Hi Prof. Crow, I am a fan and a subscriber, so I offer the following with the greatest of respect -- I am so glad that you mentioned signs of grouping symbols, because here is the real issue with these order of operations problems on RU-vid: they are using the symbol for division (the line with the two dots) instead of the actual operator for division. That's right, they are not the same thing. Division is one of the few math operations that has both a symbol and an operator. There are two, and only two operators for division: the venculum (like the line between a numerator and denominator) and the long division box. By definition, these operators function as grouping symbols also. As signs of grouping, any ambiguity as to order of multiplying or dividing is generally removed. I am guessing that the use of the symbol as an operator comes from grade school when we are first learning how to divide - and that works as long as it is understood that only the number in front of the symbol and the number immediately after the symbol are the only operands to be considered. As to the problem here, the answer is definitely 1, as malcolmtaylor pointed out. This is verified by Casio and how they treat the notation. For example, when I enter the problem exactly as written, my Casio fx-991EX Classwiz calculator will automatically put parentheses around the factor 3(2) so that it looks like this: (3(2)). This is the juxtaposition that malcolm is talking about. Here's a quote: "Latest models of EDUCATION APPROVED SCIENTIFIC CALCULATORS Like CASIO, SHARP & CANON used in board exams give higher priority to implicit multiplication or multiplication by juxtaposition." Now, if you enter the problem on the calculator using 3*(2) or 3*2, you will get the answer of 4. Even so, forget about the juxtaposition for a moment and look at the order of operations. Parentheses must be cleared before exponentiation, and both of those before doing multiplication, division, addition, and subtraction. You are absolutely correct in saying that multiplication and division are done in the order they are encountered going from left to right, but what is the first operation that you see in the problem? By order of operations rule, you have to multiply the 3 times the 2 to clear the parentheses, then you must do the exponentiation. This leaves you with 36 divided by 6 divided by 6, which results in an answer of 1. The real problem here is not the math or the order of operations; it is the syntax used in writing the problem. The syntax being used here must generate 1 as the answer because of the precedence given by implicit multiplication . Change the syntax by using explicit multiplication and it must generate the answer of 4. If anyone is interested, just enter "juxtaposition in multiplication" in a search box and you will find any number of entries that talk about and explain the idea and its precedence.
I just did a little investigation. I entered the expression in google, wolfram alpha, and on ti 30x II calculator and all of them gave me an answer of 4. I also found 2 videos with very similar problems. Links are provided below. www.google.com/gasearch?q=6%5E2/3(2)/6&source=sh/x/gs/m2/5#fpstate=ive&vld=cid:4f73abd2,vid:tvjJppmx9Hk,st:0 www.google.com/gasearch?q=6%5E2/3(2)/6&source=sh/x/gs/m2/5#fpstate=ive&vld=cid:fdd744f0,vid:gBd-qYaEWq8,st:0
@@crowsmathclass Yes, I get the same answer of 4 on my TI. Only Casio and the others that are mentioned in my quote seem to pay any attention to the change. By the way, I very much enjoyed your series of videos on applied calculus - you're a great teacher!
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 4: Casio FX 50FH, Casio FX 82ES, Casio FX 83ES, Casio 991ES, Casio 570ES, TI 86, TI 83 Plus, TI 84 Plus, TI 30X, TI 89. Calculator manufacturers like CASIO have said they took expertise from the educational community in choosing how to implement multiplication by juxtaposition and mostly use the academic interpretation which implies grouping (1). Just like Sharp does. TI who said implicit multiplication has higher priority to allow users to enter expressions in the same manner as they would be written (TI knowledge base 11773) so also used the academic interpretation (1). TI later changed to the programming/literal interpretation (4) but when I asked them were unable to find the reason why. Some commenters have said it was pressure form American teachers but I've no confirmation of that. It's well known using multiplication by juxtaposition after division is ambiguous. The American Mathematical Society stated it was ambiguous notation (Mike Breen, official spokesperson). Modern international standards like ISO-80000-1 mentions about writing division on one line with multiplication or division directly after and that brackets are required to remove ambiguity. You can see more of the Harvard and Berkeley maths ambiguity pages on this ambiguity and The PEMDAS Paradox, a paper by a PhD student. It's simply ambiguous notation. A trick. Academically, multiplication by juxtaposition implies grouping but the programming/literal interpretation does not. That's the issue. You can't prove either answer since it comes from notation conventions, not any rules of maths. Wolfram Alpha's Solidus article mentions the a/bc ambiguity and you'll see it interprets 6²÷3(2) as 24 6²÷3a @a=(2) as 24 (unexpected) 6²÷ba @a=(2) and b=3 as 6 (expected). So, online calculators aren't clear either and certainly aren't evidence for one answer over the other. DESMOS, for example, doesn't allow one line division by default, removing the ambiguity from being possible. An excellent feature. It promotes good notation writing. Multiple professors and mathematicians have said it's ambiguous 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. 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 So, both answers are acceptable. The expression itself is not, it's terrible terrible writing.
This video is a waste of time. It doesn't address the source of the controversy and declares an answer (because I say so). The source of the controversy is that different people use different conventions. The SOLUTION is to understand that it may be ambiguous and write the question more clearly. Personally I feel that the answer is 1, and I think that most people who do a graduate level degree in math/physics/engineering would agree. Here are three videos that actually teach you something rather than just spreading more confusion. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-lLCDca6dYpA.htmlsi=EiaMkC92SJYZEhOD ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-4x-BcYCiKCk.htmlsi=2S8YyB2c0GPftMve ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Q0przEtP19s.htmlsi=ioZYUHrSgI5HleyX
@@pandadrumming Except it IS a use of parentheses. if they had wanted to represent it as multiplication, they would have. Mathematics is not a game, it is a precise use of numbers and variables with specific sets of operators (and other symbols) to represent the types of actions to be performed.
@@crowsmathclass Perhaps you should do the topic justice and delve into the nuance like some other well researched videos. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-lLCDca6dYpA.htmlsi=EiaMkC92SJYZEhOD ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-4x-BcYCiKCk.htmlsi=2S8YyB2c0GPftMve ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Q0przEtP19s.htmlsi=ioZYUHrSgI5HleyX I disagree with your answer, and think people with degrees in math/physics/engineering would too. The real problem is that the question is ambiguous since the convention is so heavily disputed. And your video only adds to the confusion rather than resolving it.
