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

MForster · 2026-03-15T10:09:18 1773569358

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

notarget137 · 2026-03-15T11:40:35 1773574835

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

kccqzy · 2026-03-16T01:43:57 1773625437

> 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 · 2026-03-15T11:08:15 1773572895

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 · 2026-03-15T14:02:16 1773583336

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

So no, it’s not unconditionally correct either.

oh_my_goodness · 2026-03-15T17:29:21 1773595761

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.