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Comment by hnarayanan | Hacker News Reader

This is a general pattern in CAS. For a more basic case, it’s not obvious sqrt(square(x)) will simplify to x without any further assumptions on x.

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I think you would get sqrt(x^2) = x, if x belonged to the natural domain of sqrt, which is a Riemann surface, that may also be defined using the language of "sheaves". I don't know how to connect this to the article or Mathematica.

it's literally the prototypical example for `Assuming`

https://reference.wolfram.com/language/ref/Assuming.html

That's not what it simplifies to using a real or complex number domains for x, it's abs(x). CAS need type inference assumptions and/or type qualifiers to be more powerful.

Edit: Fixed stuff.

Right, that's why you need further assumptions on x in order for that simplification to hold.

It's not a simplification, it's wrong. Sqrt(square(x)) equals abs(x).

It also equals x with appropriate assumptions (x > 0).

so there's an unconditionally correct answer (it's also equal to abs(x) for x>0), and then there is an answer that is only correct for half the domain, which requires an additional assumption.

sqrt(square(i)) != abs(i)

So no, it’s not unconditionally correct either.

Well, then sin(x) = x if x is infinitely small

> Assuming[x == 0, Simplify[Sin[x] == x]]

Mathematica returns True. And any middle schooler will also tell you it's true.

The only reasonable interpretation of "infinitely small" is that it's zero.

Not in general. As people have pointed out elsewhere, it's true if x is real. That isn't always a helpful assumption. (When x is real you can plug that assumption into Mathematica. Then Mathematica should agree with you.)

But consider sqrt(i) = sqrt(exp(i\pi/2)). That's exp(i\pi/4). Your rule would give 1 as the answer. It's not helpful for a serious math system to give that answer to this problem.

When I square 1 I don't get i.

For x = -i, square(x) = -1, sqrt(square(x)) = i. Meanwhile, abs(x) = 1. You're right that it simplifies to abs(x) for real x, but that no longer holds for arbitrary complex values.

for arbitrary complex values sqrt() gives 2 answers with +- signs

so sqrt(square(-i)) = +-i, one of which is x

I've never seen a CAS that gives two answers for sqrt. Mathematica doesn't, sympy doesn't, and IIRC Maxima also doesn't.

The sqrt function returns the principle square root, not both. That’s true for all numbers, positive, negative, and complex alike.

It's abs(x) only over the reals, for complex numbers it's more complicated.

That abs(x) (or |x| as we wrote it) used to catch out so many of us in HS trig and algebra.

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