Proof of x^2n beeing even and other fundamental proofs

[moriheru]

Is there a book containing fundamental proofs such as any number of the form x^2n beeing even and such.
I know this is very vague, so I must apologize.
Thanks for any help.

 

[andrewkirk]

What sort of proofs did you have in mind? x^2n ($x^{2n}$)is not necessarily even. It is always odd for x odd, so there is no proof of it.

 

[moriheru]

Sorry I meant a proof concerning all numbers with a even exponent beeing positive and vice versa.

 

[SrVishi]

To add in my two cents, this sounds like something you may find in a "transition to mathematics" book. I'm pretty sure at least some of them conatin many proofs like what you displayed above. If you can't acquire such books, it may be a great habit to, when encountering something like your theorem that powers of even numbers being even functions, tryproving it on your own first, and then look up a proof online, say proofwiki.

 

[slider142]

There are many proofs of this type of statement in Chapter 1 of Spivak's Calculus. Other fundamental results can be found in Stillwell's Mathematics and its History. Propositions particular to basic number theory can be found in Hardy and Wright's An Introduction to the Theory of Numbers. Fundamental results in algebra can be found in Artin's Algebra.