I've always hated that it's called an “inverse”. It's not an inverse. The inverse of square root is square. If they had called it “reciprocal”, it would have been clear to me what it does, but “inverse” confused the hell out of me when I first saw it.
It doesnt though, inverse strictly speaking means the multiplicative inverse. The idea was extended in inverse functions eventually which for sake of brevity we also call the inverse. Now f^{-1}(x) and f(x)^{-1} are clearly different beasts the first would describe the square (for square root) while the second describes what quake was doing. Wether you call something an inverse or its full name (multiplicative/functional - inverse) depends on if its unclear from context which was meant.
the function inverse is the multiplicative inverse in the group of automorphisms over sets (when the multiplication operation is functional composition).
I think it's clear we both know that but for the sake of the commenter arguing the two things are different, it's does not help to simply say they the same without giving example, no?
Right, the bang per syllable loss is it's ambiguous, not that it's incorrect. Inverse square root could also mean -(x^0.5) if it meant the additive inverse, and it could mean x^2 if it meant the functional inverse, as said here.
Since calling the square an inverse square root makes zero sense in any practical context, the other common meaning being applied is obvious. In this case, it's not the inverse of a _function_, it's the inverse of a _number_, which is usually considered to be the same as the reciprocal.
The inverse of multiplying by 3 is dividing by 3, and vice-versa.
The inverse of squaring is squart root, and vice-versa.
The inverse of a number or other element (rather than an operation / function) means whatever number would form the inverse if you use it with whatever binary operation you're talking about. So the additive inverse of 3 is –3, the inverse of "rotate clockwise 90 degrees" in the group of geometric operations is "rotate anticlockwise 90 degrees", and the multiplicative inverse of 3 is 1/3.
Saying "the inverse of 3" to mean 1/3, i.e. its multiplicative inverse, is sloppy but acceptable so long as everyone knows what you mean (and it's fairly common). Saying "the inverse square root" to mean 1/square root is just wrong.
But sqrt · square = 1, where "sqrt: R⁺ → R⁺" is the square root operation, "square: R⁺ → R⁺" is the squaring operation, "1: R⁺ → R⁺" is the identity operation, and (·) is function composition: i.e., working in the monoid of functions R⁺ → R⁺. So x² and sqrt(x), as elements of R, are not inverses, but sqrt and square, as elements of R⁺ → R⁺, are.
It depends on how you parse the phrase "fast inverse square root".
This is true, but interpreting it one way is clearly absurd (nobody would call a square the “inverse square root”), which implicates that the other meaning was intended.
