) instead of as 2*(1+2) there has to be a reason for it, and common sense (mine at least) dictates it's because you mean the order of operations to be different. Real world math isn't a puzzle designed by someone to fool you, it's an objective way to state things and should be written accordingly. The problem here is the question, not the answer. Just write it as (6÷2)(1+2) or as a fraction and the problem is solved.
@@afsdfsadhasfh Conclusively, the experts say this. The equation is ambiguous and indeed, it can yield two different answers. Like the use of language, to convey something such that it can't be misinterpreted, it must be delivered with clarity, the intention should be made clear. The same with maths equations. To yield only one result, the equation should not be written with ambiguity and the intention of the writer must be clear. If it does or can, the equation should be re-written.
What is the correct answer? I don't know man, math isn't typically fully divorced from reality, let's look at the reasons why you're crunching these numbers and we can re-write it so it makes sense!
No, the first premise in PEMDAS, is to solve for the answer within parentheses. You never distribute into parentheses first because you would then misapply the order of operations. PEMDAS: Parenthesis, Exponents, Multiplication And Division, Addition And Subtraction (IF the same precedence, then left to right). Any order with And in between has the same precedence! Since the problem is 6/2*3 or 6/2(3), we must follow the premise regarding left to right because the problem involves only multiplication and division, orders of the same precedence. Parenthesis is only a symbol of multiplication when a number or expression is adjacent to it. If the problem were 6/(2*3), then the logical answer is 1, because we solve for the answer within parentheses first, as according to the first order of the order of operations. The answer to 6/2(2+1) is not 1.
We shouldn't change things like the order of operations, it's incredibly dangerous in things like engineering to have two different people unknowingly using two different standards.
order of operations never changed, it's always been the same. He just explained that that specific symbol for division meant something very specific other than just division over 100 years ago but the actual order of operations has never changed.
That's why for any serious communication of mathematics you have to be more explicit than this ambiguous problem. Hence why peer-reviewed papers use fractional notation and make copious use of parenthesis to remove ambiguity.
I am 45 years old and have honours degrees in Engineering and Science. We were always taught that the answer should be 1, because of the order of operations rule that we were taught to use. If you change the rule, you change the answer. I was not aware that the rules had changed!
It seems that multiplication by juxtaposition, ab or a(b) etc., may impliy grouping, or it may not, so the notation is ambiguous making both answers valid. It depends on context (e.g. academic or programming). It's just bad writing. Modern international standards, ISO-80000-1, mention that brackets are required to remove ambiguity if you use division on one line with multiplication or division directly after it. The American Mathematical Society's official spokesperson literally says "the way it's written, it's ambiguous" even though they use the explicit interpretation. Wolfram Alpha's Solidus article mentions this ambiguity also. Microsoft Math gives both answers. Many calculators, even from the same manufacturer, don't agree on how to interpret multiplication by juxtaposition. No consensus. Other references are: Entry 242 in Florian Cajori's book "A History of Mathematical Notation (1928)" (page 274) "The American Mathematical Monthly, Vol 24, No. 2 pp 93-95" mentions there was multiplication by juxtaposition ambiguity even in 1917 (and not the ÷ issue) "Common Core Math For Parents For Dummies" p109-110 addresses this problem, states it is ambiguous. "Twenty Years Before the Blackboard" (1998) p115 footnote says "note that implied multiplication is done before division". "Research on technology and teaching and learning of Mathematics: Volume 2: Cases and Perspectives" (2008) p335 mentions about implicit and explicit multiplication and the different interpretations they cause. Other credible sources are: - The PEMDAS Paradox (a paper by a PhD student on this ambiguity) - The Failure of PEMDAS (the writer has a PhD in maths) - Harvard Math Ambiguity (Cajori's book above is talked about here) - Berkeley Arithmetic Operations Ambiguity - PopularMechanics Viral Ambiguity (AMS's statement is here) - Slate Maths Ambiguity - Education Week Maths Ambiguity - The Math Doctors - Implicit Multiplication - YSU Viral Question (Highly decorated maths professor says it's ambiguous) - hmmdaily viral maths (Another maths professor says it's ambiguous) The volume of evidence highly suggests it's ambiguous.
@@bigbadlara5304 The answer is one because this video makes a mistake by ignoring that these equations require the distributive property. If you "just graduated" I'm not at all surprised that no one taught this...
@@nixboox Distribution can give both answers as it is a notational ambiguity. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. I.e. does 2(1+2) = (2×(1+2)) or 2×(1+2)? Implicit: 6÷(2×(1+2)) = 6÷(2+4) = 1 which is used by academic writing. Explicit: 6÷2×(1+2) = (6÷2×1 + 6÷2×2) = (3 + 6) = 9 which is used by modern programming and also by the American Mathematical Society according to their statement on the matter. That's why it's ambiguous. The rules can't help when the problem is the notation which has to be interpreted first. It's just written poorly and not in line with modern international standards. It should be (6/2)(1+2) for 9 or 6/(2(1+2)) for 1. Those are unambiguous and follow the guidelines.
Edit: I was wrong, operator precedence makes the answer clearly 9. A way to avoid this confusion from people like me who got lost in the order of operations would be to set up the equation as (6/2)(1+2) or (6/2) * (1+2). Note: Contrary to popular belief in this thread, I did graduate with my bachelors and also complete Basic Calculus with high marks. I am capable of error and my original comment was one of those errors. Thank you for the correction. Original comment: I graduated with my Bachelors in 2019, the answer according to the way I was taught throughout my education is 1. Because I was instructed by my professors to visualize this problem as 6/(2(1+2)) or 6/6 which equals 1. The person who wrote this did so in a way that is designed, purposefully or ignorantly so, to cause confusion. Dr. Trefor Bazett has an insightful video on this topic
Are you saying that you took university level math within the past 10 years and your professors taught you that in the case of 6➗2(1+2) you’d make 6 the numerator with the 2(1+2) being the denominator? Ima have to throw the bs flag on that one. It doesn’t even make sense that your professors would have even been instructing you on this when this is just basic math that young kids learn. It’d be like saying “When I was pursuing my master’s degree and my professor was teaching me my times tables…” If you took this stuff recently, you’d have been taught to solve left to right 6/2x3 =3x3 =9
Dr. Trevor Bassett is wrong and so are you... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ---------- = 6÷(2(1+2))= 1 2(1+2) The vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator. ________ 2(1+2) = (2(1+2)) Two grouping symbols each ________ 2(1+2) has two grouping symbols (2(1+2)) has two grouping symbols
@@trickortrump3292the bigger question would be why a University would be using the grade school obelus to teach higher level math... We have reviewed the video and the penalty flag stands... Good call Ref....LOL
@@RS-fg5mf Yeah I deserved that. When I first looked at it, I solved it your way and then the video told me I was wrong. I bought into the reasoning for why I was wrong. This question is just a mess! I went down the rabbit hole yesterday after my comment. It’s insane to me that so many experts seem to say that the right answer is “there is no right answer” because it can be correctly solved two different ways, yielding two different answers. I can’t accept that. If both answers are correct, that makes both answers wrong too. I’ve removed the bs flag I originally threw. 👍😉
@@trickortrump3292 don't remove it. LOL The red flag stands on the play because you are absolutely correct... The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended is 9 I was agreeing with you. Don't let these mathematical numpties change your mind. Those who understand and apply the basic rules and principles of math correctly as intended will get the correct answer 9 Those who fail to understand and apply the basic rules and principles of math correctly as intended will get the wrong answer 1 Those who can't prove 1 and can't accept 9 will argue ambiguity... Failure to understand and apply the basic rules and principles of math correctly as intended doesn't make the expression ambiguous and isn't a valid argument against the expression...
I prefer using ÷ over /. I only use / with fractions, but use ÷ when dividing numbers. Using improper fractions instead of using the division symbol is something that I rarely ever do. I never find it confusing when using ÷, and it never confuses me.
@@MarkQub. What do you mean 'nope'? I just stated that I prefer using this: *÷* of this: */,* when dividing. The person said that nobody uses ÷, because it's confusing, so I said that I do use ÷, and that it doesn't confuse me.
The programmer's wife sends him to the store. She says "Get one carton of milk, and if they have eggs, get a dozen". The programmer came home with 12 cartons of milk, because they did have eggs.
Agreement with Brian Fedelin. 13 cartons of milk. One carton, and if there are eggs, get a dozen. So 1 + 12 = 13. And the issue with computers is not the logic of them, it is how a human evaluates a human expression and then programs the computer. In this case, the issue was with the wife, since the expression was not clearly defined from the start by defining a dozen of WHAT was desired, the milk or the eggs. See, it is actually a trap by wife against the husband. No matter what he were to bring home, it would be incorrect since she could then change WHAT was the dozen to be of.
I'm pretty sure you missed a different confusion. I get that some would interpret the division sign as you did, but there is also the belief that implied multiplication has priority over other division and multiplication, because it was implied, it has to be resolved. You can't just change 2(3) into 2x3, because they are bound. 2x(1+2) does not just equal 2(1+2), because 2(1+2) = (2(1+2)). I realize it doesn't make a difference until (÷) gets put in front of them. Say we wanted to divide energy by 12. Would we write 12 ÷ mc^2 or would we write 12 ÷ (mc^2). We all recognize mc^2 as energy, m and c^2 are bound by implied multiplication. a completely different thing that 12 ÷ m x c^2. Or divide 12 by the area of a circle: 12 ÷ πr^2. π x r^2 is implied and therefore bound. this is the real argument implied multiplication has priority of not.
Yup all other videos on pemdas of any person in STEM agree that it's one. No person in STEM would say the answer is 9. Although almost all of them prefer fraction bars over the division symbol.
Your math teacher has issues but as long as he is grading you I suppose you need to do what is expected... There is nothing wrong with the way the expression is written just the ignorance people have about parenthetical implicit multiplication...
@@RS-fg5mf Isn't that the whole point - it is perfectly valid but makes it unclear and you have to think about it - are you mad because you got it wrong? I would be a bit concerned about your math's teacher.
@@justcheck6645 I am a math teacher and I didn't get it wrong. LMAO When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly you get the correct answer 9.... Did you get it wrong??
@@babyyoda7749 no teachers are the ones. Teachers are teaching about gender and sexuality for example. The education system is not telling them to teach that.
@@MrGamecatCanaveral Teachers change the way they teach things because we discover new things over time. Before, it was thought that the earth is the center of the solar system. Copernicus discovered that it is actually the sun. Now, do we need to change the way we teach about the solar system? Yes. It's crazy that what we believe in the present will never be entirely 'true' as it could be proven false in the future.
@@aeroljameslita4975 from a subjective point of view, isn't the point of perception the center of your reality? So the Earth is the center of the universe for everyone on it.?
Some people were taught that multiplication by juxtaposition takes precedence over explicit operations… hence why 3/2n is 3/(2n) and not (3/2)n The same juxtaposition glue applies to parenthetical coefficients… and in this case, 2 is that parenthetical coefficient. So using PEMDAS, but assigning multiplication by juxtaposition a higher priority than explicit division, the answer is 1. Additionally, if you use the distributive property from the get-go to resolve the parentheses, you get 1.
As you climb higher in math, virtually 100% of physicists, engineers and mathematicians will interpret the answer as 1. There is no debate over this at all. The implicit multiplication of 2 on the bracket is a SINGLE quantity that takes precedence prior to division. Most physicists/engineers/mathematicians would never even write such a potentially ambiguous expression. They would instead write 6/2(1+2) where the / is a horizontal line. Alternatively, they would write 6/(2(1+2)) leaving NO ROOM FOR AMBIGUITY. PEMDAS is NOT universally accepted. The implicit multiplication on the bracket does indeed take precedent. You are doing a disservice to kids trying to learn mathematical protocols. PEMDAS isn't the total protocol.
It is really funny indeed, because it gives a hint from "where people are coming". I studied physics for some time and it was completely obvious to me, that a juxtaposition has a higher order than "read left to right". It's "obviously" 1. As mentioned; 6/2y with y=1+2 is 3/y, not 3y.
Tom Yes. You give a great example. According to PEMDAS, x/yz = xz/y which is OBVIOUSLY unconventional. The implied multiplication of yz binds the two components of 'y' and 'z' together.
Eventually, yes. This is a fifth-grade expression used to teach and reinforce the order of operations. This is pretty much ground zero. From there, we stop using the obelus in favor of the solidus and vinculum and go into fractions, as well as teaching reciprocals and the multiplicative inverse. People just forget how to evaluate expressions using the order of operations due to lack of practice. Sometimes, all they remember is an acronym and then convince themselves that there are six steps instead of four and that multiplication always comes first when it doesn't.
@@pirilon78 Who says I did? I never even hinted that we don't use the order of operations beyond junior high. It should be common knowledge that we do.
After learning calculus, this answer is 1. Visualize the division line, 6 is the numerator, the 2(1+2) is the denominator. From there solve the denominator however you want, you’ll end up with 6. 6/6 = 1
Incorrect, and this has nothing to do with calculus, it's fundamental algebra. Division is division, not an implied fraction. If anything, it's the other way around: a fraction is implied division
If you write it as algebraic equation you can clearly see how it’s supposed to be done. X/Y(A+B) = X/(YA+YB), since you need to distribute property of Y among entire parenthesis first, and fully evaluate that before going back to division of X. Using numbers it’s: 6/2(1+2) = 6 / (2*1 + 2*2) = 6/6 = 1 This is how math works. People outside of America aren’t thought any of PEDMAS, BODMAS or whatever bdsm acronym is used. People are thought how order of operations works in practice, often explained by definitions, and orders, and with a help of algebraic equations, since when you remove numbers it’s clearer to see how things are evaluated.
@@admiralvirhz Incorrect. The distributing property is multiplication, which has no precedence over division. It would be wrong to distribute 2(1+2) before doing 6/2. The first operation would leave you with 3(1+2) and then you can distribute to get 3+6=9
@@paulblart7378you’re making logical error here. Multiplication doesn’t have priority over division, you’re right about this and it’s set in stone, but to fully value what’s inside parenthesis you need to distribute 2 over it. There’s no sign of multiplication, so you need to understand that it is 6 divided by double parenthesis. You see your logical mistake here? It’s not 2 multiply parenthesis since there’s no multiplication sign. It’s double parenthesis. It’s really bad written problem to deal with, I no wonder why so many people get this wrong.
@@admiralvirhz It's an implicit multiplication. It can be rewritten as 6/2*(1+2), the fact that there isn't an explicit sign doesn't change the problem. I don't know what you mean by "6 divided by double parenthesis", but there is no rule that implicit multiplication groups the operands together. You would do 6/2 first, then multiply that by (1+2)
As a trained engineer in his forties, I immediatey turned the expression into a fraction. I also have to say I don’t think I’ve ever seen that division sign used anywhere after fourth or fifth grade.
And in 4th or 5th grade arithmetic the correct answer is 9 .... The symbol is found on almost any calculator. Best to understand it than to be confused by it...
Funny, I just left a similar comment. I’m an engineer (39 yrs old) and did same as you. That’s the reason engineers and physicists don’t use that silly division symbol.
@@Superdada i don't understand the debate about the division symbol. what difference does it make whether you use : or / ? they do mean the same, don't they?
I'm 40 y/o and was taught the historical way in school. I don't feel historical though. I feel f*cked over because somewhere along the line people decided to change the rules of the game (and didn't inform me!!)
I hate order of operation squabbles. That is not math, it is convention. If there is a governing body for math they should get together and design a convention that is definite, obvious, and universally agreed upon and taught. I was taught the historical method, but knew the current method, so I knew there were two possible answers depending on which system you used. (Not counting the latest anti-racist belief that every answer is correct because saying there is a definite answer would be racist.)
In France i've been taught it in a way, that this equation equals 1. Basically 6/2(1+2) has brackets. We were taught that brackets were always a priority with the number infront of it. So what we would do is first 2*1 + 2*2 = 6, and once we got the brackets completely gone, we can finish the equation which would be 6/6 = 1. Also even if i added the numbers, it was always important to clear the brackets. Here 6/2(3) still has a bracket and doesnt just dissapear. So i would multiply 2 and 3 to get rid of the bracket. Thus we still receive 6/6 = 1 I was always taught this way and was surprised seeing that the correct answer was 9. This blew my mind
I'm pretty sure that in germany we were taught the second answer as well (equasion equaling 1) for the exact same reason you describe here (getting rid of the brackets first) and then finally dividing anything on the left side by what is left on the right side. From my point of view the answer 9 is "wrong". And even if it's just a "rule" thing, we'd better universalise that rule. To me, somehow, the answer "1" also makes more sense in a mathematical- asthethical way.
People in Europe, born before 1970, learned, that multiplication goes before division. Just a fact. I mentioned 1917, because in that year, in the USA it became official that multiplication and division are equal and You start from left to right. In 1980 is was commpn practice all around the globe. ( In the Netherlands it took till 1992 to use the 1917-method). Mathematics is about agreements and those changed over the years to an (new) international standard...
@@j.r.arnolli9734 Thank you for this insight. Anyhow I was born in 1980 and I'm pretty sure that if I showed this "problem" to my old schoolmates/ peers here in germany 99% would come up with the anwer "1". Yet again maybe I'm wrong. If this really is new international standard it still doesn't make a whole lot of sense to me in terms of logical usage of mathematical language.
Yes you are correct the answer is 1. You solve the brackets first to get a number on its own then you finish off by 6 ÷ the answer in the brackets. If your answer is 9 then you are inventing your own mathematics !
Clearing the parens is not simply performing the operation within but also performing the operation dictated by the parens. Therefore the operation requires multiplying 2x3 to get 6 prior to the next operation. If the equation was: 6 divided by 2y there would be no ambiguity that it would be 6/(2y)not (6/2) x3.
Nope, if you got 6÷2y you do 6/2 times y. Its just the current rules, i agree its weird and maybe confusing because we never use "÷", we always use fractions, but the rules are the rules and they say that if theres no parenthesis, you only divide by the first number, the closer to the "÷" symbol. Which is 2, therefore 6/2 × 3 = 9
You are using PEJMDAS like in some calculators (not all of them). J meaning Juxtaposition. But this is not PEMDAS which is the official math rule for instance in USA.
Check Wiki on the order of operation, it is indication that there is an ambiguity/confusion with expression like 1/2x for some it is (1/2)*x = x/2 and for other it is 1/(2*x) Here we have the same type of problem : a/bc, so same problem : is it (a/b)*c or a/(b*c) If for you it is not confusing, then you do not know math enough, because to remove the confusion in that sort of expression, there is a rule that apply to in-line math expression : "Always add parentheses to delineate compound denominator" So here the first thing to say is that "that expression do not follow the rule for in-line math, so It can't be solved using the order of operation; It has to be corrected first" And the problem is that it seems that a lot of people do not know that rule, so they give the result corresponding to one interpretation or the other ... making it viral Should all of those people go back to school ? Or should only the one that wrote that ambiguous expression go back to school ?
Seems we all historical and the new version only rules in special areas, clearly the areas where I’m not. I live in South Africa and here the answer is still 1🤣 should you want the answer to be 9 it would be written as a fraction not a division sign(which can’t even be found on my keyboard, so let’s just all retire the devision symbol and I’d be happy to concede that the answer is 9😂
Basically the answer isn't "wrong" if you use the historical version... they're just asking different things... in modern math, it you wanted to ask the exact same question as the historical you would have to write is 6÷[2(1+2]
@@willwalker24601 It comes down to "just use brackets to make clear what you mean". Mathematics is supposed to be a universal language, but there are still a lot of dialects, aka different notations. I see that a lot lately as I am german but using english youtube videos to review some things since I am studying for a new profession. They are doing a lot of things differently than I learned them at school 20 years ago. Maybe they do them that way in schools now too, I don't know. But since such differences exist, one should strive to write expressions as clearly and unambiguously as possible. Most of those "puzzles" thrive on their ambiguouty.
✔️✔️✔️👍👍 Correct answer is surely 1 To those who are telling it 9 Dont know how? For this xy ÷ xy = 1 But Its not y²(according to those who are telling answer to be 9) Similarly, 6÷2(3)=6/(2*3)=1 As simple as that...
The Distributive Property supports 9 not 1 The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division. The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them... The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication... The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral... 6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property Parentheses removed... 6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property. Inner parentheses REMOVED This can be further demonstrated using the vinculum.... 6 ------(1+2)= 6÷2(1+2)= 9 2 6 ------------ = 6÷(2(1+2))= 1 2(1+2) A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.... ________ 2(1+2) = (2(1+2)) two grouping symbols each That over bar (vinculum) is a grouping symbol _______ _________ 2(1+2) = 2×1+2×2 one grouping symbol each Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation...
It’s 9.. how is this even viral, it’s 5th grade math.. Also, I’m referring to PEMDAS which is taught in 5th grade. Watch the video if the answer you got wasn’t 9..
But if this is on a test, you want to know that your students actually paid attention and learned correctly, writing the way you did removes the so called "ambiguity" (which there is none) and then there will be no way to actually know if they have learned correctly
When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended you get the only correct answer 9 If you don't apply the basic rules and principles of math correctly then you are already confused.
@@lolmom3590 BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = (1/2) 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....
@@jonnel4038 x÷2y = x÷(2y) by Algebraic Convention... BUT x÷2(y)= x÷2*y by the Distributive Property... Parenthetical implicit multiplication does not have priority over division. When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses not just the number next to it. The correct answer is 9
Hi Zak, I'm starting to think that algebraic characters are simpler to process with than math with arithmetic (numeric) characters. (Actually, the order of processing should be the same for both worlds but alphabetic or algebraic characters are a bit more limiting. But that's just my opinion.) Oh, well... I've made some clarifications in my own mind. That's good enough for me. Also, the medical world has given me a wake-up call regarding my expiration date. The range where it may reside is no longer unlimited. But I have a number of years left to ponder challenges like this. It's fun and interesting. Even history is interesting now; not like it was back in high school. I started liking history when Glen Beck brought out his chalk boards on his Fox News channel program years ago. Now, I'm moving on to photography. I don't think I'll ever fully understand analytic geometry and calculus. Sigh... Too bad we can't start over, isn't it? Oops! I'm rambling... Happy trails, Wes
@Jure Lukezic That only works for very large values for 0. I was representing numbers in base-2; however, if we're talking string concatenation then yaaaaaaaaassssss!!!
@Jure Lukezic So how does it feel that your joke went over our heads? Don't you feel bad for us smug little pedantic bastards? We could have strung that out, like "I was writing in binary" ... "no you weren't" ... "yes I was" ... "no" ...
dont worry, this issue will never show up in important engineering situations because the division symbol would never be used. instead using a fraction would make everything a lot more clear
The real-life solution, as per the ISO recommendation, is just to use brackets to disambiguate. (6/2)(1+2) is totally clear regardless of division symbol used and works for handwriting, calculators, typed documents etc.
@@charliedallachie3539 thats not using the numerator and denominator, when you use an actual numerator or denominator you would have a certain part be under it. Either 6/(2*3) or (6/2)*3
@@o_sch yea I understand the two answers but in other problems which is which? I’ve always wondered PEMDAS in general I’m sure there’s a complex mathematical proof of it out there somewhere Edit* there is no proof it’s a convention.
@@JakobSchade Sure, but that's if you use PEMDAS or whatever else. There's still plenty of books where they don't use PEMDAS and have a difference between implicit and explicit multiplication. 2*3 is explicit (a * sign) and 2(3) is implicit. In that case, implicit is many times higher of importance than explicit. So 6/2(1+2) would simply be 6/6=1.
`Hi Zak, Thank you for you contributions here. I'm sure other viewers appreciate them too. I also wonder if anyone else found the TableClass Math post to which I tried to point them. Happy trails, Wes
@@zakelwe Hi Zak No, the thread was started by one of the regular instructors, John of TabletClass Math.I think the post was a 're-post' but it was recent. He made another post about putting parenthesis around each of the numerator and denominator in a post today; but I didn't note it.
@@wes4139 Let me have a wander over. Wish me luck. PS Nice to chat to someone who is civil , it can get quite heated on these. These types of problem have been occurring since 2011 and what happens is that 50% of people get one answer 50% get the other. Two tribes form. Both tribes throw arrows at each other. The current state of teaching gets blamed 😀 The fact it has been going on so long and educated people differ roughly equally shows there is an underlying problem, which of course there is.
Was it this one from about 4 days back? Why don't the numerator and denominator have separate groupings? Isn't 6 ÷ 2(3) = 6/2(3), or {6} ÷ {2(3)} implied? I thought numerators and denominators had their own groupings... The answer on my calculator = 1. Or, how about this --> 6 / 2(3) = 2x3/2(3)? Don't the 2's and 3's cancel out? Ok 6 ÷ 2(3) = 6/2(3) but that does not equal {6} ÷ {2(3)) The former two are both equal to 6 ÷ 2 x 3 where as the latter is 6 ÷ (2 x 3). The first two are ambiguous, you can do the M or D first, the latter is a well written equation, you have to do the 2x3 first so no issue, answer is 1. In effect 6 ÷ 2 x 3 written on two lines can be done as 6 6 -------- or ----- x 3 both meanings are clear on what the person writing it wants you to do first. However with 6 / 2 x 3 that meaning has been lost. 2 x 3 2 To do both options above on one line by using brackets correctly then we have 6 / ( 2 x 3 ) or (6/2) x 3 respectively. Most people would be able to work it out easily and there is no ambiguity. This was noted back in the 1920s when text books did it both ways and the Committee on the Teaching of Arithmetic in Public Schools was called to adjudicate on the matter. They said "(the) committee recommends the use of brackets to avoid ambiguity in such cases. And 100 years later .....
@@zakelwe This is another try at getting the message to you 4÷8 1) 1 The explanation I want seen is at 11:24 in... ___ 2) ¼ He explains the Parenthesis around each of 2x4 3) 1/16 the numerator and the denominator. 4) 4
The issue is, I agree that with the same precedence you go left to right so if it said 6 ÷ 2 × 3 I would correctly answer that as 9. However by wording it as 6 ÷ 2(1 + 2), my mind goes to expand the bracket first which gives 6 ÷ 6 = 1.
This. I was taught (in the US) completing the parentheses/brackets meant you did all involved with the parentheses/brackets. Here, the parenthesis is what symbolizes the 2x3 so you still do that before the division.
The rule is called BODMAS or BIDMAS It is the order of what you do first Brackets Indices (or other) Division & Multiplication Addition & Subtraction So here first we do the brackets 6 ÷ 2 (1+2) 6÷ 2 (3) 6 ÷ 2*3 Next we do division 6÷2*3 3*3 Next we do multiplication 3*3 9
@@RS-fg5mf Where the HELL did you get THOSE numbers?! PEMDAS(or a few other things that say the same basic thing): Parentheses, 1+2=3; Exponents, there are NONE so we move on; Multiplication and Division DO THEM AS YOU FIND THEM STARTING AT THE VERY BEGINNING(from left to right, you know the way we read things in most cultures!), 6/2(The first one you find when you start at the beginning) is 3, then we have *3, so 3*3 that's NINE! You didn't even come up with 1 which was my first thought but then I realized that I was wrong and redid the problem, and got NINE!
@@JacksonOwex I never said the correct answer was 1. I absolutely understand the correct answer is 9 I get pissed off when people say they were taught the historical method when they fail to even understand the context of this video and what the historical method was... The historical method was a misuse of the obelus by some text book printing companies who pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math so the ERROR was corrected post 1917... This ERROR means that 1 is not and has never been the correct answer. BUT this ERROR i.e. method of using the obelus would have made 6÷2+4=1 by this incorrect use of the obelus... So when someone says they were taught the historical method I them what 6÷2+4 is equal to and when they answer 7 that's proof that they were not taught the historical method mentioned in this video... The real confusion is the false and misleading information and willful ignorance people have about parenthetical implicit multiplication. They incorrectly believe that 2(3) is a parenthetical priority and that the implicit multiplication gives it priority over the division which is FALSE.
@@JacksonOwex That wasn't how we were taught. I also graduated in 2006, and we were taught PEMDAS and to do them precisely in that order. My teacher never told us that Multiplication and Division were on the same level and Addition and Subtraction were on the same level. We were taught to multiply first and then divide. I only recently discovered that I was taught incorrectly. By reading through the comments, I realize that I am not alone and many of us were taught incorrectly.
I have an issue with this. I fully understand it, but hear me out really quick. Lets say : n = 1 + 2 6 / 2n = x Solve for x. This is the same as the original equation in the video: 6 / 2(1+2) Essentially "2n" is just "2 x n". So essentially the equation becomes: 6 / 2 * n = x But we wouldn't solve it as: ( 6 / 2 ) * n = x We would solve it as: 6 / (2 * n) = x Now we could probably agree that "2n" isn't just "2 x n", but instead "(2 x n)". Which would always make the equation: 6 / (2 * n) = x Which sort of leads to my issue, the way a calculator would solve the original question: 6 / 2(3) = (6 / 2) * 3 It does make logical sense, but seems counter-intuitive to me. The following equation: 6 / 2(3) = 6 / (2 * 3) Seems to make more reasonable sense to me, based on principles of alegebra. Also, in a weird way it wouldn't violate our current order of operations. The equation: 2(3) Could still be looked at as a paranthesis operation, instead of just multiplication, making its order take precedence.
As soon as you write 6 / 2n you are introducing a notational convention that is not present in 6/2(1+2). Namely, you are introducing the algebraic convention that "2n" should be treated like "(2n)". Another way to say it is that a coefficient-with-variable pair is considered grouped. It appears to me that this convention is universally accepted. Therefore, 6 / 2n can be replaced by 6 / (2n) but not by 6 / 2 * n. The latter introduces an error. The same notational convention does not apply to 6 / 2(3). That is, 6 / 2(3) can not be replaced by 6 / (2(3)). However, not everyone agrees. Some people say 6 / 2(3) SHOULD be replace by 6 / (2(3)). Other people say 6 / 2(3) SHOULD be replaced by 6 / 2 * 3, Then, 6 / (2(3)) = 1, whereas 6 / 2 *3 = 9 The debate is sometimes fierce, but it has not been settled in over 10 years of UA-cam debates. I would not expect it to be settled anytime soon!
6/2n = 3/n by Algebraic Convention 6/2(n)= 3n by the Distributive Property All variables have a coefficient written or not. Constants can be coefficients but constants do not have coefficients. x/x = 1x/1x = 1 x/1(x)= 1x/1(1x)= x^2 x/x(x)= 1x/1x(1x)= x
@@RS-fg5mf 6/2n = 3/n, yes, by the convention I described, namely that "2n" is treated as if it were "(2n)". Call it "Algebric Convention" if you want, no problem. But it is, as you say, a convention, not a rule or law. It's a convention of how to interpret the notation "2n". 6/2(n) = 3n, I'd be inclined to agree. But I bet a lot of UA-cam commenters would disagree. I think it suffers from the same lack of consensus as 6/2(3). If 6/2(n) = 3n ia true, it's absolutely NOT by the Distributive Property. The distributive property is a relationship between multiplication and addition; it refers to distributing multiplication across (or over) addition. There is no addition in 6/2(n) so the distributive property does not apply. The notion that all variables have a coefficient written or not is just plain silly. It introduces an over-intellectual complication into the notation and evaluation of expressions. For example, a simpler and more direct evaluation of x/1(x) is x/1(x) = x(x) = x^2, for those who do not believe that the so-called "implicit multiplication" is special and thus do not believe that the multiplication is done first in x/1(x), Replacing x by 1x is pointless and unnecessary.
@@donmacqueen just because you fail to understand it doesn't make it pointless and isn't a valid argument against it.... 6/2(a+b)= 6/2×a+6/2×b 6/2(a+b)= 6/2(c)= 6/2*c The point you're missing is that the Distributive Property is the application of MULTIPLING the term outside the parentheses across each TERM inside the parentheses.... 6÷2×3 is no different mathematically as 6÷2×1+6÷2×2= 6÷2(1+2) ... You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol... 6 ------------ = 6÷(2(1+2)) 2(1+2) two grouping symbols each 6. 6 ------------ = -------------- one grouping symbol 2(1+2) 2×1+2×2. Removed... 6÷(2(1+2))= 6÷(2×1+2×2) one grouping symbol removed... The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division.... 6÷2(3)= 6÷2×3 the same way as 6÷2(1+2)= 6÷2×1+6÷2×2 6/2(n) the n can represent a single value or a set of operations that represent a single value... Let's take 6/2(a+b) and let a=3×4 and b= 2×5 Then 6/2(3×4+2×5)= 6/2(12+10)= 6/2(22)= 3(22)= 66 OR.... 6/2(3×4+2×5)= 6/2×3×4+6/2×2×5= 3×3×4+3×2×5= 9×4+6×5= 36+30= 66 Would you disagree that 3(3×4+2×5) is the same as 3×3×4+3×2×5 ?? The Distributive Property is explained as Multiplication over Addition but technically it is the Distribution of the TERM outside the parentheses across one or more TERMS inside the parentheses, is a much better description. As TERMS are separated by addition and subtraction....
@@RS-fg5mf You are replying to my comment in which I said, "The distributive property is a relationship between multiplication and addition; it refers to distributing multiplication across (or over) addition. " In your reply you said, "... the Distributive Property is the application of MULTIPLING the term outside the parentheses across each TERM inside the parentheses.... " These two statements are describing the same thing. Your suggestion that I don't understand distribution, is badly mistaken. Oh, and when you state, 6/2(a+b)= 6/2×a+6/2×b I will point out that to evaluate either of the terms on the right hand side, one has to do the division first within that term. Which supports my earlier asertion that the difference between 6/2(1+2)=9 and 6/2(1+2)=1 is whether the division is done first or the multiplication is done first. Everything you say just comcplicates the issue. Evaluating 6/2(1+2) is actually very simple. It's completely valid to evaluate the 1+2 first, since it's within a grouping notation. This gives 6/2(3) Now there are only two operations remaining. Which one should be done first? One of them has to be done first. The result is either 9 or 1 depending on which one is done first. That's all there is to it. Which one comes first is the only decision needed to evaluate 6/2(3).
@@PuzzleAdda we cant. All of these numbers are in the form 2+4k where k is any number from { 0, 1, 2, ... , 14 }. The equation would be (2+4k)+(2+4l)+(2+4m)=60. After we simplify this we obtain k+l+m=27/2 but all of k, l and m are whole numbers. Therefore it is impossible to obtain 27/2 by suming k+l+m and the equation does not hold.
The comments section is amazing. One of the top comments concludes that you are doing a disservice to kids trying to learn mathematical protocols. I bet you didn't see that coming.
I came up with 1 as my solution. After listening to the explanation I found it logical that 9 would be correct. Now I tested my calculators to see what their solution would be and the first one (used as a standard calculator at schools) came up with 1. The second one (a more sophisticated graphic calculator also used at schools) showed me two different solutions, depending on the writing: 1. 6÷2x(1+2)= 9 2. 6÷2(1+2) = 1 but it changed the writing into 6÷(2(1+2)) It's such a simple arithmetic problem but even calculators are challenged. I love it!
When you understand and apply the Order of Operations and the various properties and axioms of math correctly you get the only correct answer 9. Some calculators are not programmed to handle parenthetical implicit multiplication correctly....
I'd still say 1 is the right answer because PEDMAS is skipping the priority of multiplication by juxtaposition. TI calculator use PEMDAS but Casio and hp have returned to PEJMDAS. It's mainly only North American "teachers" (note that, teachers not mathematician) who insist on PEMDAS.
@@Faux_Dieu you and many others are confusing and conflating an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing... 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... 6÷2y the coefficient of y is 2 BUT 6/2(y) the coefficient of y is 3 ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9
@@RS-fg5mf I think, what's going on here is how you understand your question. When I see 6÷2(1+2) I see as the question that if there are 6 apple and you have to divide amongst two groups in which each group has one adult and two children. So how many apple do each person get. Now you tell me how do you interpret this question in real life? You are taking PEMDAS too literally and forgetting that the multiplication by juxtaposition takes priority over multiplication and division. The answer that 6÷2(1+2)=9 doesn't make sense in reality. What is the situation?
@@Faux_Dieu you don't interpret a math expression you evaluate a math expression following the basic rules and principles of math... Your word problem would be correctly written as 6÷(2(1+2))= 1 You have 6 bags divided between 2 groups and each bag contains 1 red apple and 2 green apples. How many apples did each group get?? 6÷2(1+2)=9 apples per group.
I have found the counterpart to my mantra „6/2(1+2) is an ambiguous and poorly written expression and the result is meaningless“. It is from the university of Berkeley. It debunks the claim 1 is the distinct solution for 6/2(1+2). GanonTEK is right. A problem that hit the Internet in early 2011 is, "What is the value of 48/2(9+3) ?" Depending on whether one interprets the expression as (48/2)(9+3) or as 48/(2(9+3)) one gets 288 or 2. There is no standard convention as to which of these two ways the expression should be interpreted, so, in fact, 48/2(9+3) is ambiguous. To render it unambiguous, one should write it either as (48/2)(9+3) or 48/(2(9+3)). This applies, in general, to any expression of the form a/bc : one needs to insert parentheses to show whether one means (a/b)c or a/(bc).
"It is from the university of Berkeley. It debunks the claim 1 is the distinct solution for 6/2(1+2)." - other way around you mean - MATHS TEXTBOOKS DEBUNK the Berkley blog. 🤣🤣🤣🤣🤣🤣 "GanonTEK is right." - nope, he's wrong. That's why he NEVER CITES ANY MATHS TEXTBOOKS, DUUUHHH!!!! 🤣🤣🤣🤣🤣🤣 "Depending on whether one interprets the expression as (48/2)(9+3) " - it's NOT WRITTEN LIKE THAT, so NO YOU DON'T, duh! 🤣🤣🤣🤣🤣🤣 "as 48/(2(9+3)) " - as per the rules/definitions of Terms, Products, Expressions, The Distributive Law, and Left Associativity, as found in MATHS TEXTBOOKS, and also Cajori, so that's all settled then! 🤣🤣🤣🤣🤣🤣 "There is no standard convention as to which of these two ways the expression should be interpreted"
@@michi9816 "do you honestly think I would believe a guy flat out lying about the distributive law ?" - I don't know - do you believe YOURSELF? Only YOU can answer that question, not me! 🤣🤣🤣🤣🤣🤣 I don't know if you trust yourself or not - I sure don't! If anything I trust you to LIE ABOUT EVERYTHING 🤣🤣🤣🤣🤣🤣
@@smartmanapps5588sure, I trust my memory, the math textbook from our schools on Highschool / college level, the TU, wikipedia, Encyclopedia Brittanica and the us math textbook from the sixties and finally my reasoning about math. tell me what you have to offer ?
@@michi9816 "the math textbook from our schools on Highschool / college level, the TU" - which you LIE about "wikipedia" - which is WRONG. You wanna explain how you trust these sources which CONTRADICT each other?? 🤣🤣🤣🤣🤣🤣 "Encyclopedia Brittanica" - which you LIE about "the us math textbook from the sixties" - which you LIE about "finally my reasoning about math" - which we've seen time and time again is woefully bad. "tell me what you have to offer ?" - actual direct quotes from the Maths textbooks which you LIE about, like Pages 36, 37, 282, and 577, duh!
When programming, the correct answer is to never leave any ambiguity, so always add enough parenthesis to ensure that anyone reading it will understand your intention. So write 6 / (2 * (1 + 2)) OR write (6 / 2) * (1 + 2). Both are correct, but only one would be correct depending on what your intention is. So always make sure that you enter something that cannot be misinterpreted.
So you are saying that we should be forced to write 5+(2×10) because too many people fail to understand the basic rules and principles of math and incorrectly believe that 5+2×10= 70
@@RS-fg5mf Yes, for the same reason we add comments. Make sure that we know what is happening and that people reading it in the future know that we know what it means.
@@StuartLynne then the Order of Operations and the various properties and axioms of math become redundant if you're going to add crutches for people who fail to understand and apply the basic rules and principles of math correctly
@@RS-fg5mf When math is being taught completely differently between generations, it’s bound to be misinterpreted. I’d rather them be in-depth and redundant so that people in the future won’t have to just assume anything. Assuming things causes a lot of problems.
@@onemorelisa3785 math is only being taught differently if the prrson teaching it is incompetent.... The Order of Operations and the various properties and axioms of math were established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses... The basic rules and principles of math have been the same for over 400 years... Math is based on rules not popularity or personal opinion. Failure to understand and apply the basic rules and principles of math correctly as intended is not a valid argument against them...
There ain't nobody arguing from historical context so explaining historical context to an audience that doesn't know history, or doesn't learn from history is pointless. The Order of Operations we all know stands. The answer is 9.
I am a retired chemistry, physics and math teacher, including calculus. Also worked as a field engineer for 4 years. I think the answer is 1 because I would rewrite it in it's algebraic form before solving. I don't think you'd find much confusion in engineering. I can not remember ever seeing the division symbol used in any formula or equation in advanced work. Also don't remember it used past Algebra I in high school.
I also agree. Looking at this I believed the answer to be 1 as I don't believe that such an expression would be written using the division symbol and also cannot recall seeing the division symbol used in an equation.
Please read this comment, thank you. Solve for the 2 in parentheses, it is not a bracket [ ] 6/2(1+x)=9 3(1+x)=9 3+3x=9 3x=6 X=2 The problem is that people who think that it is 1 believe that after simplifying 2(1+2) is that they think it is the denominator of the fraction. For that to be true, there must be a parentheses in front of the 2.... (2(1+2)). You will do that first if that was in the problem, but it isn’t. 6/2(1+x)=1 6/2+2x=1 Now you see that there is a fraction, but what can it be. If it is 2+2x, you get 2 as your final answer, which is correct. If it is just 2, you get -1, which is incorrect. However, you do division before addition, so you do 6/2 to get 3, eventually getting -1 as the solution. 3+2x=1 2x=-2 X=-1 This is incorrect, because we are trying to solve for 2 in the parentheses... 6/2(1+x)=1 X should equal 2. People think that after distributing the 2 into (1+x), the whole thing stays in the parentheses. It disappears after you distribute. Thank you for your time.
It depends on which interpretation of multiplication by juxtaposition you use. 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.
@@Lucian24 What I meant is that we now use fractions so we don't need to specify which goes first. You operate all on the top and divide it by all on the bottom. So you can differenciate: (6/2)×3 = 3×3 = 9 From 6/(2×3) = 6/6 = 1
As far as I'm concerned, "2y" represents a single value, and should be treated as such. To deliberately separate the values, you write "2 × y" or "2 · y". Therefore, if the expression were given as "6 ÷ 2 × (1+2)", I would agree and say it's 9. But the ligature of "2" and "(1+2)" in the form "2(1+2)" represents a single value, making the answer 1.
The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y. This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2... You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You CAN factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression... Many people, *including you*, confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial Factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 So please stop with the illusory facts and stop spreading false and misleading information. You're part of the problem.
@@jack-xf6il The thing is, no one has yet provided evidence that a(b) is not multiplication by juxtaposition and not equivalent to ab. If the wasn't ambiguity here why do modern international standards like ISO-80000-1 mention about writing division on one line with multiplication or division directly after and says that brackets are required to remove ambiguity or why does the American Mathematical Society say it's ambiguous as well as many maths professors etc.?
@@GanonTEK I was just making a joke about the misspelling of juxtapose, which I thought made it sound like 'just suppose'. But while I'm here, if y = 3 then does everyone agree that 6÷2y = 9? Because I have to say that did not seem intuitive to me. But is 6/2y = 9 also, or 6/2y = 1? Do '÷' and '/' have different meanings or are they the same?
@@RS-fg5mf Let's take a look what is stated... Statement 1 : 6/2y = 6/(2y) = 3/y by Algebraic Convention Statement 2 : 6/2(a+b)= 3a+3b not 6/(2a+2b) , because 6/2(y)= 3y by the Distributive Property... let's take y = (a + b), in statement 1 : 6/2y = 6/(2y) = 3/y What is stopping us from writing 6/2y as 6/2(a + b) , 6/(2y) as 6/(2(a + b)) and 3/y as 3/(a + b) using direct substitution in Algebraic Convention? We see 6/2y = 6/(2y) and 2y is taken as whole denominator, but notation have ambiguity and we default to seeing 6/2y where y = (a + b) will look like 6/2(a+b) but it is actually also 6/(2(a+b)) which still follows Algebraic Convention 6/2y = 6/(2y) and by observation 2(a+b) seems like (2(a+b)) and this is where the context comes in to operate canonically with higher priority to evaluate 2(a+b) because 2(a+b) is also (2(a+b)) . In Statement 2 : 6/2(a+b) = 3a+3b by Distributive Property, 6/2 as scalar to (a + b) In this case , by observation both 6/2(a + b) become 3/(a + b) or 3a+3b., which also proof there is a lack of information in the notation to distinguish your answer. By words is very certain because there is semantic you can infer priority. The notation used here alone is not distinctive to avoid ambiguity in this case. Conclusion like some have mention is it is better to rewrite the notation to denote the scalar part (6/2)(a + b), 6/2 dot (a + b) or 6/2 * (a+b). Like many Engineers will likely used often is 6/2(a+b) with parenthesis taking higher priority and evaluate as 6/(2(a+b)) or 6÷2(1+2) =1. This is why there is 2 answer to this video 1 or 9 as the video also shows just because the question is presented without context. Really this just boil down to notation syntax and interpretation. To those who say 1 of the answer is wrong, is the person who is ignorant as you are stuck in their own confirmation biases.
What does attending school in Appalachia have to do with it? Yes, I did and I was taught the 1917 way I guess from 1996-2013. WCU was still using in it in 2013, and so was all the other kids from other parts of the US.
honestly i dont know how long ago people didnt use the order of operations but im sure that in the 80s all mathematicians used it. id go as far as to say it probably existed at least a thousand years ago
realistic dan Appearantly my wife was retaught the correct way when she went to UMiss. Guess that why my kids always come home with the wrong answers when I help them do their school work
To make this easy look at the division symbol, it’s a dot over a line over a dot. This tells you that what is on the left of the symbol goes on top and what’s on the right goes on the bottom.
@@godelnahaleth No, you were not taught to follow PEMDAS as 6 exact steps... SMDH Own your mistakes and stop blaming your teachers for your failure to pay attention in class and learn correctly...
@@RS-fg5mf Nope. 2(3) is not the same as 2*3. Anyway it's been 4 years since I came across ÷ sign. I only use fractions and never had to come accross controversial problems like this one.
Even when i went to school the way we were taught to do math, at a mere 2 second glance i got the answer of 1. In figuring out the problem the parentheses when it becomes 2(3) we still solved 2(3) before we did the division. Anyone who graduated in the early 2000's will get 1 as the answer not 9 because we wouldn't have replaced the () with 2x3 subsequently changing the order of operations
You're wrong and You're the one changing the expression... There is no mathematical difference between 6÷2(3) and 6÷2×(3) or 6÷2×3... The multiplication SYMBOL is implicit rather than explicit. Grouping symbols only group and give priority to operations INSIDE the symbol not outside the symbol. The Order of Operations and the various properties and axioms of math were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses. The correct answer is and always has been 9 not 1. 6÷2(1+2)= 3(1+2) Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial Factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9
@@RS-fg5mf I think what Joshua was getting at was that we were taught a completely different order of operations in the public education system than what the orator of this video claims is correct. We were specifically taught that multiplication always takes precedence over division and that addition always takes precedence over subtraction regardless of their location/order in the equation with the order of precedence of operations being mneumonically described with the phrase "My dear Aunt Sally," which stands for "multiplication, division, addition, and subtraction." Now they are arbitrarily changing the rules and/or they deliberately miseducated millions of children. Take your pick, but something is seriously f^cked up here.
@@wesbaumguardner8829 Then you were taught wrong or more likely don't remember correctly... There isn't a mathbook on the planet that lists the Order of Operations as 6 exact steps... So your telling me you think 10-7+2=1 ?? Not now, not ever... Multiplication and Division share equal priority and can be evaluated equally from left to right as they are *inverse operations* by the reciprocal... Addition and Subtraction share equal priority and can be evaluated equally from left to right as they are *inverse operations* as subtraction is just addition of a negative number.... 6÷2×3= 6×0.5×3 now multiply in any order you want... 10-7+2 = 10+(-7)+2 now add in any order you want you still get 5
@@wesbaumguardner8829 the rules have not been changed. The Order of Operations and the various properties and axioms of math were established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses... P E M/D equally left to right A/S equally left to right There isn't a math book on the planet that states PEMDAS represents 6 exact steps...
@@wesbaumguardner8829 M and D have equal priority so the order doesn't matter. Same with A and S. Like with 10 - 8 + 3 - 1 S first: 2 + 3 - 1 = 2 + 2 = 4 A first: 10 - 5 - 1 = 5 - 1 = 4 The issue with 6÷2(3) is that multiplication by juxtaposition can imply grouping, giving it higher priority than division and regular multiplication. So it's ambiguous because it's not clear if 2(3) means 2×3, the explicit interpretation used my programming and in America, or (2×3), the implicit interpretation used by academic writing. 6÷2×3 = 9 6÷(2×3) = 1 Although, 6÷2×3 isn't good writing. Modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. ISO-80000-2 says ÷ should no longer be used also. It's just a badly written expression.
I understand both methods. It will be interesting what my books say when I get to this type of problem. I am studying from pre-1900 Maths books right now.
@@bk40907 sure it is the same thing. but with fractions the error couldn't happen as the order is directly visible. I personally haven't seen that operator once in university. If you are forced to write in one line (e.g. in programming) people use "/"
If you dont carefully listen to what your teacher says then you will get it wrong. I remember my teacher saying that if multiplication and division are the only ones left, you'll solve them from left to right. Same goes to addition and subtraction (If they're the only ones left)
Pretty nice way of saying it. It's like "A union B intersection C" in sets and expecting a certain answer. You can't write that either because it's ambiguous.
@@GanonTEK PEMDAS is not ambiguous. 6/2(1+2) Parentheses first 6/2(3) Multiplication and division left to right 3(3) 9 There is no ambiguity. The ambiguity is people not recognizing 6/2(3) = 6/2*(3) = 6/2*3 = 9. Implied multiplication is treated the same as regular multiplication. The it’s the same as “I didn’t read the question correctly, therefore I am not wrong”
@@Owen_loves_Butters There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not. I.e. does 2(1+2) = (2×(1+2)) or 2×(1+2)? Both are widely used. 6÷(2×(1+2)) = 1 (using PEMDAS) 6÷2×(1+2) = 9 (also using PEMDAS) PEMDAS isn't the problem. The notation used is. That's the cause of the ambiguity. That's why there is such a large disagreement and even calculators from the same manufacturer don't agree. You shouldn't write a/bc or a/b(c) anymore. It's not acceptable notation. ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove any ambiguity. A PhD student wrote a paper on the ambiguity called The PEMDAS Paradox if you want to look it up.
I don’t think the problem here is the division sign. I think the problem is the “implicit multiplication”. In my experience as a scientist and teacher, many people would say 5/2x = 5/(2*x), similar with the number in front of the parentheses, without the multiplication sign.
Algebraic equation vs. simple arithmetic. 2x is a variable and it's coefficient, which implies multiplication. You don't have any implied multiplication in arithmetic because there are no coefficients since you have no variables.
BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....
6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator...
I agree with Wotcher Mystic here. I learned math practices in California public school in the 1950s and 60s. Where I went to school, we did not practice math with this video's rules change. Had I pursued math in college, they may have become known to me.
That being said, wow, so many retards who can’t do simple arithmetic in the comments. It’s not that I mind, but the fact that they all say it with so much certainty.
Excuse me, but I know someone who got 1 as an answer and even if they were wrong, they happen to be a highly intelligent person who knows several programming languages and writes code very well. People make mistakes and some people have been taught outdated information. That doesn't mean they're "retards" or that you have to worry about the state of society **eyeroll**. An 8th grader will get the correct answer because they're currently learning the material. My friend who got the wrong answer still knows a shit ton more valuable information than any 8th grader lol. Why do people who perform well in math always go around calling everyone retards if they don't get an answer right? You never see an English major call a mathematics inclined person stupid if they couldn't figure out how to write a proper MLA citation lol.
I think it’s more so people disliking that he solved literally a 4th grade problem on a channel based around more advanced math... although at least he gave it SOME substance with the whole historical bit. Still, kind of out of place on this channel.
I would have thought the presenter of the equation asked us to "Divide six by twice whatever value is within the brackets". The equation can be written as A divided by B where B=2(1+2). So if [A divided by B] = 1 and A =6, then B=6, This implies 2(1+2)=6 which is correct. If [A divided by B] = 9, and A=6, then B= 6/9or 2/3. This implies 2(1+2)=2/3 which is incorrect. The rule 'brackets first' is short for "Solve the brackets first" This implies the removal of the brackets from the equation first. In adding 1+2, you do not get rid of the brackets! You are still left with 2(3), and one cannot remove brackets from the equation without solving them. In this case you must multiply 3 by 2 to the brackets. You cannot just add in or take out brackets ad lib! In applying values, 2(3) is a single Value and is expressly implied in the equation. 2 x (1+2) is two values and is not implied in the equation. It appears to me the problem lies in the computerisation of the equation. To me the equation is simple and unambiguous and means the same now as it did 50 years ago!
In the 80s, I was taught the same. Wonder if country of origin makes a difference in how one is taught this. My algebra and geometry teacher (same woman) was Vietnamese. Is it possible that in other parts of the world how that 2(3) is treated is different? Could that explain the differences in how some of us learned it?
The ignorant leading the ignorant... All you remember is being taught that 2(1+2)= 2×1+2×2 or that a(b+c)= ab+ac What you failed to learn or have forgotten is that the TERM outside the parentheses is to be multiplied by the value of the parentheses or Distributed across the TERMS inside the parentheses... TERMS are separated by addition and subtraction not multiplication or division. 6 is a single TERM 6÷2 is a single TERM 6÷2×3 is a single TERM 6÷2(1+2) is a single TERM with two TERMS inside the parentheses. 6÷2(1+2) = 6÷2×3 = 3×3= 9 The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is part of a single TERM... FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9
I just took college trig this past semester and the 2(3) is converted to 6 first. So this old 1917 rule of order of operations is still in effect to this day and being taught to new engineers likes myself.
@@rosewarrior706 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 The notation is ambiguous. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not Even online calculators don't agree. Microsoft Math shows both answers on screen at the same time in different places. I think DESMOS won't even let you type this on one line, which is great.
Yeah I'm still studying my fifth year in civil engineering and we always do it that way cause 2(1+2) it's supposed to be all together like just one and not 2*(1+2) which is not the same if you have to solve a problem like the one presented in the video. So I'm just so confuse right now.
Just ask them if they've taken advanced functions or calculus, and then tell them if they ever used the ÷ symbol instead of /. I think thats some pretty solid evidence I should say
What is interesting is that four different calculators give me four different answers. Two of the calculators can’t handle 2(1+2) or even 2(3). The Radio Shack EC-4030 gives 2(3) = 23. The Hewlett-Packard 20S gives 2(3)=3. The two calculators that can handle the equation are both Texas Instruments. The TI-30X IIS gives 6/2(1+2)=9 while the TI-85 graphing calculator resolves 6/2(1+2)=1. I can still hear my algebra teacher and physics teacher both stating that you must eliminate all parentheses before advancing to the next operation (hence 1 is correct for them). I do agree with those who say that the equation is poorly written.
BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations 6÷2(1+2)= 6÷2(3)= 3(3)= 9 2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside .... There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more. Commutative Property 6÷2(1+2)= 6(1+2)÷2= 6(3)÷2= 18÷2= 9 Distributive Property 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9 The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction. 6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2 Operational inverse of division by the reciprocal 6÷2(1+2) 6(1/2)(1+2)= 6(1/2)(3)=? Multiply in any order you want you still get 9 Proper use of grouping symbols 6 -----(1+2) = 6÷2(1+2)=9 2 6 -------- = 6÷(2(1+2))=1 2(1+2) A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator... Another argument people tend to use incorrectly is factoring.... 6 = 2+4 No parentheses required BUT 6÷(2+4) parentheses required 2+4= 2(1+2) only one set of parentheses required. 6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set... The 2(1+2) must be placed within the first set of parentheses containing the (2+4) 6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2) Let y = 0.5 6y(1+2)=? 6y*1+6y*2= ? 6/y⁻¹*1+6/y⁻¹*2= ? If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9 The rules of math have to remain logical and consistent across the board... THESE ARE THE FACTS....
without placing an explicit multiplication sign between them. A person is left wondering whether to use the sophisticated convention for implicit multiplication from algebra or to fall back on the elementary PEMDAS convention from middle school. It's poorly written
@@MrElvis1971 Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing... Convention doesn't trump LAW and the Distributive Property is a LAW. 6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property... ABC/ABD = C/D by Algebraic Convention ABC/AB(D) = CD by the Distributive Property 6/2(a+b)= 3a+3b not 6/(2a+2b) The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication.... Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right.... The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it... TERMS are separated by addition and subtraction not multiplication or division... 6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2 FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done... A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it... A=6÷2 = 3 Monomial Factor B= 1 C= 2 6÷2(1+2)= 6÷2×1+6÷2×2= 3×1+3×2= 3+6= 9
As someone doing a masters degree in mathematics, I and most people I know would read that expression it in a way that answered 1. Not because division is some all powerful "everything on the left, over everything on the right" operator. But because the "2y" notation (without the multiplication symbol) generally implies "(2×y)" i.e. bracketed, so long as all other orders "PEDMAS" are satisfied. No one would ever read "sin2y" as "sin(2)×y" because the 2y is implicitly bracketed. This isn't so much of set rule, more a general consensus of an exception. Mathematics is a language, and like any other language, it alters to suit the people who use it and the way they use it. If the equation had been written: 6÷2×(1+2) Then I dont doubt that more people would arrive at 9.
Aidan Retallick said, "No one would ever read "sin2y" as "sin(2)×y" because the 2y is implicitly bracketed" That is precisely what this problem points out and otherwise very intelligent people like yourself have fallen into bad habits. Mathematically there is no difference between 6÷2×(1+2) and 6÷2(1+2) Just because you are reading brackets but not using them in your notation does not make it "correct". Much like language, you can use slang, shorthand or idiom if you choose and are understood but there is a proper way to speak and in math, there is a proper Order of Operations and a proper notation. That is why some very smart people can look at this and get 1, but if we put it into Wolfram or any strong computational algorithm that will follow the Order of Operations precisely as the notation dictates, the answer is 9.
The question is valid and simple arithmetic as taught at school and the answer is correct, The sort of question thousands of kids get every week in tests. I would say that strictly speaking your counter example is not well posed and you have used "lazy" ambiguous notation. You mean Sin(2y) and you should not omit the brackets.
John Silvestri, how are you exactly saying 6÷2(1+2) is the same as 6÷2X(1+2)? Just because some wolfram/Google calculator said so? Neither the PEDMAS nor any other rule of order of precedence explicitly states that ab has the same precedence as a*b. They all state that they are equal, not of the same precedence. Some elementary kids who are still reciting PEDMAS on solving every line of the equation (without realizing why) along with some inconsistent calculators are the reasons this ambiguity exists now. And finally, even if all the masters students in the world are wrong, do you really believe that the solution of some elementary students will be regarded correct that are opposed to the honors/masters students? (I believe otherwise, but that's my opinion) 1. sin 2y is not necessarily a shorthand. It IS legally used for meaning sin (2y). I can give you reference if you ask for 2. PEDMAS certainly doesn't state the precedence of sine, log or any other function. So... how do you exactly solve them by following PEDMAS?
@@InsanityoftheSanitiesthere is no rule in math that says you have to open, clear, remove, take off, eliminate, get rid of or dissolve parentheses. The RULE is to evaluate operations WITHIN the symbol of INCLUSION as a priority and nothing more... (1+2) is a parenthetical priority. 2(3) is not a parenthetical priority and is mathematically the same as 2×3 There is no mathematical difference between 6÷2(1+2) and 6÷2×(1+2) despite the false and misleading information and willful ignorance people have about parenthetical implicit multiplication...
@@DadgeCity The expression 6/2(1+2) will not evaluate to that. You are violating the Distributive Property. In order to fully understand this I will impose a set of parenthesis that does not change the expression. (6/2)(1+2) which will evaluate to (3)(3) = 9 or (3 + 6) = 9. In order for you to have the 6 solely in the numerator and the expression 2(1+2) in the denominator you would have to impose this set of parenthesis which will change the expression 6/(2(1+2)). Then this will evaluate to 1. Therefore (6/2)(1+2) != 6/(2(1+2)) and if you don't believe me put both expressions into a TI Graphing Calculator!
So you're telling everyone that taking a college calculus class that they are utilizing the obelus instead of using a vinculum?? That's kinda hard to believe... The correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly is 9
@@RS-fg5mf look man, this is the way that I was taught since I started learning math as a kid. It’s also the way it’s been taught to me in college. If you take issue with the way I’m being taught, take it up with the faculty, not me. Also, *you’re.
@@wolfwarren6376 yes, a typo. I do it quite often when text swiping. Thank you. So, if you're taught wrong, you just choose to remain wrong instead of doing something about it?? My point was that you're referencing a college calculus class and I'm asking, are they using the obelus instead of a vinculum in a College Calculus class?? I find that hard to believe but I'm waiting on an answer for clarification.... ARE they using an obelus in your college calculus class????
@@RS-fg5mf frankly, they’ve had us evaluate both ways. Early on in the semester, we were tasked with doing algebra and simple precalc problems. Perhaps it was just to get us thinking, since order of operations problems like this haven’t come up recently.
@@wolfwarren6376 let me try and explain something to you... People incorrectly confuse and conflate an ALGEBRAIC CONVENTION given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing... 6/2y = 6/(2y)= 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property. When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses not just the factor next to it. Terms are separated by addition and subtraction not multiplication or division. The TERM or TERM value is attached to and to be multiplied by the parenthetical value of the parentheses.... The TERM value outside the parentheses is 3 The parenthetical value of the parentheses is 3 AND 3×3= 9
I did grow up learning that any number butted up to a parentheses meant immediately multiply that grouping as one value. in computer code you would add outer parentheses where any ambiguity exists.
Atomicninja - It is really 1 the ( ) go first. So it's 1+2 first which equals 3 obviously. Then the equation is 6/2 x 3 (the / is a division symbol). Then you multiply 2 into 3 then it's 6/6 and then your final answer is one. Simple to learn in school easy math.
first off 6/2=3 is already wrong because there is no multiplication 6/2(3) is not the same as 6/2*3 or 6/2*(1+2) if you want to elimate (1+2) the equation should be (6/3) / (2(1+3)/3) then you get 2 / 2 =1 or simply just 6/6=1 The correct answer is 1 because 2(3) is somewhat like y(x) which means the value of y is multiplied by x time.. going that approach 2(3) is interpreted as 2+2+2 = 6 6 / 6 = 1 the algebraic expression is z / y(x)=
graduated in russia in 2010. In US in 2015. In both countries was taught the historical way. I took a few minutes to analyze this problem and I would have never even thought that you could do it any other way so this contemporary way is news to me
I made it through Vector Calculus in the US university system. I was taught the correct “historical” way. That and we never used the dumbass division symbol (which served its purpose); we also used parentheses and entire numerators and denominators. I’ll rest my case with the “new” way is pushed by the same hacks that forced Common Core upon this newest generation. I tutored k-12 math for a time after graduating college, and saw enough of this garbage. You take a subject which students typically struggle with, and make it more difficult, labor intensive and structured in a way that none but their peers or teachers are able to assist them in learning an already arduous subject.
IMO the real answer is "don't write expressions this way." You write expressions so that they can't be easily misinterpreted. If there's any possible confusion, add parentheses to resolve the confusion. This is particularly so when dealing with non-commutative operations such as subtraction and division. This probably comes from a background in computer programming; you don't want to make assumptions about order of operations, because while the people who implement the languages usually have standards to work from, they sometimes screw up. (To make things even more complicated, in computer programs sometimes operations have side effects - like "x++ + x" in C - "x++" evaluates x then increments it, so the value depends on whether the left or right side of the addition is interpreted first. The solution is "don't write it that way.". Eliminates confusion and therefore bugs.)
@@benvergus1573 The fact that it's debated at all indicates that it's found confusing. This whole kerfuffle would not EXIST if people didn't find it confusing. The implicit multiplication when concatenating terms - as is done in this particular problem - complicates matters further. What is 4x/2x ? Left to right you get 4x, divided by 2, multiplied by x, so 2x^2. But I read it habitually as 4x divided by 2x and get 2, and I'm betting that 90% of people trained with strict left to right in evaluating PEMDAS/BODMAS do the same. Writing with clarity in mind, and avoiding any possible confusion, is never a bad thing. Particularly when dealing with a field where precision matters, such as mathematics or engineering.
Wrong. Because in PEMDAS/BODMAS (notice the M and D placement) multiplication and division are on equal level. Write the ÷ in fraction form and it becomes less ambiguous: (6 / 2)(1 + 2) 6 (1 + 2) / 2 6(3) / 2 18 / 2 9 This also works with PEMDAS where × or ÷ is done on a first come first encountered basis from left to right after doing the parenthesis: 6 ÷ 2 (1 + 2) 6 ÷ 2 (3) 3(3) 9
@@_Just_Another_Guy try that with money. Let say you have 6$ in your pocket « a », You want to give it to your 2 friends that are with you that day « b », Each one of them you give 1$ in the left hand « c » and 2$ in the right hand « d ». Answer is how many times can you do that on that day. a ÷ b(c+d) So the answer is?
By the time I reached the level in math where knowing the order of operations became necessary, I don’t think I ever saw division expressed in any way besides numerator and denominator. It’s pretty easy to get confused with this.
@luvlanadelrey no. bodmas is taught to elememtary students. mathematics in higher education no longer uses this symbol ÷ as division is expressed as a numerator and denominator and it is also implied that all values before ÷ sign are numerator and all values after are denominator
@@EvK_27 i did but i kinda forgot the contents already xd so yeah mybad, it does say it in the description however he division symbol ÷ isnt formal so in the end this is a poorly written question like the top commenter said
I got 9 According to my legend "PEMDAS" Edit: many ppl don't know how to pemdas so Lemme say it's more accurate to say "Gemdas" Grouping, Exponents, Multiplication/Division left to right, Addition/Subtraction left to right. Paranthesis in this equation is used as MULTIPLICATION not grouping
@@dnhn.design I multiplied the sum which is 3 and 2 to get 6 then I divided it by 6. I figured if I got 9 that's what everybody got and that was going to be considered the wrong answer so I tried something different.
People who know no maths are making this a viral problem. If you dont believe me than go ask a grade 5 child and the child will tell you the answer is 9. Life is simple. We complicate it
Historical, eh? Hehe, that makes me feel old ;-) I was a school kid in the 1990s, and as I remember it, the rule we were taught for such situations was: "multiply before division".
Same here! I began to wonder if I even could get the correct answer on a simple ecuation now.. 🤔 Have to show this to a teacher in any of our schools here in Sweden! I think none of the would go with the "new" answer that is 9.
Yeah same here.😕 and here I thought I was going CRAZY/Senile and remembering it wrong🤔 either that or I've have travelled to some parallel universe because when I was taught the order of operations in maths...I'm PRETTY sure we called it the "BOMDAS"...NOT 'BODMAS'🤨 which means we'd "multiply before division" AND for me, the "O" stood for "Of" NOT "Order", as in a "Fraction 'OF' something". and I was also told that "Of means to multiply" for example 3/4 of 12, which means you MULTIPLY by the top of the fraction and THEN divide by the bottom SO, if "Of means to multiply" then SURELY it MUST be true, that we ""multiply before division"...RIGHT??🤔🤨😕😵 SERIOUSLY when did it change...and WHY? Im just glad Im NOT the ONLY one who was taught the same way as me, so now Im feeling reassured that Im not misremembering. ...EITHER THAT, or all us 90s kids are just MENTAL lol
I just googled this and double checked problem and double checked with my calculator, and BOTH say the answer is 9😯, and yet everything I was taught tells me that it SHOULD be 1🤨😕 Seriously I have an Advanced Higher in Mathematics and yet according to this...IM WRONG😧 a calculator can NEVER be wrong...so I guess "I" MUST be😨🤯 Now this video has got me questioning what I know and what everything I've learnt😲 What is life? Is this real?😕😵
I graduated in 2020 and I calculated it as 1 with the “old way” which is just how I remember being taught, but realized quickly during the video what the other answer was.
@@unsolve9162 If you were being taught wrong or your teacher wanted to get rid of you then of course you would have passing grades.. Wrong is still wrong, regardless of the reason. The correct answer is 9 not 1
Quick thought experiment: What is (4+2)? Easy, 6. But why? (4+2)= (6)= 6 Here lies the problem. The parenthesis does not simply go away. There is a hidden expression overlooked. (6) actually means (6)= 1(6)= 6 In fact, for any number by itself, you can express it differently. 8=4(2) or 1(8). The parenthesis bounds the expression as a term. Any manipulation (i.e. functions) outside the expression changes the meaning. Regardless, the problem is poorly written to draw up ambiguity.
@@raymendez3403 idk about the west but according to how we read the question 6 is in the numerator while the rest is in the denominator so by simplifying the answer would be 1. Generally the bracket thing has to be done first but in these cases it is so better if they give questions in fractions rather than this confusing method.....
@@jayvishnuvenkatesh870 you need parenthesis to make everything on the right be the denominator. If there is no parenthesis then no priority between multiplication and division hence left to right. There is nothing confusing about this equation only confused people with different answers with all respect. The equation implies there's a parenthesis, no exponents then MD left to right. No need to make parenthesis on the left when it's already left to right
@@raymendez3403 I said idk about the western countries. According to the eastern countries parentheses actually is multiplication and it means the things r together which means in denominator, to differentiate we use the symbol × instead which of course is how u told
Schools were teaching this way (getting the answer 1) far later than 1914 (I graduated in 86), regardless of whatever the text book said. It was never directly addressed/explained, but the correct answer would have been 1 in all classes I took in high school. Without explicit explanation, we just learned to do it that way. Made it through college with a math heavy degree (one class short of a math minor), aced every math class, and never had any trouble using this approach (maybe the instructors tolerated both approaches, or avoided the ambiguity, don't recall?) I've even taught my own kids this, with a smug "trust me I know." Ooops.
They are uhm still teachinf it lmao. In the early 2000s i was taught to follow pemdas and you do every word order first. So if there were multiplication and division left. You would complete multiplication first then complete division and then keep following pemdas until you solve the equation.
You were taught incorrectly. We don't teach it that way, at least not those of us qualified to teach math. You could easily determine that 1 is not a correct answer by using distribution.
@@sirtykai3821 Regardless of the acronym you choose, multiplication and division are Inverse operations with equal priority that are solved in the order in which they appear from left to right. Same with addition and subtraction. There are 4 steps to the order of operations, not 6.
@@jamesfiddler1976 Apparently, but that doesn't help as much as you might think. Do you interpret the expression to mean 6/(2*(1+2)) or 6/2*(1+2)? Distribution doesn't help here, that is the crux of the issue. I agree that the later is simpler and more logically consistent, and apparently the way we should have been interpreting it since 1914, but these sorts of conventions have a way of sticking around. It also strikes me as weird and inelegant, but that's probably my own normalcy bias. There were (and apparently are, based on the comments) teachers teaching it this way for a long time after 1914, I suspect because their teachers taught them that way, and so on going back several generations. If I were to put it into words, without an explicit * operand, the 2 is treated as part of the parenthesized expression. Which is fine, if that's the notation that everyone follows.
20 million views!
Really a good video!
lesgo
It has millions of views because the problem initially looks too simple to have a video (and it is an excellent video).
I wondered if I missed something and chose to watch.
So, the order of operations rules were revised and both 9 and 1 are correct answers.
I thought it was 1 (the algebraic grouping of terms as you noted).
Great to know the rules changed.
Thanks for making the video.
@@idontclickbait8453 It is a good video
Great video, still slightly confused because I am taught that x(y) is one term and should be treated as 1 number but glad to learn that there are 2 different systems
1960 we will have flying cars in the future
2020: world debate over 5th grade math
daniel rushing this is probably 5th or 6th
Ya true but either way
Nope, it's 2nd (in my country)
adomnibest lol I learned it at 5, #homeschooliscool
5th grade too
The correct answer is 9 but the way I was taught math makes me keep saying 1
MOLO 27 yes 🤔
MaxMisterC Both of them state that multiplication and division have the same importance, and are some left to right. Put it in a calculator if you disagree.
@@MaxMisterC Heck it wasn't even that for me. I always just ASSUMED (don't remember if it was actually in the education I got) that anything next to a bracket was in itself inside "invisible" brackets. So if you had 2(1+2), it would simply be read as (2(1+2)) = 6 regardless of what was put in front of it. I guess I never really bothered searching up if this was wrong, OR that my teacher might have been in the same group that still insists on this sort of thinking. Either way the new method (answer of 9) is correct and there just isn't much you can do about it. That's the rule and that's how maths works I guess.
@@berdyie there is no such a rule in Maths.
@@raynatumbeva780 For my incorrect way of thinking or for the correct method in the video?
This is not a math problem... this is a rule problem....
The rules support the correct answer 9
Yeah, technically this rule doesn’t have to be a thing. Just for convenience.
Blubber Beast um it definitely DOES have to be a thing. It’s there for a reason
@@jude3426 it doesn't give priority to multiplication over division...
It is for convenience and less clutter....
@@RS-fg5mf Can we just agree to use the fraction sign when diving? It makes the intended outcome a whole lot clearer
as an engineer who has done advanced university level maths for about 7 years now, I would get 1. its the convention usually followed in physics/engineering textbooks to solve as terms and let implicit multiplication (brackets esp) go first
I'm in a similar position and I 100% agree. It's disingenuous by the video to imply there's one correct order, when so many physics and engineering books do operations in the 2nd way. The video is also wrong in stating that calculators all calculate in the same way. Mine doesn't.
I guarantee that if any engineers I know saw something like "6÷2x" they'd calculate the 2x first. It has nothingo to do with the division symbol. Implied multiplication (for example, 2x rather than 2*x) in all the engineering I've learned always takes priority over normal multiplication. If you write it as 2(1+2) instead of as 2*(1+2) there has to be a reason for it, and common sense (mine at least) dictates it's because you mean the order of operations to be different.
Real world math isn't a puzzle designed by someone to fool you, it's an objective way to state things and should be written accordingly. The problem here is the question, not the answer. Just write it as (6÷2)(1+2) or as a fraction and the problem is solved.
@@afsdfsadhasfh absolutely spot on. couldnt put it better myself
@@afsdfsadhasfh the question is deliberately misleading
@@afsdfsadhasfh Conclusively, the experts say this. The equation is ambiguous and indeed, it can yield two different answers. Like the use of language, to convey something such that it can't be misinterpreted, it must be delivered with clarity, the intention should be made clear. The same with maths equations. To yield only one result, the equation should not be written with ambiguity and the intention of the writer must be clear. If it does or can, the equation should be re-written.
Jesus. This is dangerous. Hopefully you stay in school and don't go actually build something one day.
the correct answer is this is a poorly written problem.
You are 100% correct, this is the proper answer.
What is the correct answer? I don't know man, math isn't typically fully divorced from reality, let's look at the reasons why you're crunching these numbers and we can re-write it so it makes sense!
+James Crawford yeah realistically equations would never be written this way but I think the majority of math rules indicate the answer is nine
In mathematics, there is no such thing as bad problems. Only bad rules and the misuse of good ones.
No, the first premise in PEMDAS, is to solve for the answer within parentheses. You never distribute into parentheses first because you would then misapply the order of operations.
PEMDAS: Parenthesis, Exponents, Multiplication And Division, Addition And Subtraction (IF the same precedence, then left to right).
Any order with And in between has the same precedence!
Since the problem is 6/2*3 or 6/2(3), we must follow the premise regarding left to right because the problem involves only multiplication and division, orders of the same precedence. Parenthesis is only a symbol of multiplication when a number or expression is adjacent to it.
If the problem were 6/(2*3), then the logical answer is 1, because we solve for the answer within parentheses first, as according to the first order of the order of operations.
The answer to 6/2(2+1) is not 1.
We shouldn't change things like the order of operations, it's incredibly dangerous in things like engineering to have two different people unknowingly using two different standards.
Explain that to a teacher. Go ahead.
They only thing common about school now is that every child is getting left behind.
That’s why no mathematically inclined individual worth their salt uses the division symbol.
order of operations never changed, it's always been the same. He just explained that that specific symbol for division meant something very specific other than just division over 100 years ago but the actual order of operations has never changed.
That's why for any serious communication of mathematics you have to be more explicit than this ambiguous problem. Hence why peer-reviewed papers use fractional notation and make copious use of parenthesis to remove ambiguity.
I am 45 years old and have honours degrees in Engineering and Science.
We were always taught that the answer should be 1, because of the order of operations rule that we were taught to use.
If you change the rule, you change the answer.
I was not aware that the rules had changed!
I just graduated high school and my answer to this problem was 9. I guess it's taught correctly now atleast :)
It seems that multiplication by juxtaposition, ab or a(b) etc., may impliy grouping, or it may not, so the notation is ambiguous making both answers valid. It depends on context (e.g. academic or programming).
It's just bad writing.
Modern international standards, ISO-80000-1, mention that brackets are required to remove ambiguity if you use division on one line with multiplication or division directly after it.
The American Mathematical Society's official spokesperson literally says "the way it's written, it's ambiguous" even though they use the explicit interpretation.
Wolfram Alpha's Solidus article mentions this ambiguity also.
Microsoft Math gives both answers.
Many calculators, even from the same manufacturer, don't agree on how to interpret multiplication by juxtaposition. No consensus.
Other references are:
Entry 242 in Florian Cajori's book "A History of Mathematical Notation (1928)" (page 274)
"The American Mathematical Monthly, Vol 24, No. 2 pp 93-95" mentions there was multiplication by juxtaposition ambiguity even in 1917 (and not the ÷ issue)
"Common Core Math For Parents For Dummies" p109-110 addresses this problem, states it is ambiguous.
"Twenty Years Before the Blackboard" (1998) p115 footnote says "note that implied multiplication is done before division".
"Research on technology and teaching and learning of Mathematics: Volume 2: Cases and Perspectives" (2008) p335 mentions about implicit and explicit multiplication and the different interpretations they cause.
Other credible sources are:
- The PEMDAS Paradox (a paper by a PhD student on this ambiguity)
- The Failure of PEMDAS (the writer has a PhD in maths)
- Harvard Math Ambiguity (Cajori's book above is talked about here)
- Berkeley Arithmetic Operations Ambiguity
- PopularMechanics Viral Ambiguity (AMS's statement is here)
- Slate Maths Ambiguity
- Education Week Maths Ambiguity
- The Math Doctors - Implicit Multiplication
- YSU Viral Question (Highly decorated maths professor says it's ambiguous)
- hmmdaily viral maths (Another maths professor says it's ambiguous)
The volume of evidence highly suggests it's ambiguous.
@@bigbadlara5304 The answer is one because this video makes a mistake by ignoring that these equations require the distributive property. If you "just graduated" I'm not at all surprised that no one taught this...
That is correct. This video made a mistake when it ignored the distributive property. The entire problem is wrongly represented here.
@@nixboox Distribution can give both answers as it is a notational ambiguity.
There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not.
I.e. does 2(1+2) = (2×(1+2)) or 2×(1+2)?
Implicit: 6÷(2×(1+2)) = 6÷(2+4) = 1 which is used by academic writing.
Explicit: 6÷2×(1+2) = (6÷2×1 + 6÷2×2) = (3 + 6) = 9 which is used by modern programming and also by the American Mathematical Society according to their statement on the matter.
That's why it's ambiguous. The rules can't help when the problem is the notation which has to be interpreted first. It's just written poorly and not in line with modern international standards.
It should be
(6/2)(1+2) for 9 or
6/(2(1+2)) for 1.
Those are unambiguous and follow the guidelines.
Edit: I was wrong, operator precedence makes the answer clearly 9. A way to avoid this confusion from people like me who got lost in the order of operations would be to set up the equation as (6/2)(1+2) or (6/2) * (1+2).
Note: Contrary to popular belief in this thread, I did graduate with my bachelors and also complete Basic Calculus with high marks. I am capable of error and my original comment was one of those errors. Thank you for the correction.
Original comment:
I graduated with my Bachelors in 2019, the answer according to the way I was taught throughout my education is 1. Because I was instructed by my professors to visualize this problem as 6/(2(1+2)) or 6/6 which equals 1. The person who wrote this did so in a way that is designed, purposefully or ignorantly so, to cause confusion. Dr. Trefor Bazett has an insightful video on this topic
Are you saying that you took university level math within the past 10 years and your professors taught you that in the case of 6➗2(1+2) you’d make 6 the numerator with the 2(1+2) being the denominator? Ima have to throw the bs flag on that one. It doesn’t even make sense that your professors would have even been instructing you on this when this is just basic math that young kids learn. It’d be like saying “When I was pursuing my master’s degree and my professor was teaching me my times tables…”
If you took this stuff recently, you’d have been taught to solve left to right 6/2x3
=3x3
=9
Dr. Trevor Bassett is wrong and so are you...
6
------(1+2)= 6÷2(1+2)= 9
2
6
---------- = 6÷(2(1+2))= 1
2(1+2)
The vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator.
________
2(1+2) = (2(1+2))
Two grouping symbols each
________
2(1+2) has two grouping symbols
(2(1+2)) has two grouping symbols
@@trickortrump3292the bigger question would be why a University would be using the grade school obelus to teach higher level math...
We have reviewed the video and the penalty flag stands... Good call Ref....LOL
@@RS-fg5mf Yeah I deserved that. When I first looked at it, I solved it your way and then the video told me I was wrong. I bought into the reasoning for why I was wrong. This question is just a mess! I went down the rabbit hole yesterday after my comment. It’s insane to me that so many experts seem to say that the right answer is “there is no right answer” because it can be correctly solved two different ways, yielding two different answers. I can’t accept that. If both answers are correct, that makes both answers wrong too.
I’ve removed the bs flag I originally threw. 👍😉
@@trickortrump3292 don't remove it. LOL The red flag stands on the play because you are absolutely correct...
The only correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended is 9
I was agreeing with you. Don't let these mathematical numpties change your mind.
Those who understand and apply the basic rules and principles of math correctly as intended will get the correct answer 9
Those who fail to understand and apply the basic rules and principles of math correctly as intended will get the wrong answer 1
Those who can't prove 1 and can't accept 9 will argue ambiguity...
Failure to understand and apply the basic rules and principles of math correctly as intended doesn't make the expression ambiguous and isn't a valid argument against the expression...
That's why nobody uses the division symbol. It's confusing and it leads to errors. Both in math and physics, fractions are the way to go.
I prefer using ÷ over /.
I only use / with fractions, but use ÷ when dividing numbers.
Using improper fractions instead of using the division symbol is something that I rarely ever do.
I never find it confusing when using ÷, and it never confuses me.
@@MarioLandscape nope
@@MarkQub.
What do you mean 'nope'?
I just stated that I prefer using this: *÷* of this: */,* when dividing.
The person said that nobody uses ÷, because it's confusing, so I said that I do use ÷, and that it doesn't confuse me.
@@MarioLandscape you see how you lost. You got mad/serious over someone who just simply said “nope”
@@MarkQub.
What on Earth?
Lost what?
What did I lose? Please explain.
All I did was ask you what you meant by nope.
How was that getting mad?
think of it like a fraction. there's a reason why in higher math '÷' isn't used.
dont forget that 2(3) is one term
I know right... I'm suprised the education system is failing this hard to teach math.
but it's not a fraction. there is the discrepancy in the solution
all division is a fraction lol
Exactly Erick!! we use the division sign just to teach kids but it is wrong in advanced math.
The programmer's wife sends him to the store. She says "Get one carton of milk, and if they have eggs, get a dozen". The programmer came home with 12 cartons of milk, because they did have eggs.
And that's why I used to hate computers so much.
It would be 13 cartons because of the and boolean logic instead of or.
I'm truly thankful for the opportunity to give thumbs-up #42.
tf
Agreement with Brian Fedelin. 13 cartons of milk. One carton, and if there are eggs, get a dozen. So 1 + 12 = 13. And the issue with computers is not the logic of them, it is how a human evaluates a human expression and then programs the computer. In this case, the issue was with the wife, since the expression was not clearly defined from the start by defining a dozen of WHAT was desired, the milk or the eggs. See, it is actually a trap by wife against the husband. No matter what he were to bring home, it would be incorrect since she could then change WHAT was the dozen to be of.
I'm pretty sure you missed a different confusion. I get that some would interpret the division sign as you did, but there is also the belief that implied multiplication has priority over other division and multiplication, because it was implied, it has to be resolved. You can't just change 2(3) into 2x3, because they are bound. 2x(1+2) does not just equal 2(1+2), because 2(1+2) = (2(1+2)). I realize it doesn't make a difference until (÷) gets put in front of them.
Say we wanted to divide energy by 12. Would we write 12 ÷ mc^2 or would we write 12 ÷ (mc^2). We all recognize mc^2 as energy, m and c^2 are bound by implied multiplication. a completely different thing that 12 ÷ m x c^2. Or divide 12 by the area of a circle: 12 ÷ πr^2. π x r^2 is implied and therefore bound.
this is the real argument implied multiplication has priority of not.
Exactly. 6 ÷ 2 x (1+2) isn't the same as 6÷2(1+2). It's like 6 ÷ 2x, where x= 1+2
Yup all other videos on pemdas of any person in STEM agree that it's one. No person in STEM would say the answer is 9. Although almost all of them prefer fraction bars over the division symbol.
Just graduated and I was legitimately taught that "historical way" all through school
Same here
I graduated high school 15-16 yrs ago and got my associate degree in '08, that's how I was taught, the old way.
That's terrible.
Ditto, and I aced math in school
Old technique is correct because
a=6/2(1+2)is not equal to.
b=6/2*(1+2)
In this video he is using second technique
No doubt 1 answer
As a math student, I’m mad at the way this is written. My teacher said he would fail anyone who wrote math problems like this 😂😂😂
Your math teacher has issues but as long as he is grading you I suppose you need to do what is expected...
There is nothing wrong with the way the expression is written just the ignorance people have about parenthetical implicit multiplication...
@@RS-fg5mf Isn't that the whole point - it is perfectly valid but makes it unclear and you have to think about it - are you mad because you got it wrong? I would be a bit concerned about your math's teacher.
@@justcheck6645 I am a math teacher and I didn't get it wrong. LMAO
When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly you get the correct answer 9....
Did you get it wrong??
@Christopher Butler if you don't care, why bother replying?? LOL
Enjoy your show. 😁😁
I graduated math in 80s, cannot believe people are discussing primary school math
The problem isint the equation itself, it's whoever wrote it.
No the problem is teachers changing they way they teach things.
@@MrGamecatCanaveral
Not teacher the education system.
@@babyyoda7749 no teachers are the ones. Teachers are teaching about gender and sexuality for example. The education system is not telling them to teach that.
@@MrGamecatCanaveral Teachers change the way they teach things because we discover new things over time. Before, it was thought that the earth is the center of the solar system. Copernicus discovered that it is actually the sun. Now, do we need to change the way we teach about the solar system? Yes. It's crazy that what we believe in the present will never be entirely 'true' as it could be proven false in the future.
@@aeroljameslita4975 from a subjective point of view, isn't the point of perception the center of your reality? So the Earth is the center of the universe for everyone on it.?
Some people were taught that multiplication by juxtaposition takes precedence over explicit operations… hence why 3/2n is 3/(2n) and not (3/2)n
The same juxtaposition glue applies to parenthetical coefficients… and in this case, 2 is that parenthetical coefficient. So using PEMDAS, but assigning multiplication by juxtaposition a higher priority than explicit division, the answer is 1. Additionally, if you use the distributive property from the get-go to resolve the parentheses, you get 1.
Nah it should be 1.5n
It's just in computer writing, it's hard to position the number & variable
3/2n = 1.5n
not
3
_
2n
@@coreyramstein9778
3/2n = (3/2)(n)
3 ÷ 2n = 3 ÷ (2n)
As you climb higher in math, virtually 100% of physicists, engineers and mathematicians will interpret the answer as 1. There is no debate over this at all. The implicit multiplication of 2 on the bracket is a SINGLE quantity that takes precedence prior to division. Most physicists/engineers/mathematicians would never even write such a potentially ambiguous expression. They would instead write 6/2(1+2) where the / is a horizontal line. Alternatively, they would write 6/(2(1+2)) leaving NO ROOM FOR AMBIGUITY. PEMDAS is NOT universally accepted. The implicit multiplication on the bracket does indeed take precedent.
You are doing a disservice to kids trying to learn mathematical protocols. PEMDAS isn't the total protocol.
Totally wrong and nonsense.
What is the inverse of "implicit multiplication"?
I agree with ZoeCat
It is really funny indeed, because it gives a hint from "where people are coming". I studied physics for some time and it was completely obvious to me, that a juxtaposition has a higher order than "read left to right". It's "obviously" 1.
As mentioned; 6/2y with y=1+2 is 3/y, not 3y.
Tom Yes. You give a great example.
According to PEMDAS, x/yz = xz/y which is OBVIOUSLY unconventional. The implied multiplication of yz binds the two components of 'y' and 'z' together.
Zoe TheCat Agree with you 100%. I'm an engineer and my first response was to say the answer is 1. Everything on the right is just a factored 6.
I solved this in 5 seconds this shouldn't be a problem for anyone who attended school.
Same
Exactly, I always loved doing really long order of operations math problems.
Same, idk why they are making fuss over this? I mean this is taught in school....
@@_mahiii lol he got 14 million views it served its purpose
Unless your History teacher doubled as your Maths teacher!
After all this debate and discussion, I think we can all agree that this is why we use fractions instead.
Eventually, yes. This is a fifth-grade expression used to teach and reinforce the order of operations. This is pretty much ground zero. From there, we stop using the obelus in favor of the solidus and vinculum and go into fractions, as well as teaching reciprocals and the multiplicative inverse. People just forget how to evaluate expressions using the order of operations due to lack of practice. Sometimes, all they remember is an acronym and then convince themselves that there are six steps instead of four and that multiplication always comes first when it doesn't.
All my homies hate ÷
@@jamesfiddler1976 how the hell do you not use order of operations im highschool? You need to use them for literally any equation
@@godlikefish1193 - Me too my friend. All forms.
@@pirilon78 Who says I did? I never even hinted that we don't use the order of operations beyond junior high. It should be common knowledge that we do.
After learning calculus, this answer is 1. Visualize the division line, 6 is the numerator, the 2(1+2) is the denominator. From there solve the denominator however you want, you’ll end up with 6. 6/6 = 1
Incorrect, and this has nothing to do with calculus, it's fundamental algebra. Division is division, not an implied fraction. If anything, it's the other way around: a fraction is implied division
If you write it as algebraic equation you can clearly see how it’s supposed to be done. X/Y(A+B) = X/(YA+YB), since you need to distribute property of Y among entire parenthesis first, and fully evaluate that before going back to division of X.
Using numbers it’s:
6/2(1+2) = 6 / (2*1 + 2*2) = 6/6 = 1
This is how math works. People outside of America aren’t thought any of PEDMAS, BODMAS or whatever bdsm acronym is used. People are thought how order of operations works in practice, often explained by definitions, and orders, and with a help of algebraic equations, since when you remove numbers it’s clearer to see how things are evaluated.
@@admiralvirhz Incorrect. The distributing property is multiplication, which has no precedence over division. It would be wrong to distribute 2(1+2) before doing 6/2. The first operation would leave you with 3(1+2) and then you can distribute to get 3+6=9
@@paulblart7378you’re making logical error here. Multiplication doesn’t have priority over division, you’re right about this and it’s set in stone, but to fully value what’s inside parenthesis you need to distribute 2 over it. There’s no sign of multiplication, so you need to understand that it is
6 divided by double parenthesis.
You see your logical mistake here? It’s not 2 multiply parenthesis since there’s no multiplication sign. It’s double parenthesis.
It’s really bad written problem to deal with, I no wonder why so many people get this wrong.
@@admiralvirhz It's an implicit multiplication. It can be rewritten as 6/2*(1+2), the fact that there isn't an explicit sign doesn't change the problem. I don't know what you mean by "6 divided by double parenthesis", but there is no rule that implicit multiplication groups the operands together. You would do 6/2 first, then multiply that by (1+2)
As a trained engineer in his forties, I immediatey turned the expression into a fraction. I also have to say I don’t think I’ve ever seen that division sign used anywhere after fourth or fifth grade.
And in 4th or 5th grade arithmetic the correct answer is 9 .... The symbol is found on almost any calculator. Best to understand it than to be confused by it...
Funny, I just left a similar comment. I’m an engineer (39 yrs old) and did same as you. That’s the reason engineers and physicists don’t use that silly division symbol.
@@Superdada i don't understand the debate about the division symbol. what difference does it make whether you use : or / ?
they do mean the same, don't they?
@@alxlej They mean using fractions instead of a symbol and having everything next to eachother.
@@borismuller1086 but the meaning and therefore the resulting operation are still the same, aren't they?
what am i missing?
I'm 40 y/o and was taught the historical way in school. I don't feel historical though. I feel f*cked over because somewhere along the line people decided to change the rules of the game (and didn't inform me!!)
Good ol' "Meneer Van Dalen Wacht Op Antwoord" for the Dutch viewers...
@@manofculture9051 Your mom sends her regards! And dinner's at six, be on time please.
The problem isn't that the rules were changed; the problem is that they are being misapplied.
I hate order of operation squabbles. That is not math, it is convention. If there is a governing body for math they should get together and design a convention that is definite, obvious, and universally agreed upon and taught. I was taught the historical method, but knew the current method, so I knew there were two possible answers depending on which system you used. (Not counting the latest anti-racist belief that every answer is correct because saying there is a definite answer would be racist.)
@@HQBergeron Agreed. It's mathematical semantics.
#planetpluto
In France i've been taught it in a way, that this equation equals 1.
Basically 6/2(1+2) has brackets. We were taught that brackets were always a priority with the number infront of it. So what we would do is first 2*1 + 2*2 = 6, and once we got the brackets completely gone, we can finish the equation which would be 6/6 = 1.
Also even if i added the numbers, it was always important to clear the brackets. Here 6/2(3) still has a bracket and doesnt just dissapear. So i would multiply 2 and 3 to get rid of the bracket. Thus we still receive 6/6 = 1
I was always taught this way and was surprised seeing that the correct answer was 9. This blew my mind
I'm pretty sure that in germany we were taught the second answer as well (equasion equaling 1) for the exact same reason you describe here (getting rid of the brackets first) and then finally dividing anything on the left side by what is left on the right side. From my point of view the answer 9 is "wrong". And even if it's just a "rule" thing, we'd better universalise that rule. To me, somehow, the answer "1" also makes more sense in a mathematical- asthethical way.
People in Europe, born before 1970, learned, that multiplication goes before division. Just a fact. I mentioned 1917, because in that year, in the USA it became official that multiplication and division are equal and You start from left to right. In 1980 is was commpn practice all around the globe. ( In the Netherlands it took till 1992 to use the 1917-method).
Mathematics is about agreements and those changed over the years to an (new) international standard...
@@j.r.arnolli9734 Thank you for this insight. Anyhow I was born in 1980 and I'm pretty sure that if I showed this "problem" to my old schoolmates/ peers here in germany 99% would come up with the anwer "1". Yet again maybe I'm wrong.
If this really is new international standard it still doesn't make a whole lot of sense to me in terms of logical usage of mathematical language.
Yes you are correct the answer is 1. You solve the brackets first to get a number on its own then you finish off by 6 ÷ the answer in the brackets.
If your answer is 9 then you are inventing your own mathematics !
I was taught that way also. In the US, but three generations ago. I’m old.
Clearing the parens is not simply performing the operation within but also performing the operation dictated by the parens. Therefore the operation requires multiplying 2x3 to get 6 prior to the next operation. If the equation was: 6 divided by 2y there would be no ambiguity that it would be 6/(2y)not (6/2) x3.
Nope, if you got 6÷2y you do 6/2 times y. Its just the current rules, i agree its weird and maybe confusing because we never use "÷", we always use fractions, but the rules are the rules and they say that if theres no parenthesis, you only divide by the first number, the closer to the "÷" symbol. Which is 2, therefore 6/2 × 3 = 9
The parentheses are just around 3 though, not 2*3
@@wadabid6165: 2y is grouped.
Just like 2π, 2π, or 24.
You are using PEJMDAS like in some calculators (not all of them).
J meaning Juxtaposition.
But this is not PEMDAS which is the official math rule for instance in USA.
@@whoff59: PEMDAS is not an official rule anywhere. It's not even a rule.
I did this in 10 seconds... how can it be viral?
Check Wiki on the order of operation, it is indication that there is an ambiguity/confusion with expression like 1/2x
for some it is (1/2)*x = x/2 and for other it is 1/(2*x)
Here we have the same type of problem : a/bc, so same problem : is it (a/b)*c or a/(b*c)
If for you it is not confusing, then you do not know math enough, because to remove the confusion in that sort of expression, there is a rule that apply to in-line math expression :
"Always add parentheses to delineate compound denominator"
So here the first thing to say is that "that expression do not follow the rule for in-line math, so It can't be solved using the order of operation; It has to be corrected first"
And the problem is that it seems that a lot of people do not know that rule, so they give the result corresponding to one interpretation or the other ... making it viral
Should all of those people go back to school ?
Or should only the one that wrote that ambiguous expression go back to school ?
@@ghislainmaury2065 English translation please?
This is so easy, I solved it in 5 seconds
Defaulty Boi lol.
Letucces Satan yeah... I’ll pass thanks
I’d gladly take yours though
Its gonna go viral again because its in my recommended
6 / (2*1 + 2*2) = 1
@@miguelgm808 did u even watch the video?
Ya
Nobody care
@@miguelgm808
(4/2+2/2)(3) = 9
I used the historical version all my life. I must be rather ancient.
Seems we all historical and the new version only rules in special areas, clearly the areas where I’m not. I live in South Africa and here the answer is still 1🤣 should you want the answer to be 9 it would be written as a fraction not a division sign(which can’t even be found on my keyboard, so let’s just all retire the devision symbol and I’d be happy to concede that the answer is 9😂
Basically the answer isn't "wrong" if you use the historical version... they're just asking different things... in modern math, it you wanted to ask the exact same question as the historical you would have to write is 6÷[2(1+2]
@@willwalker24601 It comes down to "just use brackets to make clear what you mean". Mathematics is supposed to be a universal language, but there are still a lot of dialects, aka different notations. I see that a lot lately as I am german but using english youtube videos to review some things since I am studying for a new profession. They are doing a lot of things differently than I learned them at school 20 years ago. Maybe they do them that way in schools now too, I don't know. But since such differences exist, one should strive to write expressions as clearly and unambiguously as possible. Most of those "puzzles" thrive on their ambiguouty.
I think the difference is people forget about the brackets so they just disappear
✔️✔️✔️👍👍
Correct answer is surely 1
To those who are telling it 9
Dont know how?
For this xy ÷ xy = 1
But Its not y²(according to those who are telling answer to be 9)
Similarly, 6÷2(3)=6/(2*3)=1
As simple as that...
The answer is 1. Especially since the distributive property applies to the expression in parenthesis. 2(1+2) = (1x2+2x2) = (2+4) = 6
The Distributive Property supports 9 not 1
The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division.
The Distributive Property is congruent with the Order of Operations it doesn't supercede the Order of Operations... The Order of Operations work because of the Properties and Axioms of math not in spite of them...
The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in... If you can't draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven't applied the Distributive Property correctly...
The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication...
The axiom a(b+c)= ab+ac however the variable "a" represents the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that "a" is a single numeral...
6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property
Parentheses removed...
6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property.
Inner parentheses REMOVED
This can be further demonstrated using the vinculum....
6
------(1+2)= 6÷2(1+2)= 9
2
6
------------ = 6÷(2(1+2))= 1
2(1+2)
A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator....
________
2(1+2) = (2(1+2)) two grouping symbols each
That over bar (vinculum) is a grouping symbol
_______ _________
2(1+2) = 2×1+2×2 one grouping symbol each
Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation...
It’s 9.. how is this even viral, it’s 5th grade math..
Also, I’m referring to PEMDAS which is taught in 5th grade. Watch the video if the answer you got wasn’t 9..
It's 1
It’s 7
Noob, its one
Not even 5th grade math. It’s like 2nd grade math
It depends on pemdas or bodmas
oh sorry i’m late, UA-cam just recommended me this video 3 YEARS LATER
chris yeah me too
same
isnt it 2 years
Onur Akar technically, yes
Me2
Interpretation is the key word. The problem should, in my opinion, always be written as: (6÷2)(1+2) so there is no more confusion on interpretation
But if this is on a test, you want to know that your students actually paid attention and learned correctly, writing the way you did removes the so called "ambiguity" (which there is none) and then there will be no way to actually know if they have learned correctly
When you actually understand and apply the Order of Operations and the various properties and axioms of math correctly as intended you get the only correct answer 9
If you don't apply the basic rules and principles of math correctly then you are already confused.
@@RS-fg5mf the answer is 1…
@@lolmom3590 BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations
6÷2(1+2)=
6÷2(3)=
3(3)=
9
2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside ....
There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more.
Commutative Property
6÷2(1+2)=
6(1+2)÷2=
6(3)÷2=
18÷2=
9
Distributive Property
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction.
6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2
Operational inverse of division by the reciprocal
6÷2(1+2)
6(1/2)(1+2)=
6(1/2)(3)=?
Multiply in any order you want you still get 9
Proper use of grouping symbols
6
-----(1+2) = 6÷2(1+2)=9
2
6
-------- = 6÷(2(1+2))=1
2(1+2)
A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in a linear format extra brackets are required to maintain the grouping of operations within the denominator...
Another argument people tend to use incorrectly is factoring....
6 = 2+4 No parentheses required BUT
6÷(2+4) parentheses required
2+4= 2(1+2) only one set of parentheses required.
6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set...
The 2(1+2) must be placed within the first set of parentheses containing the (2+4)
6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2)
Let y = (1/2)
6y(1+2)=?
6y*1+6y*2= ?
6/y⁻¹*1+6/y⁻¹*2= ?
If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9
The rules of math have to remain logical and consistent across the board...
THESE ARE THE FACTS....
@@jonnel4038 x÷2y = x÷(2y) by Algebraic Convention... BUT x÷2(y)= x÷2*y by the Distributive Property...
Parenthetical implicit multiplication does not have priority over division. When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses not just the number next to it.
The correct answer is 9
Hi Zak,
I'm starting to think that algebraic characters are simpler to process with than math with arithmetic (numeric) characters. (Actually, the order of processing should be the same for both worlds but alphabetic or algebraic characters are a bit more limiting. But that's just my opinion.)
Oh, well... I've made some clarifications in my own mind. That's good enough for me.
Also, the medical world has given me a wake-up call regarding my expiration date. The range where it may reside is no longer unlimited. But I have a number of years left to ponder challenges like this. It's fun and interesting. Even history is interesting now; not like it was back in high school. I started liking history when Glen Beck brought out his chalk boards on his Fox News channel program years ago. Now, I'm moving on to photography. I don't think I'll ever fully understand analytic geometry and calculus. Sigh... Too bad we can't start over, isn't it? Oops! I'm rambling...
Happy trails,
Wes
Enjoy your day Wes ! Yes it is good to keep the old noggin a tickin.
Next viral problem..
1+1 = 2 or 11.. 🤔🤔🤔
1+1 = 10 ;) 1+1+1 = 11, 1+1+1+1 = 100.
@Jure Lukezic binary smh
@Jure Lukezic That only works for very large values for 0. I was representing numbers in base-2; however, if we're talking string concatenation then yaaaaaaaaassssss!!!
@Jure Lukezic So how does it feel that your joke went over our heads? Don't you feel bad for us smug little pedantic bastards? We could have strung that out, like "I was writing in binary" ... "no you weren't" ... "yes I was" ... "no" ...
10
dont worry, this issue will never show up in important engineering situations because the division symbol would never be used. instead using a fraction would make everything a lot more clear
Yep i remeber i got so used to fraction that when i saw the division simbol at first i thought it was percentage xD
100%
The real-life solution, as per the ISO recommendation, is just to use brackets to disambiguate. (6/2)(1+2) is totally clear regardless of division symbol used and works for handwriting, calculators, typed documents etc.
@@anonymes2884 100% and ISO-80000-2 says that ÷ should no longer be used also.
I don't know algebra
in short don’t use the outdated division symbol, just use the typical numerator and denominator removes all uncertainty
It still becomes 6/2*3. So if it’s multiplication before division that’s 6/6 =1
@@charliedallachie3539 thats not using the numerator and denominator, when you use an actual numerator or denominator you would have a certain part be under it. Either 6/(2*3) or (6/2)*3
@@o_sch yea I understand the two answers but in other problems which is which? I’ve always wondered PEMDAS in general I’m sure there’s a complex mathematical proof of it out there somewhere
Edit* there is no proof it’s a convention.
@@charliedallachie3539 but it isn't multiplication before division. they are equal, so it is left to right.
@@JakobSchade Sure, but that's if you use PEMDAS or whatever else. There's still plenty of books where they don't use PEMDAS and have a difference between implicit and explicit multiplication. 2*3 is explicit (a * sign) and 2(3) is implicit. In that case, implicit is many times higher of importance than explicit. So 6/2(1+2) would simply be 6/6=1.
`Hi Zak, Thank you for you contributions here. I'm sure other viewers appreciate them too.
I also wonder if anyone else found the TableClass Math post to which I tried to point them.
Happy trails,
Wes
I have not located you on there yet, did you start the thread and when ? I should be able to have a look.
@@zakelwe Hi Zak No, the thread was started by one of the regular instructors, John of TabletClass Math.I think the post was a 're-post' but it was recent. He made another post about putting parenthesis around each of the numerator and denominator in a post today; but I didn't note it.
@@wes4139 Let me have a wander over. Wish me luck.
PS Nice to chat to someone who is civil , it can get quite heated on these. These types of problem have been occurring since 2011 and what happens is that
50% of people get one answer
50% get the other.
Two tribes form.
Both tribes throw arrows at each other.
The current state of teaching gets blamed 😀
The fact it has been going on so long and educated people differ roughly equally shows there is an underlying problem, which of course there is.
Was it this one from about 4 days back?
Why don't the numerator and denominator have separate groupings? Isn't 6 ÷ 2(3) = 6/2(3), or {6} ÷ {2(3)} implied? I thought numerators and denominators had their own groupings... The answer on my calculator = 1. Or, how about this --> 6 / 2(3) = 2x3/2(3)? Don't the 2's and 3's cancel out?
Ok 6 ÷ 2(3) = 6/2(3) but that does not equal {6} ÷ {2(3)) The former two are both equal to 6 ÷ 2 x 3 where as the latter is 6 ÷ (2 x 3). The first two are ambiguous, you can do the M or D first, the latter is a well written equation, you have to do the 2x3 first so no issue, answer is 1.
In effect 6 ÷ 2 x 3 written on two lines can be done as
6 6
-------- or ----- x 3 both meanings are clear on what the person writing it wants you to do first. However with 6 / 2 x 3 that meaning has been lost.
2 x 3 2
To do both options above on one line by using brackets correctly then we have 6 / ( 2 x 3 ) or (6/2) x 3 respectively. Most people would be able to work it out easily and there is no ambiguity.
This was noted back in the 1920s when text books did it both ways and the Committee on the Teaching of Arithmetic in Public Schools was called to adjudicate on the matter. They said "(the) committee recommends the use of brackets to avoid ambiguity in such cases.
And 100 years later .....
@@zakelwe This is another try at getting the message to you
4÷8 1) 1 The explanation I want seen is at 11:24 in...
___ 2) ¼ He explains the Parenthesis around each of
2x4 3) 1/16 the numerator and the denominator.
4) 4
The issue is, I agree that with the same precedence you go left to right so if it said 6 ÷ 2 × 3 I would correctly answer that as 9. However by wording it as 6 ÷ 2(1 + 2), my mind goes to expand the bracket first which gives 6 ÷ 6 = 1.
This. I was taught (in the US) completing the parentheses/brackets meant you did all involved with the parentheses/brackets. Here, the parenthesis is what symbolizes the 2x3 so you still do that before the division.
@@timelyspirit I was taught the same thing
Inside the parenthesis, outside the parentheses, then L to R.
The rule is called BODMAS or BIDMAS
It is the order of what you do first
Brackets
Indices (or other)
Division & Multiplication
Addition & Subtraction
So here first we do the brackets
6 ÷ 2 (1+2)
6÷ 2 (3)
6 ÷ 2*3
Next we do division
6÷2*3
3*3
Next we do multiplication
3*3
9
@@WokeVeganLiberal wait wasn’t it pemdas?
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
So, the problem that yielded an answer of 1 in 1917 yields 9 today. Wow - inflation is everywhere!
Trevor Keen You are correct. I was taught to do the the parenthetical expression, multiply and then divide.
The world parenthesis should be eliminated from all human language
Trevor Keen j
you were taught wrong. the end.
It's because the creator of this video is a moron
I graduated in 2006. I also got the answer of 1. The "historical" version of the order of operations was still being taught 80 years after 1917.
6÷2+4 ???
@@RS-fg5mf Where the HELL did you get THOSE numbers?! PEMDAS(or a few other things that say the same basic thing): Parentheses, 1+2=3; Exponents, there are NONE so we move on; Multiplication and Division DO THEM AS YOU FIND THEM STARTING AT THE VERY BEGINNING(from left to right, you know the way we read things in most cultures!), 6/2(The first one you find when you start at the beginning) is 3, then we have *3, so 3*3 that's NINE!
You didn't even come up with 1 which was my first thought but then I realized that I was wrong and redid the problem, and got NINE!
If you got the answer 1 then you were taught WRONG, and as my father says, "You should go back and slap your teachers!"! The correct answer is NOT 1!
@@JacksonOwex I never said the correct answer was 1. I absolutely understand the correct answer is 9
I get pissed off when people say they were taught the historical method when they fail to even understand the context of this video and what the historical method was...
The historical method was a misuse of the obelus by some text book printing companies who pushed the use of the obelus in a manner similar to the vinculum because the vinculum took up too much vertical page space, was difficult to type set and more costly to print with the printing methods at that time. However, this was in direct conflict with the Order of Operations and the various properties and axioms of math so the ERROR was corrected post 1917...
This ERROR means that 1 is not and has never been the correct answer. BUT this ERROR i.e. method of using the obelus would have made 6÷2+4=1 by this incorrect use of the obelus... So when someone says they were taught the historical method I them what 6÷2+4 is equal to and when they answer 7 that's proof that they were not taught the historical method mentioned in this video...
The real confusion is the false and misleading information and willful ignorance people have about parenthetical implicit multiplication. They incorrectly believe that 2(3) is a parenthetical priority and that the implicit multiplication gives it priority over the division which is FALSE.
@@JacksonOwex That wasn't how we were taught. I also graduated in 2006, and we were taught PEMDAS and to do them precisely in that order. My teacher never told us that Multiplication and Division were on the same level and Addition and Subtraction were on the same level. We were taught to multiply first and then divide. I only recently discovered that I was taught incorrectly. By reading through the comments, I realize that I am not alone and many of us were taught incorrectly.
I have an issue with this. I fully understand it, but hear me out really quick.
Lets say :
n = 1 + 2
6 / 2n = x
Solve for x.
This is the same as the original equation in the video:
6 / 2(1+2)
Essentially "2n" is just "2 x n". So essentially the equation becomes:
6 / 2 * n = x
But we wouldn't solve it as:
( 6 / 2 ) * n = x
We would solve it as:
6 / (2 * n) = x
Now we could probably agree that "2n" isn't just "2 x n", but instead "(2 x n)".
Which would always make the equation:
6 / (2 * n) = x
Which sort of leads to my issue, the way a calculator would solve the original question:
6 / 2(3) = (6 / 2) * 3
It does make logical sense, but seems counter-intuitive to me.
The following equation:
6 / 2(3) = 6 / (2 * 3)
Seems to make more reasonable sense to me, based on principles of alegebra. Also, in a weird way it wouldn't violate our current order of operations.
The equation:
2(3)
Could still be looked at as a paranthesis operation, instead of just multiplication, making its order take precedence.
As soon as you write 6 / 2n you are introducing a notational convention that is not present in 6/2(1+2). Namely, you are introducing the algebraic convention that "2n" should be treated like "(2n)". Another way to say it is that a coefficient-with-variable pair is considered grouped. It appears to me that this convention is universally accepted. Therefore, 6 / 2n can be replaced by 6 / (2n) but not by 6 / 2 * n. The latter introduces an error.
The same notational convention does not apply to 6 / 2(3). That is, 6 / 2(3) can not be replaced by 6 / (2(3)).
However, not everyone agrees. Some people say 6 / 2(3) SHOULD be replace by 6 / (2(3)). Other people say 6 / 2(3) SHOULD be replaced by 6 / 2 * 3, Then, 6 / (2(3)) = 1, whereas 6 / 2 *3 = 9
The debate is sometimes fierce, but it has not been settled in over 10 years of UA-cam debates. I would not expect it to be settled anytime soon!
6/2n = 3/n by Algebraic Convention
6/2(n)= 3n by the Distributive Property
All variables have a coefficient written or not. Constants can be coefficients but constants do not have coefficients.
x/x = 1x/1x = 1
x/1(x)= 1x/1(1x)= x^2
x/x(x)= 1x/1x(1x)= x
@@RS-fg5mf
6/2n = 3/n, yes, by the convention I described, namely that "2n" is treated as if it were "(2n)". Call it "Algebric Convention" if you want, no problem. But it is, as you say, a convention, not a rule or law. It's a convention of how to interpret the notation "2n".
6/2(n) = 3n, I'd be inclined to agree. But I bet a lot of UA-cam commenters would disagree. I think it suffers from the same lack of consensus as 6/2(3).
If 6/2(n) = 3n ia true, it's absolutely NOT by the Distributive Property. The distributive property is a relationship between multiplication and addition; it refers to distributing multiplication across (or over) addition. There is no addition in 6/2(n) so the distributive property does not apply.
The notion that all variables have a coefficient written or not is just plain silly. It introduces an over-intellectual complication into the notation and evaluation of expressions. For example, a simpler and more direct evaluation of x/1(x) is
x/1(x) = x(x) = x^2,
for those who do not believe that the so-called "implicit multiplication" is special and thus do not believe that the multiplication is done first in x/1(x), Replacing x by 1x is pointless and unnecessary.
@@donmacqueen just because you fail to understand it doesn't make it pointless and isn't a valid argument against it....
6/2(a+b)= 6/2×a+6/2×b
6/2(a+b)= 6/2(c)= 6/2*c
The point you're missing is that the Distributive Property is the application of MULTIPLING the term outside the parentheses across each TERM inside the parentheses....
6÷2×3 is no different mathematically as 6÷2×1+6÷2×2= 6÷2(1+2) ...
You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol...
6
------------ = 6÷(2(1+2))
2(1+2) two grouping symbols each
6. 6
------------ = -------------- one grouping symbol
2(1+2) 2×1+2×2. Removed...
6÷(2(1+2))= 6÷(2×1+2×2) one grouping symbol removed...
The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division.... 6÷2(3)= 6÷2×3 the same way as 6÷2(1+2)= 6÷2×1+6÷2×2
6/2(n) the n can represent a single value or a set of operations that represent a single value...
Let's take 6/2(a+b) and let a=3×4 and b= 2×5
Then 6/2(3×4+2×5)= 6/2(12+10)= 6/2(22)= 3(22)= 66 OR....
6/2(3×4+2×5)=
6/2×3×4+6/2×2×5=
3×3×4+3×2×5=
9×4+6×5=
36+30=
66
Would you disagree that 3(3×4+2×5) is the same as 3×3×4+3×2×5 ?? The Distributive Property is explained as Multiplication over Addition but technically it is the Distribution of the TERM outside the parentheses across one or more TERMS inside the parentheses, is a much better description. As TERMS are separated by addition and subtraction....
@@RS-fg5mf You are replying to my comment in which I said,
"The distributive property is a relationship between multiplication and addition; it refers to distributing multiplication across (or over) addition. "
In your reply you said,
"... the Distributive Property is the application of MULTIPLING the term outside the parentheses across each TERM inside the parentheses.... "
These two statements are describing the same thing. Your suggestion that I don't understand distribution, is badly mistaken.
Oh, and when you state,
6/2(a+b)= 6/2×a+6/2×b
I will point out that to evaluate either of the terms on the right hand side, one has to do the division first within that term. Which supports my earlier asertion that the difference between 6/2(1+2)=9 and 6/2(1+2)=1 is whether the division is done first or the multiplication is done first.
Everything you say just comcplicates the issue. Evaluating
6/2(1+2)
is actually very simple. It's completely valid to evaluate the 1+2 first, since it's within a grouping notation. This gives
6/2(3)
Now there are only two operations remaining. Which one should be done first? One of them has to be done first. The result is either 9 or 1 depending on which one is done first. That's all there is to it. Which one comes first is the only decision needed to evaluate 6/2(3).
for me as a programmer, this was very clearly a 9.
I'm not even a programmer, I just use basic bodmas knowledge
Yea
pog
How can we get 60 by adding only three numbers out of these:
2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, & 58?
@@PuzzleAdda we cant. All of these numbers are in the form 2+4k where k is any number from { 0, 1, 2, ... , 14 }. The equation would be (2+4k)+(2+4l)+(2+4m)=60. After we simplify this we obtain k+l+m=27/2 but all of k, l and m are whole numbers. Therefore it is impossible to obtain 27/2 by suming k+l+m and the equation does not hold.
10 million views!
Gratz
I can't believe people don't know this. Congrats on the views.
The comments section is amazing. One of the top comments concludes that you are doing a disservice to kids trying to learn mathematical protocols. I bet you didn't see that coming.
MindYourDecisions wow nice
So 6÷2a = 9 if a=3?
I came up with 1 as my solution. After listening to the explanation I found it logical that 9 would be correct. Now I tested my calculators to see what their solution would be and the first one (used as a standard calculator at schools) came up with 1. The second one (a more sophisticated graphic calculator also used at schools) showed me two different solutions, depending on the writing:
1. 6÷2x(1+2)= 9
2. 6÷2(1+2) = 1 but it changed the writing into 6÷(2(1+2))
It's such a simple arithmetic problem but even calculators are challenged. I love it!
When you understand and apply the Order of Operations and the various properties and axioms of math correctly you get the only correct answer 9. Some calculators are not programmed to handle parenthetical implicit multiplication correctly....
I'd still say 1 is the right answer because PEDMAS is skipping the priority of multiplication by juxtaposition. TI calculator use PEMDAS but Casio and hp have returned to PEJMDAS. It's mainly only North American "teachers" (note that, teachers not mathematician) who insist on PEMDAS.
@@Faux_Dieu you and many others are confusing and conflating an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing...
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
6÷2y the coefficient of y is 2 BUT 6/2(y) the coefficient of y is 3
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is part of a single TERM...
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
@@RS-fg5mf I think, what's going on here is how you understand your question. When I see 6÷2(1+2) I see as the question that if there are 6 apple and you have to divide amongst two groups in which each group has one adult and two children. So how many apple do each person get.
Now you tell me how do you interpret this question in real life? You are taking PEMDAS too literally and forgetting that the multiplication by juxtaposition takes priority over multiplication and division. The answer that 6÷2(1+2)=9 doesn't make sense in reality. What is the situation?
@@Faux_Dieu you don't interpret a math expression you evaluate a math expression following the basic rules and principles of math...
Your word problem would be correctly written as 6÷(2(1+2))= 1
You have 6 bags divided between 2 groups and each bag contains 1 red apple and 2 green apples. How many apples did each group get?? 6÷2(1+2)=9 apples per group.
I have found the counterpart to my mantra „6/2(1+2) is an ambiguous and poorly written expression and the result is meaningless“. It is from the university of Berkeley. It debunks the claim 1 is the distinct solution for 6/2(1+2). GanonTEK is right.
A problem that hit the Internet in early 2011 is, "What is the value of 48/2(9+3) ?"
Depending on whether one interprets the expression as (48/2)(9+3) or as 48/(2(9+3)) one gets 288 or 2. There is no standard convention as to which of these two ways the expression should be interpreted, so, in fact, 48/2(9+3) is ambiguous. To render it unambiguous, one should write it either as (48/2)(9+3) or 48/(2(9+3)). This applies, in general, to any expression of the form a/bc : one needs to insert parentheses to show whether one means (a/b)c or a/(bc).
"It is from the university of Berkeley. It debunks the claim 1 is the distinct solution for 6/2(1+2)." - other way around you mean - MATHS TEXTBOOKS DEBUNK the Berkley blog. 🤣🤣🤣🤣🤣🤣
"GanonTEK is right." - nope, he's wrong. That's why he NEVER CITES ANY MATHS TEXTBOOKS, DUUUHHH!!!! 🤣🤣🤣🤣🤣🤣
"Depending on whether one interprets the expression as (48/2)(9+3) " - it's NOT WRITTEN LIKE THAT, so NO YOU DON'T, duh! 🤣🤣🤣🤣🤣🤣
"as 48/(2(9+3)) " - as per the rules/definitions of Terms, Products, Expressions, The Distributive Law, and Left Associativity, as found in MATHS TEXTBOOKS, and also Cajori, so that's all settled then! 🤣🤣🤣🤣🤣🤣
"There is no standard convention as to which of these two ways the expression should be interpreted"
@@smartmanapps5588do you honestly think I would believe a guy flat out lying about the distributive law ?
@@michi9816 "do you honestly think I would believe a guy flat out lying about the distributive law ?" - I don't know - do you believe YOURSELF? Only YOU can answer that question, not me! 🤣🤣🤣🤣🤣🤣 I don't know if you trust yourself or not - I sure don't! If anything I trust you to LIE ABOUT EVERYTHING 🤣🤣🤣🤣🤣🤣
@@smartmanapps5588sure, I trust my memory, the math textbook from our schools on Highschool / college level, the TU, wikipedia, Encyclopedia Brittanica and the us math textbook from the sixties and finally my reasoning about math. tell me what you have to offer ?
@@michi9816 "the math textbook from our schools on Highschool / college level, the TU" - which you LIE about
"wikipedia" - which is WRONG. You wanna explain how you trust these sources which CONTRADICT each other?? 🤣🤣🤣🤣🤣🤣
"Encyclopedia Brittanica" - which you LIE about
"the us math textbook from the sixties" - which you LIE about
"finally my reasoning about math" - which we've seen time and time again is woefully bad.
"tell me what you have to offer ?" - actual direct quotes from the Maths textbooks which you LIE about, like Pages 36, 37, 282, and 577, duh!
Maybe that’s why no one past fifth grade uses that division symbol
Finally someone with logics
Yup, you're right.
Yes thank you
Looks like someone didn’t pass grade 3 English. passed*
@@sageight818 if you're gonna roast someone on spelling, please atleast be right next time. Thanks
Congrats, this just became topical again. Expect another influx of views my man.
i just came to check if im braindead turns out nah
I dont know why people think this is hard
@@kolowar6600 because they failed second grade
Anf another influx of ignorants disliking this video again.
I’m here to see if I’m brain dead this is basic I’m not through the video yet so I’m pretty sure it’s 1
When programming, the correct answer is to never leave any ambiguity, so always add enough parenthesis to ensure that anyone reading it will understand your intention.
So write 6 / (2 * (1 + 2)) OR write (6 / 2) * (1 + 2). Both are correct, but only one would be correct depending on what your intention is. So always make sure that you enter something that cannot be misinterpreted.
So you are saying that we should be forced to write 5+(2×10) because too many people fail to understand the basic rules and principles of math and incorrectly believe that 5+2×10= 70
@@RS-fg5mf Yes, for the same reason we add comments. Make sure that we know what is happening and that people reading it in the future know that we know what it means.
@@StuartLynne then the Order of Operations and the various properties and axioms of math become redundant if you're going to add crutches for people who fail to understand and apply the basic rules and principles of math correctly
@@RS-fg5mf When math is being taught completely differently between generations, it’s bound to be misinterpreted. I’d rather them be in-depth and redundant so that people in the future won’t have to just assume anything. Assuming things causes a lot of problems.
@@onemorelisa3785 math is only being taught differently if the prrson teaching it is incompetent....
The Order of Operations and the various properties and axioms of math were established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses... The basic rules and principles of math have been the same for over 400 years... Math is based on rules not popularity or personal opinion. Failure to understand and apply the basic rules and principles of math correctly as intended is not a valid argument against them...
There ain't nobody arguing from historical context so explaining historical context to an audience that doesn't know history, or doesn't learn from history is pointless. The Order of Operations we all know stands. The answer is 9.
👏👏👏👏👍👍👍
Let me guess...next viral thing is 1+1
Answer is obviously 0
Lmao
1+1=11 no? :D :D
There are no possible solutions to that equation.
wrong
I am a retired chemistry, physics and math teacher, including calculus. Also worked as a field engineer for 4 years. I think the answer is 1 because I would rewrite it in it's algebraic form before solving. I don't think you'd find much confusion in engineering. I can not remember ever seeing the division symbol used in any formula or equation in advanced work. Also don't remember it used past Algebra I in high school.
I agree. The parentheses denote not only the precedence of the contained expression but grouping of the multiplication as well.
I also agree. Looking at this I believed the answer to be 1 as I don't believe that such an expression would be written using the division symbol and also cannot recall seeing the division symbol used in an equation.
6/2(1+2) = 9 and 6 / (2*(1+2))=1
But I‘m only a german electrician.
The answer I got is also 1
That how it's always been.
History biggest question: How did a 5th grade math question create a riot on the media
It's not even 5th grade, it was 3rd or 4th grade
I learned this in 5th grade and solved this in approximately 5 seconds.
Please read this comment, thank you.
Solve for the 2 in parentheses, it is not a bracket [ ]
6/2(1+x)=9
3(1+x)=9
3+3x=9
3x=6
X=2
The problem is that people who think that it is 1 believe that after simplifying 2(1+2) is that they think it is the denominator of the fraction. For that to be true, there must be a parentheses in front of the 2.... (2(1+2)). You will do that first if that was in the problem, but it isn’t.
6/2(1+x)=1
6/2+2x=1
Now you see that there is a fraction, but what can it be. If it is 2+2x, you get 2 as your final answer, which is correct. If it is just 2, you get -1, which is incorrect.
However, you do division before addition, so you do 6/2 to get 3, eventually getting -1 as the solution.
3+2x=1
2x=-2
X=-1
This is incorrect, because we are trying to solve for 2 in the parentheses... 6/2(1+x)=1
X should equal 2. People think that after distributing the 2 into (1+x), the whole thing stays in the parentheses. It disappears after you distribute.
Thank you for your time.
Lol
norman hughmin because people are morons
Its 9, look it up. You go left to right on the multiplication and division stage of PEMDAS.
It depends on which interpretation of multiplication by juxtaposition you use.
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.
Yes, the correct answer is 9
Yes, you should look up The Distributive Law. Only 1 is correct.
For things like this we don't use the ÷ symbol anymore, it's just easier to use a fraction so you can differentiate (6/2)×3 from 6/(2×3)
That's exactly what I was thinking.
It's not its 1
You work left to right
@@Lucian24 What I meant is that we now use fractions so we don't need to specify which goes first. You operate all on the top and divide it by all on the bottom. So you can differenciate:
(6/2)×3 = 3×3 = 9
From 6/(2×3) = 6/6 = 1
That makes sense
As far as I'm concerned, "2y" represents a single value, and should be treated as such. To deliberately separate the values, you write "2 × y" or "2 · y". Therefore, if the expression were given as "6 ÷ 2 × (1+2)", I would agree and say it's 9. But the ligature of "2" and "(1+2)" in the form "2(1+2)" represents a single value, making the answer 1.
The biggest mistake that people make is incorrectly comparing 6÷2(1+2) as 6÷2y.
This is an inaccurate comparison... 6÷2(1+2) does not Algebraically equate to 6÷2y it correctly equates to y(1+2) where y is equal to the Monomial Factor of the TERM outside the parentheses. 6÷2 is juxstaposed to the parentheses as a whole not just the numeral 2...
You can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol.
You CAN factor out LIKE TERMS from an expanded expression. 6÷2×1+6÷2×2= 6÷2(1+2) as the LIKE TERM 6÷2 was factored out of the expanded expression...
Many people, *including you*, confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
Convention doesn't trump LAW and the Distributive Property is a LAW.
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2 = 3 Monomial Factor
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
So please stop with the illusory facts and stop spreading false and misleading information. You're part of the problem.
@@RS-fg5mf I juxtaspose so!
@@jack-xf6il The thing is, no one has yet provided evidence that a(b) is not multiplication by juxtaposition and not equivalent to ab.
If the wasn't ambiguity here why do modern international standards like ISO-80000-1 mention about writing division on one line with multiplication or division directly after and says that brackets are required to remove ambiguity or why does the American Mathematical Society say it's ambiguous as well as many maths professors etc.?
@@GanonTEK I was just making a joke about the misspelling of juxtapose, which I thought made it sound like 'just suppose'. But while I'm here, if y = 3 then does everyone agree that 6÷2y = 9? Because I have to say that did not seem intuitive to me. But is 6/2y = 9 also, or 6/2y = 1? Do '÷' and '/' have different meanings or are they the same?
@@RS-fg5mf Let's take a look what is stated...
Statement 1 : 6/2y = 6/(2y) = 3/y by Algebraic Convention
Statement 2 : 6/2(a+b)= 3a+3b not 6/(2a+2b) , because 6/2(y)= 3y by the Distributive Property...
let's take y = (a + b), in statement 1 : 6/2y = 6/(2y) = 3/y
What is stopping us from writing 6/2y as 6/2(a + b) , 6/(2y) as 6/(2(a + b)) and 3/y as 3/(a + b) using direct substitution in Algebraic Convention? We see 6/2y = 6/(2y) and 2y is taken as whole denominator, but notation have ambiguity and we default to seeing 6/2y where y = (a + b) will look like 6/2(a+b) but it is actually also 6/(2(a+b)) which still follows Algebraic Convention 6/2y = 6/(2y) and by observation 2(a+b) seems like (2(a+b)) and this is where the context comes in to operate canonically with higher priority to evaluate 2(a+b) because 2(a+b) is also (2(a+b)) .
In Statement 2 : 6/2(a+b) = 3a+3b by Distributive Property, 6/2 as scalar to (a + b)
In this case , by observation both 6/2(a + b) become 3/(a + b) or 3a+3b., which also proof there is a lack of information in the notation to distinguish your answer.
By words is very certain because there is semantic you can infer priority. The notation used here alone is not distinctive to avoid ambiguity in this case.
Conclusion like some have mention is it is better to rewrite the notation to denote the scalar part (6/2)(a + b), 6/2 dot (a + b) or 6/2 * (a+b). Like many Engineers will likely used often is 6/2(a+b) with parenthesis taking higher priority and evaluate as 6/(2(a+b)) or 6÷2(1+2) =1. This is why there is 2 answer to this video 1 or 9 as the video also shows just because the question is presented without context. Really this just boil down to notation syntax and interpretation.
To those who say 1 of the answer is wrong, is the person who is ignorant as you are stuck in their own confirmation biases.
The "1917" example is exactly how I was taught both in high school algebra and in college algebra. That was in the 80's, not 100 years ago, lol.
What does attending school in Appalachia have to do with it? Yes, I did and I was taught the 1917 way I guess from 1996-2013. WCU was still using in it in 2013, and so was all the other kids from other parts of the US.
same here...thank god for comments almost gave up on my math.
honestly i dont know how long ago people didnt use the order of operations but im sure that in the 80s all mathematicians used it. id go as far as to say it probably existed at least a thousand years ago
Don't be jackass Steve
realistic dan
Appearantly my wife was retaught the correct way when she went to UMiss. Guess that why my kids always come home with the wrong answers when I help them do their school work
To make this easy look at the division symbol, it’s a dot over a line over a dot. This tells you that what is on the left of the symbol goes on top and what’s on the right goes on the bottom.
Please excuse my dear Aunt Sally. I thought everyone was taught that.
PEMDAS = 9
and some people were taught other acronyms that mean exactly the same thing, like BODMAS
That's how I learned it in high school, class of 1998, and then in college in the early 00's... to do it in the exact order of the sentence.
@@godelnahaleth No, you were not taught to follow PEMDAS as 6 exact steps... SMDH
Own your mistakes and stop blaming your teachers for your failure to pay attention in class and learn correctly...
@@RS-fg5mf Nope. 2(3) is not the same as 2*3. Anyway it's been 4 years since I came across ÷ sign. I only use fractions and never had to come accross controversial problems like this one.
Exactly its easy
Oh, I had the 1917 math class, then
Me too 😂 honstly, not surprised, though
Me too.
Try my channel mathfullyexplained
I guess I didn’t.. I got 9
Me too ✋
Even when i went to school the way we were taught to do math, at a mere 2 second glance i got the answer of 1. In figuring out the problem the parentheses when it becomes 2(3) we still solved 2(3) before we did the division. Anyone who graduated in the early 2000's will get 1 as the answer not 9 because we wouldn't have replaced the () with 2x3 subsequently changing the order of operations
You're wrong and You're the one changing the expression... There is no mathematical difference between 6÷2(3) and 6÷2×(3) or 6÷2×3... The multiplication SYMBOL is implicit rather than explicit. Grouping symbols only group and give priority to operations INSIDE the symbol not outside the symbol.
The Order of Operations and the various properties and axioms of math were established in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses. The correct answer is and always has been 9 not 1.
6÷2(1+2)= 3(1+2)
Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
Convention doesn't trump LAW and the Distributive Property is a LAW.
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2 = 3 Monomial Factor
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
@@RS-fg5mf I think what Joshua was getting at was that we were taught a completely different order of operations in the public education system than what the orator of this video claims is correct. We were specifically taught that multiplication always takes precedence over division and that addition always takes precedence over subtraction regardless of their location/order in the equation with the order of precedence of operations being mneumonically described with the phrase "My dear Aunt Sally," which stands for "multiplication, division, addition, and subtraction." Now they are arbitrarily changing the rules and/or they deliberately miseducated millions of children. Take your pick, but something is seriously f^cked up here.
@@wesbaumguardner8829 Then you were taught wrong or more likely don't remember correctly... There isn't a mathbook on the planet that lists the Order of Operations as 6 exact steps... So your telling me you think 10-7+2=1 ?? Not now, not ever...
Multiplication and Division share equal priority and can be evaluated equally from left to right as they are *inverse operations* by the reciprocal...
Addition and Subtraction share equal priority and can be evaluated equally from left to right as they are *inverse operations* as subtraction is just addition of a negative number....
6÷2×3= 6×0.5×3 now multiply in any order you want...
10-7+2 = 10+(-7)+2 now add in any order you want you still get 5
@@wesbaumguardner8829 the rules have not been changed. The Order of Operations and the various properties and axioms of math were established and internationally recognized and accepted as the standard for evaluating a math expression in the early 1600's when Algebraic notation was being developed in order to eliminate ambiguity and to minimize the unnecessary and excessive use of parentheses...
P
E
M/D equally left to right
A/S equally left to right
There isn't a math book on the planet that states PEMDAS represents 6 exact steps...
@@wesbaumguardner8829 M and D have equal priority so the order doesn't matter.
Same with A and S.
Like with 10 - 8 + 3 - 1
S first: 2 + 3 - 1 = 2 + 2 = 4
A first: 10 - 5 - 1 = 5 - 1 = 4
The issue with 6÷2(3) is that multiplication by juxtaposition can imply grouping, giving it higher priority than division and regular multiplication. So it's ambiguous because it's not clear if 2(3) means 2×3, the explicit interpretation used my programming and in America, or (2×3), the implicit interpretation used by academic writing.
6÷2×3 = 9
6÷(2×3) = 1
Although, 6÷2×3 isn't good writing.
Modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. ISO-80000-2 says ÷ should no longer be used also.
It's just a badly written expression.
I understand both methods. It will be interesting what my books say when I get to this type of problem. I am studying from pre-1900 Maths books right now.
Imagine using division symbol instead of fraction.
Yeah that's how that goes...?
It's 5th grade man
@@mrpickle8959 fractions is also division 1/2 is the same as 1÷2, and this works for anything.
@@bk40907 sure it is the same thing. but with fractions the error couldn't happen as the order is directly visible. I personally haven't seen that operator once in university. If you are forced to write in one line (e.g. in programming) people use "/"
@@Langweiler11 that's my point
My answer was "1" because I was taught the "historical" way. I was not aware that the rules had changed.
🤣🤣🤣🤣🤣🤣🙄
Ong
If you dont carefully listen to what your teacher says then you will get it wrong. I remember my teacher saying that if multiplication and division are the only ones left, you'll solve them from left to right. Same goes to addition and subtraction (If they're the only ones left)
The answer is 1. There is a reason the coefficient is up against the parenthesis, you multiply first.
@@Graphenor the correct answer is and always has been 9
So it boils down to the old "I didn't give the wrong answer. You didn't ask the right question".
@MATHEMATICAL FRAUD HUNTER RESPECT TO MATHEMATICS that is not how math works bro...
Pretty nice way of saying it.
It's like "A union B intersection C" in sets and expecting a certain answer. You can't write that either because it's ambiguous.
@@GanonTEK PEMDAS is not ambiguous.
6/2(1+2)
Parentheses first
6/2(3)
Multiplication and division left to right
3(3)
9
There is no ambiguity. The ambiguity is people not recognizing 6/2(3) = 6/2*(3) = 6/2*3 = 9. Implied multiplication is treated the same as regular multiplication. The it’s the same as “I didn’t read the question correctly, therefore I am not wrong”
@@Owen_loves_Butters There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not.
I.e. does 2(1+2) = (2×(1+2)) or 2×(1+2)?
Both are widely used.
6÷(2×(1+2)) = 1 (using PEMDAS)
6÷2×(1+2) = 9 (also using PEMDAS)
PEMDAS isn't the problem. The notation used is. That's the cause of the ambiguity.
That's why there is such a large disagreement and even calculators from the same manufacturer don't agree.
You shouldn't write a/bc or a/b(c) anymore. It's not acceptable notation. ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove any ambiguity.
A PhD student wrote a paper on the ambiguity called The PEMDAS Paradox if you want to look it up.
@@GanonTEK You don’t add parentheses if there are none to begin with.
I don’t think the problem here is the division sign. I think the problem is the “implicit multiplication”.
In my experience as a scientist and teacher, many people would say 5/2x = 5/(2*x), similar with the number in front of the parentheses, without the multiplication sign.
Algebraic equation vs. simple arithmetic. 2x is a variable and it's coefficient, which implies multiplication. You don't have any implied multiplication in arithmetic because there are no coefficients since you have no variables.
@@Antelope2000
The number in front of the brackets is a coefficient.
Yes!!!
The historical usage makes so much more sense to me than the current method.
same
BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations
6÷2(1+2)=
6÷2(3)=
3(3)=
9
2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside ....
There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more.
Commutative Property
6÷2(1+2)=
6(1+2)÷2=
6(3)÷2=
18÷2=
9
Distributive Property
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction.
6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2
Operational inverse of division by the reciprocal
6÷2(1+2)
6(1/2)(1+2)=
6(1/2)(3)=?
Multiply in any order you want you still get 9
Proper use of grouping symbols
6
-----(1+2) = 6÷2(1+2)=9
2
6
-------- = 6÷(2(1+2))=1
2(1+2)
A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator...
Another argument people tend to use incorrectly is factoring....
6 = 2+4 No parentheses required BUT
6÷(2+4) parentheses required
2+4= 2(1+2) only one set of parentheses required.
6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set...
The 2(1+2) must be placed within the first set of parentheses containing the (2+4)
6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2)
Let y = 0.5
6y(1+2)=?
6y*1+6y*2= ?
6/y⁻¹*1+6/y⁻¹*2= ?
If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9
The rules of math have to remain logical and consistent across the board...
THESE ARE THE FACTS....
6
-----(1+2) = 6÷2(1+2)=9
2
6
-------- = 6÷(2(1+2))=1
2(1+2)
A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator...
I agree with Wotcher Mystic here. I learned math practices in California public school in the 1950s and 60s. Where I went to school, we did not practice math with this video's rules change. Had I pursued math in college, they may have become known to me.
The amusing thing is that lots of subjects still use the old order. Physics for example still uses the old rules in their proof.
This is why we write division problems as fractions. That symbol is elementary.
That being said, wow, so many retards who can’t do simple arithmetic in the comments. It’s not that I mind, but the fact that they all say it with so much certainty.
Excuse me, but I know someone who got 1 as an answer and even if they were wrong, they happen to be a highly intelligent person who knows several programming languages and writes code very well. People make mistakes and some people have been taught outdated information. That doesn't mean they're "retards" or that you have to worry about the state of society **eyeroll**. An 8th grader will get the correct answer because they're currently learning the material. My friend who got the wrong answer still knows a shit ton more valuable information than any 8th grader lol. Why do people who perform well in math always go around calling everyone retards if they don't get an answer right? You never see an English major call a mathematics inclined person stupid if they couldn't figure out how to write a proper MLA citation lol.
I even specifically said that I didn’t mind that they got it wrong but how they said it.
Kleo3392 no, it isn't. Just use the PEMDAS/BODMAS and you will get one correts answer
The answer is 9 because you should convert the '6÷2' to (6/2)(3) = 3(3) = 9
Presh: gives the correct answer
People: *32k dislikes*
Edit: 01/02/2020 : *33k dislikes* Edit 2: 05/05/2020 : *34k dislikes*
Fuzz Lightyear not anymore lol
Now it is 33k
😂😂😂
I think it’s more so people disliking that he solved literally a 4th grade problem on a channel based around more advanced math... although at least he gave it SOME substance with the whole historical bit. Still, kind of out of place on this channel.
mad because bad
I would have thought the presenter of the equation asked us to "Divide six by twice whatever value is within the brackets". The equation can be written as A divided by B where B=2(1+2). So if [A divided by B] = 1 and A =6, then B=6, This implies 2(1+2)=6 which is correct.
If [A divided by B] = 9, and A=6, then B= 6/9or 2/3. This implies 2(1+2)=2/3 which is incorrect.
The rule 'brackets first' is short for "Solve the brackets first" This implies the removal of the brackets from the equation first. In adding 1+2, you do not get rid of the brackets! You are still left with 2(3), and one cannot remove brackets from the equation without solving them. In this case you must multiply 3 by 2 to the brackets. You cannot just add in or take out brackets ad lib!
In applying values, 2(3) is a single Value and is expressly implied in the equation.
2 x (1+2) is two values and is not implied in the equation.
It appears to me the problem lies in the computerisation of the equation. To me the equation is simple and unambiguous and means the same now as it did 50 years ago!
Well said man!!
I was in school in the late 60’s. We were clearly taught to complete the parentheses including the multiplier of the parentheses first.
Still same in 2021
In the 80s, I was taught the same.
Wonder if country of origin makes a difference in how one is taught this.
My algebra and geometry teacher (same woman) was Vietnamese. Is it possible that in other parts of the world how that 2(3) is treated is different?
Could that explain the differences in how some of us learned it?
Work parentheses first. Answer is 1
The ignorant leading the ignorant...
All you remember is being taught that 2(1+2)= 2×1+2×2 or that a(b+c)= ab+ac
What you failed to learn or have forgotten is that the TERM outside the parentheses is to be multiplied by the value of the parentheses or Distributed across the TERMS inside the parentheses... TERMS are separated by addition and subtraction not multiplication or division.
6 is a single TERM
6÷2 is a single TERM
6÷2×3 is a single TERM
6÷2(1+2) is a single TERM with two TERMS inside the parentheses.
6÷2(1+2) = 6÷2×3 = 3×3= 9
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of eliminating the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM not just the factor outside the parentheses...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is part of a single TERM...
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
Lol everyone real quiet after you dropped this
I just took college trig this past semester and the 2(3) is converted to 6 first. So this old 1917 rule of order of operations is still in effect to this day and being taught to new engineers likes myself.
And that's how I was taught it as well: the bracket being the second operation in order, not merely a conversion of typology.
Then they are taunt wrong? Use any calculator you like heck even program your own it will get 9 not 1
University Trig would have taught it properly.
@@rosewarrior706 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
The notation is ambiguous. There is no agreed upon convention on whether multiplication by juxtaposition implies grouping or not
Even online calculators don't agree.
Microsoft Math shows both answers on screen at the same time in different places.
I think DESMOS won't even let you type this on one line, which is great.
Yeah I'm still studying my fifth year in civil engineering and we always do it that way cause 2(1+2) it's supposed to be all together like just one and not 2*(1+2) which is not the same if you have to solve a problem like the one presented in the video. So I'm just so confuse right now.
Just came to check if my brain was working and it is, I’m tired of fighting over the right answers through Twitter 🏃♀️
Just ask them if they've taken advanced functions or calculus, and then tell them if they ever used the ÷ symbol instead of /. I think thats some pretty solid evidence I should say
NOT ALL OF US COMING FROM THAT TWEET BYEAKSJSKSJ 😭
@@Whatthescallop1776
÷ and / are both division sign.
Same lol
@@Whatthescallop1776 UK don't take calculus or advanced functions as a separate subject so that wouldn't work
if you were using quarters: 0.25(3+5) = 1/4(3+5) =1/4(8)
would you have $2 or $0.03?
2
It's just freaking easy the answer is 9.
hah its 1 i know
It's 1
Mira Acharya no.... It's 9
Private_Onion Gaming ..I thought it was 5
Its 2
What is interesting is that four different calculators give me four different answers. Two of the calculators can’t handle 2(1+2) or even 2(3). The Radio Shack EC-4030 gives 2(3) = 23. The Hewlett-Packard 20S gives 2(3)=3. The two calculators that can handle the equation are both Texas Instruments. The TI-30X IIS gives 6/2(1+2)=9 while the TI-85 graphing calculator resolves 6/2(1+2)=1. I can still hear my algebra teacher and physics teacher both stating that you must eliminate all parentheses before advancing to the next operation (hence 1 is correct for them). I do agree with those who say that the equation is poorly written.
BODMAS/PEMDAS and any other acronym that is a memory tool for the Order of Operations
6÷2(1+2)=
6÷2(3)=
3(3)=
9
2(3) is not a bracketed priority and is exactly the same as 2×3 M not B or O in BODMAS. Brackets/Parentheses only GROUP and GIVE priority to operations (INSIDE) the symbol not outside ....
There is no rule in math that says you have to open, clear, remove or take off parentheses. The rule is to evaluate operations (INSIDE) the parentheses and nothing more.
Commutative Property
6÷2(1+2)=
6(1+2)÷2=
6(3)÷2=
18÷2=
9
Distributive Property
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
The Distributive Property is an act of removing the need for parentheses by multiplying all the TERMS inside the parentheses with the TERM outside the parentheses... TERMS are seperated by addition and subtraction.
6÷2 is one TERM attached to and multiplied with the two TERMS inside the parentheses 1 and 2
Operational inverse of division by the reciprocal
6÷2(1+2)
6(1/2)(1+2)=
6(1/2)(3)=?
Multiply in any order you want you still get 9
Proper use of grouping symbols
6
-----(1+2) = 6÷2(1+2)=9
2
6
-------- = 6÷(2(1+2))=1
2(1+2)
A vinculum (fraction bar) is a grouping symbol and groups operations within the denominator and when written in an inline infix format extra parentheses are required to maintain the grouping of operations within the denominator...
Another argument people tend to use incorrectly is factoring....
6 = 2+4 No parentheses required BUT
6÷(2+4) parentheses required
2+4= 2(1+2) only one set of parentheses required.
6÷(2+4) we already have a set of parentheses and the factoring must take place within that first set of parentheses. You can NOT just dismiss the first set of parentheses out of hand in favor of the second set...
The 2(1+2) must be placed within the first set of parentheses containing the (2+4)
6÷(2+4) = 6÷(2(1+2)) NOT 6÷2(1+2)
Let y = 0.5
6y(1+2)=?
6y*1+6y*2= ?
6/y⁻¹*1+6/y⁻¹*2= ?
If you answered 9 to all three algebraic expressions then it would be ILLOGICAL and INCONSISTENT as well as hypocritical to say that 6/y⁻¹(1+2) doesn't also equal 9
The rules of math have to remain logical and consistent across the board...
THESE ARE THE FACTS....
without placing an explicit multiplication sign between them. A person is left wondering whether to use the sophisticated convention for implicit multiplication from algebra or to fall back on the elementary PEMDAS convention from middle school. It's poorly written
You are correct! My algebraic teacher tought me the same.
@@RS-fg5mf You are full of nothingness.
6÷2(3)=
3(3)= WRONG first you should do the 2(3) which is 6. Go learn you greek again.
@@MrElvis1971
Many people confuse and conflate an Algebraic Convention (special relationship) between a variable and its coefficient that are directly prefixed (juxstaposed) and forms a composite quantity by this convention to Parenthetical Implicit Multiplication... They are not the same thing...
Convention doesn't trump LAW and the Distributive Property is a LAW.
6/2y = 6/(2y) = 3/y by Algebraic Convention BUT 6/2(y)= 3y by the Distributive Property...
ABC/ABD = C/D by Algebraic Convention
ABC/AB(D) = CD by the Distributive Property
6/2(a+b)= 3a+3b not 6/(2a+2b)
The Distributive Property is a PROPERTY of Multiplication, NOT Parenthetical Implicit Multiplication, and as such has the same priority as Multiplication... The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication....
Multiplication does not have priority over Division they share equal priority and can be evaluated equally from left to right....
The Distributive Property is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in. The Distributive Property REQUIRES you to multiply all the TERMS inside the parentheses with the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just the numeral next to it...
TERMS are separated by addition and subtraction not multiplication or division...
6÷2 is a single TERM juxstaposed to the parentheses as a whole not just the numeral 2
FURTHERMORE people misunderstand Parenthetical Priority... The rule is to evaluate OPERATIONS INSIDE the symbol as a priority before joining the rest of the expression outside the symbol. It does NOT literally mean that the parentheses have to be evaluated BEFORE anything else in the expression can be done...
A(B+C)= AB+AC where A is equal to the TERM VALUE i.e. monomial factor outside the parentheses not just the factor next to it...
A=6÷2 = 3 Monomial Factor
B= 1
C= 2
6÷2(1+2)=
6÷2×1+6÷2×2=
3×1+3×2=
3+6=
9
As someone doing a masters degree in mathematics, I and most people I know would read that expression it in a way that answered 1. Not because division is some all powerful "everything on the left, over everything on the right" operator. But because the "2y" notation (without the multiplication symbol) generally implies "(2×y)" i.e. bracketed, so long as all other orders "PEDMAS" are satisfied.
No one would ever read "sin2y" as "sin(2)×y" because the 2y is implicitly bracketed. This isn't so much of set rule, more a general consensus of an exception. Mathematics is a language, and like any other language, it alters to suit the people who use it and the way they use it. If the equation had been written:
6÷2×(1+2)
Then I dont doubt that more people would arrive at 9.
Aidan Retallick said, "No one would ever read "sin2y" as "sin(2)×y" because the 2y is implicitly bracketed"
That is precisely what this problem points out and otherwise very intelligent people like yourself have fallen into bad habits.
Mathematically there is no difference between 6÷2×(1+2) and 6÷2(1+2)
Just because you are reading brackets but not using them in your notation does not make it "correct".
Much like language, you can use slang, shorthand or idiom if you choose and are understood but there is a proper way to speak and in math, there is a proper Order of Operations and a proper notation.
That is why some very smart people can look at this and get 1, but if we put it into Wolfram or any strong computational algorithm that will follow the Order of Operations precisely as the notation dictates, the answer is 9.
The question is valid and simple arithmetic as taught at school and the answer is correct, The sort of question thousands of kids get every week in tests. I would say that strictly speaking your counter example is not well posed and you have used "lazy" ambiguous notation. You mean Sin(2y) and you should not omit the brackets.
John Silvestri, how are you exactly saying 6÷2(1+2) is the same as 6÷2X(1+2)?
Just because some wolfram/Google calculator said so? Neither the PEDMAS nor any other rule of order of precedence explicitly states that ab has the same precedence as a*b.
They all state that they are equal, not of the same precedence.
Some elementary kids who are still reciting PEDMAS on solving every line of the equation (without realizing why) along with some inconsistent calculators are the reasons this ambiguity exists now.
And finally, even if all the masters students in the world are wrong, do you really believe that the solution of some elementary students will be regarded correct that are opposed to the honors/masters students? (I believe otherwise, but that's my opinion)
1. sin 2y is not necessarily a shorthand. It IS legally used for meaning sin (2y). I can give you reference if you ask for
2. PEDMAS certainly doesn't state the precedence of sine, log or any other function. So... how do you exactly solve them by following PEDMAS?
John Silvestri. Solve :
6 ÷ 2x = 9
Silverfox10 x=1/3
There is no controversy. Following the order of operation, the answer is 9.
Yeah, folding the order of operations the answer is 1. You forgot to dissolve the parentheses.
@@InsanityoftheSanities I'm going by the F.O.I.L order of operation.
@@TylerKnight Are you able to reproduce to me your work? I'm interested in what the F.O.I.L order of operations is.
@@TylerKnightthere is no FOIL in this expression but the answer is 9
@@InsanityoftheSanitiesthere is no rule in math that says you have to open, clear, remove, take off, eliminate, get rid of or dissolve parentheses.
The RULE is to evaluate operations WITHIN the symbol of INCLUSION as a priority and nothing more... (1+2) is a parenthetical priority. 2(3) is not a parenthetical priority and is mathematically the same as 2×3
There is no mathematical difference between 6÷2(1+2) and 6÷2×(1+2) despite the false and misleading information and willful ignorance people have about parenthetical implicit multiplication...
Imagine a real mathematician ever using the divided by symbol
@Wubba Wubba mathematicians as in people who do it for a living generally don't. That's not gatekeeping - it's literally their langauge
Wubba Wubba r divided by iamsoverysmart?
If you write it properly: 6 over 2(1+2), surely the answer is 1?
@@DadgeCity yes if written with
6
----------
2(1+2)
Answer is 1
@@DadgeCity The expression 6/2(1+2) will not evaluate to that. You are violating the Distributive Property. In order to fully understand this I will impose a set of parenthesis that does not change the expression. (6/2)(1+2) which will evaluate to (3)(3) = 9 or (3 + 6) = 9. In order for you to have the 6 solely in the numerator and the expression 2(1+2) in the denominator you would have to impose this set of parenthesis which will change the expression 6/(2(1+2)). Then this will evaluate to 1. Therefore (6/2)(1+2) != 6/(2(1+2)) and if you don't believe me put both expressions into a TI Graphing Calculator!
*Doesn't matter what the correct answer is, the whole thing shouldn't have been written this way to begin with*
True
We got google to do this for us
It’s 1
It's 9 if you follow PEMDAS
So you divide before multiplying in PEMDAS right no wrong you multiply first before you divide
Class of 2019, in college taking calculus now. Your “historical” way is how we are taught.
So you're telling everyone that taking a college calculus class that they are utilizing the obelus instead of using a vinculum?? That's kinda hard to believe...
The correct answer when you actually understand and apply the Order of Operations and the various properties and axioms of math correctly is 9
@@RS-fg5mf look man, this is the way that I was taught since I started learning math as a kid. It’s also the way it’s been taught to me in college. If you take issue with the way I’m being taught, take it up with the faculty, not me. Also, *you’re.
@@wolfwarren6376 yes, a typo. I do it quite often when text swiping. Thank you.
So, if you're taught wrong, you just choose to remain wrong instead of doing something about it?? My point was that you're referencing a college calculus class and I'm asking, are they using the obelus instead of a vinculum in a College Calculus class?? I find that hard to believe but I'm waiting on an answer for clarification.... ARE they using an obelus in your college calculus class????
@@RS-fg5mf frankly, they’ve had us evaluate both ways. Early on in the semester, we were tasked with doing algebra and simple precalc problems. Perhaps it was just to get us thinking, since order of operations problems like this haven’t come up recently.
@@wolfwarren6376 let me try and explain something to you... People incorrectly confuse and conflate an ALGEBRAIC CONVENTION given to coefficients and variables that are directly prefixed and form a composite quantity to parenthetical implicit multiplication. They are NOT the same thing...
6/2y = 6/(2y)= 3/y by Algebraic Convention
BUT 6/2(y)= 3y by the Distributive Property.
When a constant, variable or TERM is placed next to parentheses without an explicit operator the OPERATOR is an implicit multiplication symbol meaning you multiply the constant, variable or TERM with the value of the parentheses not just the factor next to it. Terms are separated by addition and subtraction not multiplication or division. The TERM or TERM value is attached to and to be multiplied by the parenthetical value of the parentheses....
The TERM value outside the parentheses is 3
The parenthetical value of the parentheses is 3 AND 3×3= 9
I did grow up learning that any number butted up to a parentheses meant immediately multiply that grouping as one value. in computer code you would add outer parentheses where any ambiguity exists.
me too
Same here
Me: Answer is obviously 1
"Answer is 9"
Me: Well fuck me.
It's actually 1 :) The person explaining made a crucial mistake thinking that () is the same as x.
As many explained, it's not
Atomicninja - It is really 1 the ( ) go first. So it's 1+2 first which equals 3 obviously. Then the equation is 6/2 x 3 (the / is a division symbol). Then you multiply 2 into 3 then it's 6/6 and then your final answer is one. Simple to learn in school easy math.
Kai Cluster I don't agree I think the answer is 9 because you add the parentheses P, then you go from left to right so 6/2=3 and 3x3=9
first off 6/2=3 is already wrong because there is no multiplication
6/2(3) is not the same as 6/2*3 or 6/2*(1+2)
if you want to elimate (1+2) the equation should be (6/3) / (2(1+3)/3) then you get 2 / 2 =1
or simply just 6/6=1
The correct answer is 1 because 2(3) is somewhat like y(x) which means the value of y is multiplied by x time.. going that approach 2(3) is interpreted as 2+2+2 = 6
6 / 6 = 1
the algebraic expression is z / y(x)=
Divide should precede Multiply so 6/2x3 should be 3x3=9
graduated in russia in 2010. In US in 2015. In both countries was taught the historical way. I took a few minutes to analyze this problem and I would have never even thought that you could do it any other way so this contemporary way is news to me
Same. I graduated high school in 2001 and got the answer as 1
I made it through Vector Calculus in the US university system. I was taught the correct “historical” way. That and we never used the dumbass division symbol (which served its purpose); we also used parentheses and entire numerators and denominators. I’ll rest my case with the “new” way is pushed by the same hacks that forced Common Core upon this newest generation. I tutored k-12 math for a time after graduating college, and saw enough of this garbage. You take a subject which students typically struggle with, and make it more difficult, labor intensive and structured in a way that none but their peers or teachers are able to assist them in learning an already arduous subject.
I graduated high school in 2017 and did some Calculus in college back in 2021 and the answer would have been 1. I do not understand why it would be 9.
I graduated in Germany in 1984 and my answer was clearly 9.
@@lubnan08 If the problem was 6÷2×2+1= what would your answer Be?
IMO the real answer is "don't write expressions this way." You write expressions so that they can't be easily misinterpreted. If there's any possible confusion, add parentheses to resolve the confusion. This is particularly so when dealing with non-commutative operations such as subtraction and division.
This probably comes from a background in computer programming; you don't want to make assumptions about order of operations, because while the people who implement the languages usually have standards to work from, they sometimes screw up. (To make things even more complicated, in computer programs sometimes operations have side effects - like "x++ + x" in C - "x++" evaluates x then increments it, so the value depends on whether the left or right side of the addition is interpreted first. The solution is "don't write it that way.". Eliminates confusion and therefore bugs.)
100%
Amen
Its just the correct way of writing it, there isn't much to be confused about
Basically you just read from left to right and solve brackets first, whats so confusing?
@@benvergus1573 The fact that it's debated at all indicates that it's found confusing. This whole kerfuffle would not EXIST if people didn't find it confusing.
The implicit multiplication when concatenating terms - as is done in this particular problem - complicates matters further. What is 4x/2x ? Left to right you get 4x, divided by 2, multiplied by x, so 2x^2. But I read it habitually as 4x divided by 2x and get 2, and I'm betting that 90% of people trained with strict left to right in evaluating PEMDAS/BODMAS do the same.
Writing with clarity in mind, and avoiding any possible confusion, is never a bad thing. Particularly when dealing with a field where precision matters, such as mathematics or engineering.
Answer is 1
Multiplication by juxtaposition always before division, they’re connected.
6 over 2(1+2)
6 over 6 is 1
Wrong. Because in PEMDAS/BODMAS (notice the M and D placement) multiplication and division are on equal level.
Write the ÷ in fraction form and it becomes less ambiguous:
(6 / 2)(1 + 2)
6 (1 + 2) / 2
6(3) / 2
18 / 2
9
This also works with PEMDAS where × or ÷ is done on a first come first encountered basis from left to right after doing the parenthesis:
6 ÷ 2 (1 + 2)
6 ÷ 2 (3)
3(3)
9
@@_Just_Another_Guy try that with money.
Let say you have 6$ in your pocket « a »,
You want to give it to your 2 friends that are with you that day « b »,
Each one of them you give 1$ in the left hand « c » and 2$ in the right hand « d ».
Answer is how many times can you do that on that day.
a ÷ b(c+d)
So the answer is?
@@OLaro1483 making a text base question which fits your interpretation isn't an argument
By the time I reached the level in math where knowing the order of operations became necessary, I don’t think I ever saw division expressed in any way besides numerator and denominator. It’s pretty easy to get confused with this.
@luvlanadelrey no. bodmas is taught to elememtary students. mathematics in higher education no longer uses this symbol ÷ as division is expressed as a numerator and denominator and it is also implied that all values before ÷ sign are numerator and all values after are denominator
@@wisn6327 Yes but the two symbols are treated the same so the answer remains 9. However, it is pretty confusing
@@EvK_27 nope its still 1. if they're treated the same, then its going to be 6 divided by 6
@@wisn6327 I don't think you watched the video :)
Also, read the descirption, it clearly says both symbols are same and will give the answer 9
@@EvK_27 i did but i kinda forgot the contents already xd so yeah mybad, it does say it in the description however he division symbol ÷ isnt formal so in the end this is a poorly written question like the top commenter said
I got 9
According to my legend "PEMDAS"
Edit: many ppl don't know how to pemdas so Lemme say it's more accurate to say "Gemdas"
Grouping, Exponents, Multiplication/Division left to right, Addition/Subtraction left to right. Paranthesis in this equation is used as MULTIPLICATION not grouping
I got 9 at first then I decided that was wrong so now I have 1. I'm still watching as I write this.
The correct answer is 9.
@@josephwhite7960 how in rhe world u get 1
@@dnhn.design I multiplied the sum which is 3 and 2 to get 6 then I divided it by 6. I figured if I got 9 that's what everybody got and that was going to be considered the wrong answer so I tried something different.
Pemdas will in fact give you 1... which is why Pemdas is wrong.
People: nine.
Me, an intellectual: *nein*
Ivan09 Ja Ja! Ich spreche auch Deutsch
Translation: Yes yes! I also speak German!
@@rydh6zgjhbfvrwhb259 Grüße aus NRW XD
@@Brontok Heyyyyyyyyy! Ich grüße dich zurück! Natürlich auch aus NRW!
@survival pete ??
Ivan: posts a meme containing a fraction of german
The comments: iCh bIn dEmEnT
People who know no maths are making this a viral problem. If you dont believe me than go ask a grade 5 child and the child will tell you the answer is 9. Life is simple. We complicate it
Historical, eh? Hehe, that makes me feel old ;-) I was a school kid in the 1990s, and as I remember it, the rule we were taught for such situations was: "multiply before division".
Same here! I began to wonder if I even could get the correct answer on a simple ecuation now.. 🤔
Have to show this to a teacher in any of our schools here in Sweden! I think none of the would go with the "new" answer that is 9.
Must be a cool 40 year old, with that pfp lol
I was taught the same rule as you(it was before 5-6 years btw and im not sure if it is changed now)
Yeah same here.😕
and here I thought I was going CRAZY/Senile and remembering it wrong🤔 either that or I've have travelled to some parallel universe
because when I was taught the order of operations in maths...I'm PRETTY sure we called it the "BOMDAS"...NOT 'BODMAS'🤨
which means we'd "multiply before division"
AND for me, the "O" stood for "Of" NOT "Order", as in a "Fraction 'OF' something".
and I was also told that "Of means to multiply" for example 3/4 of 12, which means you MULTIPLY by the top of the fraction and THEN divide by the bottom
SO, if "Of means to multiply" then SURELY it MUST be true, that we ""multiply before division"...RIGHT??🤔🤨😕😵
SERIOUSLY when did it change...and WHY?
Im just glad Im NOT the ONLY one who was taught the same way as me, so now Im feeling reassured that Im not misremembering.
...EITHER THAT, or all us 90s kids are just MENTAL lol
I just googled this and double checked problem and double checked with my calculator, and BOTH say the answer is 9😯, and yet everything I was taught tells me that it SHOULD be 1🤨😕
Seriously I have an Advanced Higher in Mathematics and yet according to this...IM WRONG😧
a calculator can NEVER be wrong...so I guess "I" MUST be😨🤯
Now this video has got me questioning what I know and what everything I've learnt😲
What is life? Is this real?😕😵
This one seems a little more complicated than the "2+2" viral math problem.
That's all they talked about when I was younger
It’s Fish
the one that says "2+2=6"?
Hahahahhaha
It’s not uncommon today to consider division’s to be fractions.
I graduated in 2020 and I calculated it as 1 with the “old way” which is just how I remember being taught, but realized quickly during the video what the other answer was.
Then you were taught wrong or you just didn't pay attention and your teacher passed you to get rid of you...
@@RS-fg5mf was nowhere near failing any math class ever in school thanks
@@unsolve9162 If you were being taught wrong or your teacher wanted to get rid of you then of course you would have passing grades..
Wrong is still wrong, regardless of the reason. The correct answer is 9 not 1
@@unsolve9162 R S has been massively pwned on a very similar video. Ignore him/her!
@@tommy8290 what happened in the other comment?
Quick thought experiment:
What is (4+2)? Easy, 6. But why?
(4+2)=
(6)=
6
Here lies the problem. The parenthesis does not simply go away. There is a hidden expression overlooked. (6) actually means
(6)=
1(6)=
6
In fact, for any number by itself, you can express it differently. 8=4(2) or 1(8). The parenthesis bounds the expression as a term. Any manipulation (i.e. functions) outside the expression changes the meaning.
Regardless, the problem is poorly written to draw up ambiguity.
1:59 “And this gets us to the correct answer of dine”
Nicolas T-R 😂😂
You brainwashed me and now that’s all I here instead I of nine
lol
Speechless of laughter 🤭😂😆
lol
Watching 5th grade math at 2AM even though I'm 15🤦♂️
well theres only one truth.
math is attractive 😂😂
Sameee 😂
Same
Im 11
Im 15 too
In logical maths at higher levels, division is always a fraction. When simplifying, the logical answer is 1.
How
@@Adversity2175 How isn't it 1. Anyone says that it's not one, was schooled incorrectly.
Fractions have implied parenthesis
@@patricklee4581 did you just ignore the entire video?
I guess I was looking at it with old eyes, I got 1.
It doesnt matter what the answer is we just shouldnt argue over this stuff
Well if people could stop making UA-cams like this one which give the wrong answer, then people would stop arguing about it.
In my country (as of 2021 to be clear), we use that historical way because I actually got 1 as the answer.
The correct answer is 9.
6
6/2(1+2) = --- (1+2) = 9
2
6
6/(2(1+2)) = ----------- = 1
2(1+2)
@@raymendez3403 idk about the west but according to how we read the question 6 is in the numerator while the rest is in the denominator so by simplifying the answer would be 1. Generally the bracket thing has to be done first but in these cases it is so better if they give questions in fractions rather than this confusing method.....
@@jayvishnuvenkatesh870 you need parenthesis to make everything on the right be the denominator. If there is no parenthesis then no priority between multiplication and division hence left to right.
There is nothing confusing about this equation only confused people with different answers with all respect.
The equation implies there's a parenthesis, no exponents then MD left to right. No need to make parenthesis on the left when it's already left to right
@@raymendez3403 I said idk about the western countries. According to the eastern countries parentheses actually is multiplication and it means the things r together which means in denominator, to differentiate we use the symbol × instead which of course is how u told
For me, its 9
funny how i got both answers and then blamed my math teacher for making me so indecisive
Same. I got both answers and then watched the video to see what obscure rule they would pull out of their hat.
Schools were teaching this way (getting the answer 1) far later than 1914 (I graduated in 86), regardless of whatever the text book said. It was never directly addressed/explained, but the correct answer would have been 1 in all classes I took in high school. Without explicit explanation, we just learned to do it that way. Made it through college with a math heavy degree (one class short of a math minor), aced every math class, and never had any trouble using this approach (maybe the instructors tolerated both approaches, or avoided the ambiguity, don't recall?)
I've even taught my own kids this, with a smug "trust me I know." Ooops.
They are uhm still teachinf it lmao. In the early 2000s i was taught to follow pemdas and you do every word order first. So if there were multiplication and division left. You would complete multiplication first then complete division and then keep following pemdas until you solve the equation.
I graduated in 87 and I was taught with the answer being 9. I think it depends on the instructor.
You were taught incorrectly. We don't teach it that way, at least not those of us qualified to teach math. You could easily determine that 1 is not a correct answer by using distribution.
@@sirtykai3821 Regardless of the acronym you choose, multiplication and division are Inverse operations with equal priority that are solved in the order in which they appear from left to right. Same with addition and subtraction. There are 4 steps to the order of operations, not 6.
@@jamesfiddler1976 Apparently, but that doesn't help as much as you might think. Do you interpret the expression to mean 6/(2*(1+2)) or 6/2*(1+2)? Distribution doesn't help here, that is the crux of the issue. I agree that the later is simpler and more logically consistent, and apparently the way we should have been interpreting it since 1914, but these sorts of conventions have a way of sticking around. It also strikes me as weird and inelegant, but that's probably my own normalcy bias.
There were (and apparently are, based on the comments) teachers teaching it this way for a long time after 1914, I suspect because their teachers taught them that way, and so on going back several generations. If I were to put it into words, without an explicit * operand, the 2 is treated as part of the parenthesized expression. Which is fine, if that's the notation that everyone follows.