Why don't we see a $\frac{\times}{ \div}$ like we see $\pm$?

Question

It's common to see a plus-minus ($\pm$), for example in describing error $$ t=72 \pm 3 $$ or in the quadratic formula $$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $$ or identities like $$ \sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B) $$

I've never seen an analogous version combining multiplication with division, something like $\frac{\times}{\div}$

Does this ever come up, and if not why?

I suspect it simply isn't as naturally useful as $\pm$.

Answer 1

Perhaps because $$ a\frac{\times}{\div}b $$ (typographically quite horrible) is written as $$ a\cdot b^{\pm1} $$

Answer 2

As you indicated, square root can be + or -. $\pm$ shows this ambiguity.

As far as I know, there is no similar use case where the choice is to multiply or divide by an expression.

Answer 3

There are many function (square roots for ex.) where $f(x) = f(-x)$ and for $f^{-1}$, the $\pm$ is useful. If there were common functions where $f(x) = f(\frac{1}{x})$ it might be a thing. Can anyone think of example functions like this?

Answer 4

I think that this question is primarily opinion-based, so here is my opinion:

• The expression $[t=72\pm3]$ is equivalent to $[t=72+(+3)]\vee[t=72+(-3)]$
• The expression $[t=72\frac{\times}{\div}3]$ would be equivalent to $[t=72\times3]\vee[t=72\times\frac13]$

So the second operand "looks the same" in the case of $\pm$ but not in the case of $\frac{\times}{\div}$.

If we had a different notation for $\frac13$ (for example, $\color\red3$), then it might have seemed more appropriate to denote something like $[t=72\frac{\times}{\div}3]$, which would be equivalent to $[t=72\times3]\vee[t=72\times\color\red3]$.

So it's basically a matter of "backward compatibility" with our existing notation for inverse...

Answer 5

The multiplication sign $\cdot$ is usually omited in abelian groups (or even non-abelian). I think the most important thing is, that something like $\div$ does mislead the reader, because if we consider $a\div b$, then $b$ is not on the same level as $a$, but in contrary the sumbology does suggest so. This can cause serious errors in calculations whereas $\frac{a}{b}$ is much more clearer as well as $ab^{-1}$. I think this levelness is the main point of never using $\div$.