[burnt-resistor]

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.

[yorwba]

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.

[NooneAtAll3]

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

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

[syockit]

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

[Armisael16]

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

[fph]

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

[jstanley]

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

[contubernio]

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

[MForster]

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

[notarget137]

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

[kccqzy]

> 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.

[exe34]

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.

[crubier]

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

So no, it’s not unconditionally correct either.

[oh_my_goodness]

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.

[SoftTalker]

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