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I had a student write "$x^-$" as a "shorthand" for $x^{-1}$. Is anyone aware of a context where this is standard notation?

Edit: Since it appears that as far as anyone knows he did just make it up, it seems unlikely that anyone will ever have the same question. Hence I was planning on deleting the question. It's been suggested that I shouldn't delete it after it's been answered. Fine.

I may as well add this, so a reader might get something out of reading the question: I marked it wrong with a big question mark. He asked what the question was, I said I had no idea what $x^--$ meant, he said it was shorthand for $x^{-1}$.

[insert pause; timing...] So I told him his score of 40/50 was shorthand for 50/50.[rim shot]

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    $\begingroup$ Seems to be no standard notation at all. $\endgroup$
    – Wuestenfux
    Commented Feb 9, 2019 at 12:48
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    $\begingroup$ As if ${}^{-1}$ isn't already short enough! $\endgroup$
    – Clive Newstead
    Commented Feb 9, 2019 at 12:48
  • $\begingroup$ I don't see this notation nowhere. I think it's not a formal notation. $\endgroup$
    – BarzanHayati
    Commented Feb 9, 2019 at 13:04
  • $\begingroup$ Presumably that's meant to be a multiplicative analog of the notation $-x$ for additive inverses. But I'm not aware of any such single-symbol in wide use to denote multiplicative inverses. $\endgroup$
    – Bill Dubuque
    Commented Feb 9, 2019 at 13:05
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    $\begingroup$ If the student really understood what the negative exponent meant I'd have given the answer nearly full credit and used the discussion to talk about the need for clarity and precision. Lots of conventional notation is "shortcut" - he'd have been fine if he defined his. $\endgroup$
    – Ethan Bolker
    Commented Feb 9, 2019 at 13:53

2 Answers 2

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I do not believe this is standard notation.

The only place where I have seen a superscript minus sign is in contexts where we're interested in separating a function into its positive and negative parts (e.g. in Lebesgue integration).

Specifically, given $f : X \to \mathbb{R}$, define $$f^+(x) = \begin{cases} f(x) & \text{if } f(x) \ge 0 \\ 0 & \text{otherwise} \end{cases} \quad \text{and} \quad f^-(x) = \begin{cases} -f(x) & \text{if } f(x) \le 0 \\ 0 & \text{otherwise} \end{cases}$$

Then $f^+$ and $f^-$ are non-negative-valued and $f = f^+ - f^-$.

Obviously this has nothing to do with the reciprocal or inverse, but I thought I'd add it because I didn't think "I do not believe this is standard notation" was worthy of an answer on its own.

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I don't think it should be accepted, but I have an idea why a student may have made it up.

In chemistry it is standard to write for example $Cl^-$, whereas for higher charging states numbers are usually added, e.g. $P^{2-}$, $P^{3-}$ etc; although I have also seen things like $P^=$ instead of $P^{2-}$. But when charge is only 1 e, the number is never written i.e. always $Cl^-$ never $Cl^{1-}$ and certainly not $Cl^{-1}$.

Of course ion charge states are completely different from mathematical exponentiation. They both just happen to use superscripts; and in chemistry these are symbols, not numbers.

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    $\begingroup$ Excellent answer, thanks. Alas you're not going to get any points for it, because I intend to delete the question soon, since it's unlikely to be of any interest to future readers. $\endgroup$
    – David C. Ullrich
    Commented Feb 9, 2019 at 13:12
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    $\begingroup$ @David Please don't delete your question after folks have spent their time composing answers. $\endgroup$
    – Bill Dubuque
    Commented Feb 9, 2019 at 13:14
  • $\begingroup$ @BillDubuque Ok. I'm surprised that the question actually got upvotes instead of complaints about it being off-topic - I figured simply deleting it myself would save the MSE police a bit of trouble. $\endgroup$
    – David C. Ullrich
    Commented Feb 9, 2019 at 13:18
  • $\begingroup$ Well glad to help anyway :-) $\endgroup$
    – J.P.
    Commented Feb 9, 2019 at 13:20

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