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.
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.
@@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.?
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.
If you substitute all numbers with variables: a÷b(c+d) = You would get a/b(c+d) = a/(bc+bd) Not a/b × (c+d) That's how algebraic functions are ordered.
That's using implicit multiplication vs bodmas. You're coming with the assumption that b(c+d) multiplication holds higher priority than a/b part. That's all it is
@@spamspamspamspam3459it does because you first need to resolve the parenthesis. They are not resolved until you distribute them out (by multiplying the coefficient into the brackets).
@@spamspamspamspam3459my father studied to masters in mathematics, he has stated that in different areas both methods are correct. Within South Africa, and from what I have gathered much of the rest of the world) that is exactly how it is done. Simple reason, it should not matter in which order one does the multiplication or division whether right to left or left to right as long as it does not combine any addition or subtraction.
@@spamspamspamspam3459 again, differs from location. However in my country, that is the correct method to resolve the parenthesis. Notably from the logic my father gave me, this method means that it does not matter in which order you do the multiplication and division once the brackets are solved. The method you use requires that one solves the equation in a specific order lest the answer be different (I. E. If one first does multiplication before division). As was said, the purpose of the bodmas is so that regardless of the order one solves the equation (within each relative Order) in that it will arrive at the same answer. The method we use, the order in which division and multiplication is done matters not.
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??
I graduated in 2003. I was and am pretty good in math subjects. I was taught to solve this with the answer of 1. The brackets are to be dealt with b4 other division of multiplication occurs
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.
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.
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 can ensure you that in most Stem environments the symbols ÷ and / are pretty much forbidden, every division must be written as a fraction, so all formulas and expressions are just sequences of products of franctions, and the length of an horizontal line is clearer than any pemdas rule
6 ------(1+2) = 6÷2(1+2)= 9 2 6 ---------- = 6÷(2(1+2))=1 2(1+2) WHY?? Because 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 Objective facts...
@@realGBx64 Most people confuse and conflate an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity by this convention to Parenthetical Implicit Multiplication. They are not the same thing... 1/2x = 1/(2*x) by Algebraic Convention 1/2x^3= 1/(2*x^3) by Algebraic Convention 1/2(x)= (1/2)(x) by the Distributive Property 1/2(x^3)= (1/2)(x^3) by the Distributive Property... 1/2x and 1/2(x) are not the same thing.
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.
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?
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)=
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...
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 ?
"t’s just written the wrong way" - no it isn't. a(b+c) is the standard way to write a Factorised Term, solved with The Distributive Law, a(b+c)=(ab+ac).
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...
@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" ...
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.)
@@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...
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
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.
@@gajarajmaharjan881 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?
@@gajarajmaharjan881 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.
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)
@@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.
@@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.
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.
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
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.
I do not recall the "same precedence" rule. So, I have always taken PEMDAS to be performed, literally, left-to-right. But, knowing the rule now, I think a better mnemonic would be (P)(E)(MD)(AS). It will be mine from now on! Thanks for the lesson.
@@InsanityoftheSanities Jokes on you, we can assume that who asked is an imaginary number. An imaginary number that we know of is sqrt(-1) which is i. Conclusion: I asked.
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'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.
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
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..
First of all: 6/2(1+2) is the same like 6/2*(1+2). Even if it is not written, the * is between the 2 and the brackets. 6/2*(1+2) Solve the brackets first = 6/2*3 Solve the division = 3*3 Solve the multiplication = 9
It is not the same. A scientific calculator makes a difference between 6÷2(1+2) and 6÷2*(1+2). Mine gets 1 for 6÷2(1+2) and 9 for 6÷2(1+2). It's simply not the same.
after you add the numbers in the brackets, its not a bracket anymore, it simply becomea 6 ÷ 2 × 3- then because in BODMAS division comes before multiplication, you do 6÷2 which is 3, and then multiply that by 3.
I haven't read any comments yet, but you could also be wrong in your interpretation of the order of operations. once you add 1 and 2 you're left with a 3 inside parentheses, and a convincing case may be made that since it's inside a parentheses, your're supposed to resolve that operation before the division, since parentheses trump multiplication/division... so again 1
Yeah, that's the logic my father taught me. It should not matter what order you do multiplication and division so long as one does not cross addition and subtraction (once parenthesis have been solved). The method being shown means that it is neccesary to solve the division and multiplication in a specific order because you will either end up with a 6/6 or a 3*3 depending on which order you do multiplication and division.
This means you were taught incorrectly or that you remembered it wrong. There is no ambiguity for modern usage of mathematics, and the RU-vidr you just watched said it himself. DO NOT distribute before dividing because you are essentially multiplying that number to the other numbers inside the parenthesis, which breaks the rule of precedence
@@FloraLemonYT You always distribute 1st as this is part of solving the parenthesis. The value of the parenthesis is 6 not 3. (2*1+2*2)=(2+4)=(6)=6 Remove Factor 2(1+2)=(2+4)=(6)=6 combine terms 2(3)=(2*3)=(6)=6 Although the common factor has been "removed" and written before the parenthesis. It remains a part of the parenthesis. It must figure in the evaluation of the parenthesis to get the correct value of 6. This is why they say that the 2 is stuck to... or a prisoner of...- the parenthesis, it can not be used anywhere else or the parenthesis will not evaluate properly. So 6/2(1+2) = ? 6 / 2(1+2) = ? 6 / 2(3) = ? 6 / 6 = 1
@@mikestuart7674 the coefficient of a set of parentheses is NOT part of the parentheses because it is essentially a means of multiplication. Thus, you are multiplying first while there is a division operation behind it, making it a non legal method.
Not sure where you're getting your "modern interpretation" from but certainyl in the UK 6 ÷ 2(1+2) wouldn't be treated as (6 ÷ 2)(1+2) because the implicit multiplication where no dot or multiplication symbol is used takes the same priority as the bracket. So, 6 ÷ 2(1+2) would be read as 6 ÷ 2y where y=1+2 If the original were written as 6 ÷ 2 x (1 + 2) then 9 is the correct answer but when written as 6 ÷ 2(1+2) 1 is still the correct answer.
You are wrong. It's the same thing whether it's written as 6 ÷2(1+2) or 6 ÷ 2 × (1 + 2). The multiplication symbol is implicit. The only way it could be written to equal 1 is 6 ÷ (2(1+2)).
Yep very sleepy there when I wrote that. I meant that in order of operations 2y is treated there as a single unit, 6 ÷ 2y = 6/(2y) rather than (6/2)y ie 3/y vs 3y
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
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...
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.
@@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.
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.
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.
@@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.
Seems like so, since there is an implied multiplication that _normally_ implies grouping with parentheses, so it's *6 ÷ 2(1+2) = 6 ÷ (2 × (1+2)),* not simply *6 ÷ 2 × (1+2).* Though I guess some people in the comments might not agree with this position (just like the author of this video).
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.
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?
The term 2(1+2) = x(a+b) = (xa+xb) = (2x1+2x2) = (6) Thus: 6÷2(1+2) = 6÷(2x1+2x2) = 6÷(2+4) = 6÷(6) = 1 There are 4 terms in this equation, not 5: 1) the 6 2) the 2(a+b) containing: 3) the 1 4) the 2 If it were written as 6÷2*(1+2) THEN there would be 5 terms and answer would be 9. 1) the 6 2) the 2 3) the (a+b) containing: 4) the 1 5) the 2 The 2(1+2) is a SINGLE term which must be resolved first before being divided by 6. The issue is that 2(1+2) --> (2x1+2x2) --> (6) RETAINS THE PARENTHESES thus resulting in 6÷(6) = 1
You distributed. Distribution is a property of multiplication and division. So essentially you multiplied first. But you can't do that! Parentheses go first, as stated in PEMDAS. So first you must reduce (1+2) to (3).
lohphat 2(3) is not a single unresolved term, it is two different resolved terms multiplied together. You can't just divide 6 by it, you must remove the brackets first and turn it into 2*3. Then you divide, so 6/2*3 = 3*3, which is 9. Remember, just because there is no multiplication sign (it's called implicit multiplication), it does not make it a single term - after you solved 2(1+2) into 2(3), you can (and HAVE to) remove the brackets, so 2(3) = 2*3.
lohphat You're not changing anything. Who taught you that 2(3) is not 2*3? It's called implicit multiplication. It is used when the multiplication sign can be substituted by a pair of brackets. It does not change the value of the equation at all. 2(3) = 2*3 in all cases. Just because the multiplication is implicit, it does not hold any priority over the normal (explicit) multiplication, unless you want to invent a new rule or something.
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.
The American mathematical society says in its style guide that multiplication with juxtaposition is done before division. That is generally how people do it. Most calculators do it so with the exception of TI. Even at the TI they admit it should be so but US math teachers insist that multiplication is always done at the same time as division.
Ummm the AMS gives priority to ALL multiplication over division and suggests that you use multiplication by the reciprocal instead of Division... The ANS is specifically stateing a change to the STANDARD Order of Operations in their STYLE GUIDE... The AMS is also not dealing with basic 4th grade arithmetic expressions. While I don't like the idea of the AMS going against the standard rules and principles of math they have inducated it was for formatting and printing purposes and it only applies to their publication... 6÷2(1+2) = 6*2⁻¹(1+2) is how the AMS prefer you write it. The majority of scientific calculators and online math engines give you the correct answer 9. CASIO has admitted to programming different models for different markets based on popularity and opinion not the rule of math. 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
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.
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.
@@sensei5668 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 "/"
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 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.
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.
My gut instinct was 1. I assume most people who go through higher math or science courses will naturally gravitate toward 1. To get 9 as the answer you'd be limiting yourself to notational rules and not applying the formula in any way with applied meaning.
Now that I think about it a bit more, it might be illuminating to check the math by rewriting the problem as an algebraic formula: 6÷2(1+x)=1, solve for x. First you'd distribute the parenthetical expression and get 6÷(2+2x)=1. Then you'd multiply both sides of the equation to get 6=2+2x. Then subtract 2 from both sides to get 4=2x and thus through division you see that x=2.
Yeah I think they are not getting that 2(x+1) whilst it looks the same as 2*(x+1), it is NOT as the brackets are still in play. You would be required to drop the operator precedence to change 2(x+1) to 2*(x+1) The former is a single calculation and the later is two calculations.
It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and modern international standards like ISO-80000-1 mention about division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. Even over in America where the programming interpretation is more popular, the American Mathematical Society stated it was ambiguous notation too. Multiple professors and mathematicians have said so also like: Prof. Steven Strogatz, Dr. Trevor Bazett, Dr. Jared Antrobus, Prof. Keith Devlin, Prof. Anita O'Mellan (an award winning mathematics professor no less), Prof. Jordan Ellenberg, David Darling, Matt Parker, David Linkletter, Eddie Woo etc. Even scientific calculators don't agree on one interpretation or the other. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation. Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation. TI later changed to the programming interpretation but when I asked them were unable to find the reason why. A recent example from another commenter: Intermediate Algebra, 4th edition (Roland Larson and Robert Hostetler) c. 2005 that while giving the order of operations, includes a sidebar study tip saying the order of operations applies when multiplication is indicated by × or • When the multiplication is implied by parenthesis it has a higher priority than the Left-to-Right rule. It then gives the example 8 ÷ 4(2) = 8 ÷ 8 = 1 but 8 ÷ 4 • 2 = 2 • 2 = 4
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 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.
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.
Forget historical arguments. I would claim that 2(3) falls under the category of "collecting like terms", which has a higher precedence than PEMDAS/BODMAS. Nobody would argue that "2/3x = 2/3 of x" , this is exactly the same scenario because "2(3)" is an *implicit* multiplication, which makes it a term
You FAIL to understand TERMS... People confuse and conflate two different types of Implicit multiplication .... One without a delimiter and one with a delimiter.. Type 1... Implicit Multiplication between a coefficient and variable... A special relationship given to coefficients and variables that are directly prefixed i.e. juxstaposed WITHOUT a delimiter and forms a composite quantity by Algebraic Convention... Example 2y or BC This type of Implicit Multiplication is given priority over Division and most other operations but not all other operations... This can be seen in most Algebra text books or Physics book. Physics uses this type of Implicit Multiplication quite heavily.. Type 2... Implicit Multiplication between a TERM and a Parenthetical value that have been juxstaposed without an explicit operator but WITH a delimiter...The parentheses serve to delimit the two sub-expressions.. Parenthetical implicit multiplication. The act of placing a constant, variable or TERM next to parentheses without a physical operator. The multiplication SYMBOL is implicit, implied though not plainly expressed, meaning you multiply the constant, variable or TERM with the value of the parentheses or across each TERM within the parenthetical sub-expression. Parentheses group and give priority to operations WITHIN the symbol of INCLUSION not outside the symbol. Terms are separated by addition and subtraction not multiplication or division. The axiom for the Distributive Property is a(b+c)= ab+ac but what most people fail to understand is that each of those variables represents a constant value OR a set of operations that represent a constant value... A single TERM expression like 6÷2(1+2) has two sub-expressions. The single TERM sub-expression 6÷2 juxstaposed to the two TERM parenthetical sub-expression 1+2. The lack of an explicit operator implies multiplication between the TERM or TERM value outside the parentheses and the parenthetical value or across each TERM within the parenthetical sub-expression... The parentheses DELIMIT the TERM 6÷2 from the two TERMS 1+2 maintaining comparison and contrast between the two elements... Implicit multiplication is always by juxstaposition but not all juxstaposition is Implicit multiplication. Example 2½ = 2.5 not 2 times ½... There is “implicit multiplication” WITH delimiters and there is “implicit multiplication WITHOUT delimiters. Two different types of Implicit multiplication and mathematically different. 6÷2y the 2y has no delimiter.... 6÷2y=3÷y by Algebraic Convention. 6÷2(a+b) has a delimiter... 6÷2(a+b)= 3a+3b by the Distributive Property... 6y÷2y = 6y÷(2y) = 6y÷(2*y) 6y÷2(y)= (6y÷2)(y)= 6y÷2*y 6y÷2y(y)= (6y÷(2y))(y)= 6y÷(2y)*y= 6y÷(2*y)*y ÷2y the denominator is 2y ÷2(y) the denominator is 2
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
@@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!
I also got 1. But that is because I do not see 3(2+1) and 3x2+1 as being the same. It seems logical to me that if the lack of space between the "3" and the bracket means that this 3 is multiplied by whatever is in the brackets then they should be viewed as inseparable and therefore that operation comes first. Then, and only then is the product a usable number in the overall equation which is the division of 6 by whatever is on the right side of what was obviously meant to be the last operation, the division itself.
Oh, wait... I used the wrong original equation. I meant that the "2" is up against the bracket. so from 6 -:- 2(2+1) I get 6 -:- 2(3), then 6 -:- 6 = 1
Glad I'm not alone. ;-) I remember when math made logical sense.. Now they change a detail somewhere in the text and everything is off, again! How are we supposed to use this "universal language" that is Math to communicate with aliens if we can't even agree on the rules?! There's only so much you can do with tinfoil. LOL
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.
@@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
Me too! And I was born long after 1917. My scientific calculator may be old, but it too was built long after 1917 and according to that calculator the answer is 1.
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
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.
Avery C. realy? where i learned it was the multiplication and division were at the same step, to decide what comes first you ho left to right. The way i learned it, the answer would be 9
Bilal Safdar what I mean is the answer is 1. Division and multiplication are right to left, but for this you multiply the parentheses by what is right outside.
Even in Russia the fraction a over b must give the same result as a divided by b combined with the distributive law the solution must solve a/b*(c+d) = (c*a/b-d*a/b) => 6/2*(1+2) = (1*6/2+2*6/2) = 9
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
