TITLE: $\sqrt 2$ is even?
QUESTION [15 upvotes]: Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n?
Similar argument for odd numbers should give $\sqrt[n]{k}$ is even or odd when k is even odd.
My question is does any of above has been considered under a more formal subject or it is a correct/nonsensical observation ?

REPLY [15 votes]: Other answerers are answering several slightly different interpretations of your question; but taking literally what you asked,

Is it mathematically acceptable to use “Prove if $n^2$ is even, then $n$ is even.” to conclude since 2 is even then $\sqrt 2$ is even?

the answer is a quite interesting “no, but also yes”.
Formally, the answer is definitely no: as @john explained, you can’t use that theorem to conclude what you suggest, since the theorem and proof start by assuming that n is an integer, so $\sqrt{2}$ is not a valid value for $n$.
However, it is excellent and very acceptable mathematical practice to do what you did, and take that theorem as inspiration for considering a new generalisation of the concepts involved.  And, as @KeyIdeas describes, this can lead to some very nice theories.
TL;DR: you can’t conclude this from that theorem/proof, but you can certainly be inspired along these lines by them.