The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log.

Continuing upwards hyperoperationally, what is the inverse of $↑↑$?

Whats more somtimes values that are fixed are given names, some being labels such as:

$2\times = \text{double}\\ \text{^}2 = \text{squared}$

And some that are also other functions, such as:

$\text{^}0.5 = \text{square root}\\\text{^}0.33... = \text{cube root}\\\text{^}-1 = \text{reciprocal}$

What are some for ↑↑?

  • $\begingroup$ ^0.25 is fourth root. $\endgroup$ – Akiva Weinberger Apr 9 '15 at 11:03
  • $\begingroup$ Also, I'm not sure I understand the question, but this might be of interest (the section labeled "Inverse relations"). $\endgroup$ – Akiva Weinberger Apr 9 '15 at 11:06
  • $\begingroup$ Thanks, I think maybe "Super Logarithm" then. 3 + 4 = 7, 7 - 4 = 3. 3 * 4 = 12, 12 / 4 = 3. 3 ^ 4 = 81, Log3 81 = 3. 3 ↑↑ 4 = h, SuperLog3(h) = 3. $\endgroup$ – alan2here Apr 9 '15 at 23:12
  • $\begingroup$ My proposal is to use "height" from the idea, that the integer version of the tetration is often understood as "power tower" (right associative) and is also derived from number of iterations, so "height" might be the most useful name, generalizable in the context of hyperoperations "iterative height of exponentiation", "of multiplication" etc. In my software I use thus $ \operatorname{hgh}(x)$ for this and $ \operatorname{hgh}(x_1,x_0)$ if I want precisely express the iteration-"height" from one $x_0$ to another $x_1$ by the (generalized) iteration of the current operation under consideration. $\endgroup$ – Gottfried Helms Apr 10 '15 at 7:27

For starters, the inverses of addition and multiplication don't care about order, since $-a+a=a-a$ and $\frac1a\cdot a=a\cdot\frac1a$. However, when saying "inverse of exponentiation", one must be clear. It can be taken to mean roots or logarithms, depending on which inverse you seek.

The inverses which you seek to find are mentioned on Wikipedia (tetration). The particular inverse you seek appears to be the super logarithm, which has various properties you can find on the relevant wikipedia page.


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