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If division is the same thing as a fraction, the expression 8÷2(2+2) can be written as:

$\frac{8}{2(2+2)}$

$=\frac{8}{2(4)}$

$=\frac{8}{8}$

$=1$

In this case, the result would be 1. But if you solve it following the order of operations, the result would be 16.

$8\div2(2+2)$

$=8\div2(4)$

$=4(4)$

$=16$

So, is division really the same thing as a fraction??

reference: Is division the same thing as a fraction?

AdministrativeCow
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    The sooner we stop using the $\div$ symbol, the less confusion. It's unnecessary. That being said, the correct interpretation is the second one. – Rushabh Mehta Nov 27 '21 at 21:25
  • In the second example if you meant $8\div(2(2+2))$ then you need to write exactly that – Paul Nov 27 '21 at 21:28
  • @DonThousand I've heard that in Algebra the division sign (÷) is not used. Is it true?? What causes me confusion is that in Kumon, they keep using this sign even on Algebra lessons. – AdministrativeCow Nov 27 '21 at 21:39
  • It’s not obvious why, even with the fraction meaning, it isn’t $$\frac82\times(2+2).$$ Order of operation is, in general, a convention, and mathematicians rarely use the $\div$ symbol, so the rules there are vague. But the symbol, if synonymous with $/,$ usually would be read as $(8\div 2)(2+2).$ – Thomas Andrews Nov 27 '21 at 21:48

1 Answers1

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Despite what some teachers like to pretend, there is no universally observed "order of operations" convention for sequences of multiplication and division operators.

This is particularly true when the $\div$ symbol is used. Since this symbol doesn't generally appear at all in "serious" mathematical writing, there's little chance for people to converge on a definite convention for how to use it in complex situations.

Professional mathematicians generally prefer fraction notation for division; in that case the grouping of the operands is always unambiguous. For space efficiently, mathematicians also use a forward slash $x/y$, when the formula would otherwise be undesirably tall. With a forward slash there's general agreement (or so I thought; see comments for a dissent) that multiplication notated as juxtaposition of factors binds tighter than division by slash, so $x/2y$ is treated as just a typographical variant of $\frac{x}{2y}$. But you definitely cannot rely on such a convention if the division is written with $\div$ instead of $/$. Then all bets on how a random reader will interpret it are off again.

This partial agreement disappears completely if the multiplication is written with an explicit sign, either $\cdot$ or $\times$. You will find reasonable people disagreeing about how they'd interpret $x/2{\cdot}y$, for example.

Programming languages, as well as computer systems designed by programmers, generally tend to use * and / as multiplication and division signs. They will usually interpret a/b*c as $\frac ab\cdot c$ -- not because that is really an established convention in mathematics, but merely because it has to mean something (it is seen as uncharitable just to reject the input a/b*c as nonsense, even though a case could be made that it should have been). At last a/b*c=(a/b)*c is not clearly worse than the opposite interpretation, and also matches how the similar ambiguity in $a-b+c$ is (by firm mathematical convention) resolved.

Note well that this is not really a dispute about the underlying mathematics -- it is purely about how we write down our mental idea of a particular calculation with ink on paper, such that a reader can reconstruct the same idea in their head, if they know the notations and conventions being used.

When you write formulas yourself, do your reader a favor and put in explicit parentheses to make it clear which calculation you mean to specify if you're using a notation where this is not backed by a strong convention.

Troposphere
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    $x/2y$ means $\frac{x}{2y}?$ I’m not sure that is true. Wolfram Alpha definitely doesn’t interpret it this way. – Thomas Andrews Nov 27 '21 at 21:51
  • @ThomasAndrews: Well, that just underscores the lack of firm convention, then. :-) Though Wolfram Alpha is not a human reader. I wonder if anyone would differ between $x/2y$ and $x/2\pi$ ... or even $x/2\pi i$. – Troposphere Nov 27 '21 at 21:54
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    I usually ask. $x/2\pi$ could be either. For example, $\sin(x/2\pi)$ seems to me to hardly ever mean $x/(2\pi),$ but in other contexts, if $x$ was obviously in radians, I’d guess the writer meant $x/(2\pi).$ But no competent math writer would write it that way. I see this notation most commonly here by mostly-new question-askers, who don’t know where to put parens and also don’t know MathJax to write fractions. – Thomas Andrews Nov 27 '21 at 22:34