# Talk:Absolute Value Equals Square Root of Square

## My thoughts

In my opinion the best way, and most conventional/useful, to approach this would be to define $|x| := \sqrt{x^2}$ over the reals, rather than having this as a theorem. (even given a "good definition", this theorem would probably be trivial) Caliburn (talk) 13:05, 17 November 2017 (EST)

Until recently it was a definition. But the square root is too deep a notion to use it to define absolute value. See Definition talk:Absolute Value/Real Number --barto (talk) (contribs) 13:53, 17 November 2017 (EST)

## Deletion notice

It is actually explained at Definition talk:Absolute Value/Real Number what the purpose of the now reverted refactoring was. If you absolutely want to define the absolute value using $\sqrt{\cdot}$, you can, but you need an ad-hoc proof of the Intermediate Value Theorem to prove Existence and Uniqueness of Positive Square Root of Positive Real Number without mentioning continuity. Be careful. --barto (talk) (contribs) 05:56, 10 February 2018 (EST)

Okay, so if we want to go down that route we need to finish it. If the concerns you have are genuine, then the place to start would be to craft that rigorous definition of the square root that you recommend. I confess I can't see what needs to be done here.
Once that is done, and we have that area tied down, we can proceed to discuss whether the refactoring you recommend is appropriate. The changes you implemented were based on a statement that begins "I'm not sure".
That is, the problem I had, and the reason I reverted back to the existing structure, is that this page has no foundational backing.
If our current definition of "square root" based on the case-by-case analysis of the real number space axioms is inadequate (from my memory of the work done on the Munkres work, which I did a couple of years back, I'm sure this is not the case) then this needs to be addressed. There are several paths to rigor, depending on which axiomatic framework we use, and it may be a mistake to abandon them all in favour of just one. --prime mover (talk) 06:13, 10 February 2018 (EST)