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I humbly approach You and beg forgiveness, for I must ask the question: What is

$$ a/b*c $$

Note that I am very explicit in the way I write the expression. This expression, in this form, is presented on various platforms, where people (who I believe refer to themselves as "entertainers") present what seems to be an algebraic expression of the most simple kind to a unknowing person, in video form, and makes a mockery of said person should it not know the "correct" answer. By observing the comments of people who "consume" such material, there seem to be much debate of the truth of what is being presented. I seek a response to this debate that can be referred to as a definitive, community-approved answer!

My personal belief, as a failed mathematician, is that this is, at best, ambiguous due to notation, at worst explicit under some undefined set of conventions.

My reason for the first is based primarily on multiplication and division having same precedence - so the order in which they are evaluated should not matter. It is easy to prove by example that this is not the case. Another thing is that I find it unclear if the commutative law would suggest a/b should be commutative with c, or just b.

For the second, I keep hearing references to conventions such as "you always evaluate from left to right" (which I believe ignores the fundamental ideas of the associative and commutative laws) and BODMAS, where Division comes before Multiplication. But if you have been taught PEMDAS, where the order is reversed? In this case, the "truth" depends on what mnemonic device you have been taught!

If this question is not up to standard, I apologize.

J. W. Tanner
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MortenMC
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    A computer algebra system will perform it as $(a/b)\cdot c$, and not as $a/(b\cdot c)$. So usually people add brackets to make it clear what they want. – Dietrich Burde Dec 02 '22 at 10:30
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    Addition and subtraction is universally evaluated from left to right. No one is really ever confused about $a - b + c$. I am personally under the conviction that in the case of multiplication and division, however, this is ambiguous and should be avoided. – Arthur Dec 02 '22 at 10:31
  • Since multiplication and division have the same priority level , it would be reasonable to calculate from the left , but as Arthur pointed out , there is no clear convention. In the case of exponentiation, the convention is even to calculate from right : $a\uparrow b\uparrow c$ is meant as $a\uparrow (b\uparrow c)$. Best is in fact to simply avoid such ambiguities by using fractions. – Peter Dec 02 '22 at 10:37
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    This will be a duplicate of questions like https://math.stackexchange.com/questions/16502/do-values-attached-to-integers-have-implicit-parentheses and those linked to it. Since $a/bc$ is deliberately ambiguous, the question has in past not had a good answer. – Henry Dec 02 '22 at 10:48
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    Does this answer your question? Do values attached to integers have implicit parentheses? – Henry Dec 02 '22 at 10:53
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    "No one is really ever confused about $a−b+c.$" My experience tutoring students (usually ages $18$ or below) says otherwise! Students are confused about all sorts of basic things. – Adam Rubinson Dec 02 '22 at 12:20

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