The many, many ways superscript is used

Welcome to the game of guess the notation! Today, we have the following contenders:

• $f^{-1}$
• $f^n$
• $f^{(n)}$

(for $n\in\mathbb{N}$). Well, can you guess the notation?

Probably not. Out of the many abuses of notation in Mathematics I think that superscript is defiantly the one of the biggest (if not the biggest) abused notations. Just for fun, here are two completely different answers for the list above:

It seems very funny, to me, that this is just an accepted thing. Many high-school students struggle especially with the first one, and with trigonometric functions and their inverse $\sin x$ vs. $\arcsin x$ ("$\sin^{-1} x$"). But worry do not! for there are many more examples of abuse similar to this, regarding the superscript:

• When declaring an point of dimension $n$: $M(x_1, x_2, \ldots, x_n)$, declaring another point gets a bit funny: $M’(x_1 ^0, x_2 ^0, \ldots, x_n ^0)$ (no, this isn’t a point consisting of only $1$s :)
• Thankfully, only the trigonometric functions suffer the issue with ${}^{-1}$. The expression $\ln^{-1} x$ is very clear.
• Derivatives are intuitive: When you raise the derivative (operator) to a power $n$ — $\partial^n$, it means that you are taking it $n$ times! …and not raising the derivative to $n$, obviously.
• Set theory, which is the best theory, agrees that the superscript should be reserved to mean power of, so $A^n$ means $A \times A \times \underbrace{\ldots}_{n-3 \text{ times}} \times A$, and so $A^B$ means… ugh.
• Another Set theory one: while I (and many more, I hope) use $\mathcal P$ to denote a Power set, I’ve seen some use $2^A$ to denote a Power set, which I believe is the worst notation, ever (what is a scalar raised to a set??)

a never ending list. Last updated: November 26th, 2021.