Hackers 
they are when it comes to math 
Would subtraction also approximate division?
Yes.
Would negation also approximate the reciprocal?
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 heck out of me when I first saw it.
It's confusing for non-mathematicians, but (and you may know that) it is not incorrect. https://en.wikipedia.org/wiki/Inverse_element "In mathematics, the concept of an inverse element generalises the concepts of opposite (βx) and reciprocal (1/x) of numbers."
Shoulda just called it fast 
mapping()
"inverse" has a very specific meaning: inverse(x) * x = 1, x^2 * x != 1 for any x other than 1. So no, x^2 is not the inverse of sqrt(x)
The inverse is about the negative power right? Square root is the 0.5
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Two things that are (almost) monoids both behave similarly
what could explain this? 
perhaps they both...behave like monoids... 
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When I was curious about haskell I mostly wrapped my head around functors and etc, but I found it disorienting to approach the concepts from the perspective of my programming experience. My math stopped at linear 101, so I'm not even sure how to find the path that leads to practical mathematical comprehension of these terms.
I get the same feeling from church numerals and the lambda calculus. It's clear that these were useful in the math domain, but I had never seen anything like them before encountering them as a programmer, and have no idea who used them or for what purpose.
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It's just generalizing stuff more and more. Take a class on group theory or real analysis (as in properly starting from axioms of the reals and proving convergence theorems etc.) They're very fun and quite approachable subjects
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Wait, do you mean that because integer addition and float multiplication are both monoids, it would imply that float2int(x * y) β float2int(x) + float2int(y)?
That's not it at all, the relevant property here is that float2int approximates log2
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