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Where in a conventionally ordered list of constants does $\pi$ go?

For example, if you look at the attached image, you can see that in my last answer, I’ve put $h \pi$. But in the answer section of the book I’m working out of, they put $\pi h$.

I’ve read that constants ($ABCDEFG$, etc…) should be ordered alphabetically, but how about for symbolic/non-variable constants (or whatever the correct terminology is) such as $\pi$, $\phi$, $i$, and $e$?

Is there any such rule for this?

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    $\begingroup$ $3\pi$ looks better than $\pi3$, but $\pi r$ looks better than $r\pi$. $a+bi$ is the usual form for a complex number, but you'll almost always see $e^{i\pi}+1=0$, not $e^{\pi i}+1=0$. So, I suppose there isn't an actual "rule" for these things; just do whatever "looks nice", whatever that means. $\endgroup$ Oct 15, 2021 at 3:39
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    $\begingroup$ Seconding the "looks nice" rule, whatever saves mental effort is best. I suspect $e^{i\pi}$ is preferred because it is an evaluation of $e^{i \theta}$, and in general when we plug in $\theta \to \pi$ we don't commute them for many different reasons! The general orthography seems to be (general constants)(special constants)(variables) and we preserve order in function application, including those specific forms that come from more general forms. This of course all goes out the window when commutativity cannot be taken for granted. $\endgroup$ Oct 15, 2021 at 6:17
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    $\begingroup$ @JamesA But I would prefer $e^{2\pi i}$ to $e^{2i\pi}$. $\endgroup$ Oct 15, 2021 at 7:07
  • $\begingroup$ Largely if $n$ is integer, the form commonly used seems $n\pi$. OTOH, if $x$ in general could be real or complex, $\pi x$ seems to be the usage. For e.g. $\sin(n \pi x)$. But then there's the famous $e^{i\pi}+1=0$. $\endgroup$
    – Macavity
    Oct 15, 2021 at 10:00
  • $\begingroup$ It makes anyway more sense to order constants by their magnitude and not by their notation. Ordering constants alphabetically is a weird idea. $\endgroup$
    – Peter
    Oct 15, 2021 at 10:56

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Strictly speaking, it is mostly subjective; $h\pi\equiv\pi h$, so neither of them are wrong. But, some things just look nicer.

Of course, nobody is going to write $x17\ (17x)$, $e^x2\ (2e^x)$, or $\pi3\ (3\pi)$. But, it seems perfectly fine (to my eyes) to write $e^xy$ or $\pi\sqrt6$.

To make this slightly less biased, I did some tests on a Casio fx-82AU PLUS II. In the following table, I enter the expressions exactly as I entered them into the calculator. "N" means that the expression returned a syntax error and "Y" means that it evaluated. $$ \begin{array} {|r|r|} \hline \text{Expression} & \text{Evaluated?} \\ \hline 3\pi & Y \\ \hline \pi3 & N \\ \hline (\pi)(3) & Y \\ \hline (\pi)3 & N \\ \hline \pi(3) & Y \\ \hline \pi\log6 &Y \\ \hline \log(6)\pi &Y \\ \hline \pi\sqrt6 &Y \\ \hline \sqrt6\pi & Y \\ \hline \frac23\pi & Y \\ \hline \pi\frac23 & Y^1 \\ \hline |3|2 & N \\ \hline |3|(2) & Y \\ \hline \pi|3|\pi & Y \\ \hline \pi|3|\pi\sqrt3e & Y \\ \hline \pi|3|\pi\sqrt3e\frac23 & Y^1 \\ \hline \end{array} $$ $^1$The calculator automatically added brackets around the fraction when I pressed $=$, for example the screen said $\pi\left(\frac23\right)$.

$3\pi$ looks better than $\pi3$, but $\pi r^2$ looks better than $r^2\pi$. $a+bi$ is the usual form for a complex number, but you'll almost always see $e^{i\pi}+1=0$, not $e^{\pi i}+1=0$. But (as mentioned in the comments), $e^{2\pi i}$ is preferred over $e^{i2\pi}$, although we write $e^{i\theta}$ not $e^{\theta i}$.

Basically, there is no precise set of rules to follow when ordering things. (Assuming the operation is commutative, that is.) The way I usually see things done is: if you're dealing with the Latin alphabet, write in alphabetical order. Usually, integers or fractions (put the integers in the fractions (multiply) if there are fractions) go first, numbers with "operations" ($\sqrt,\ \log,\ \tan$, etc.) go next, and variables go last.

Of course, this won't always work, just write whatever looks the neatest and conveys what you're trying to say the clearest. To avoid any possibility of ambiguity, use brackets; brackets also seem to "soften the blow" of bad ordering; $x17$ just looks disgusting, whereas $(x)(17)$ is acceptable.

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