Is an irrational number odd or even? My sister just asked this question to me: "Is an irrational number odd or even?" I told her that decimals are not odd or even and that does imply that not recurring and non repeating decimals will also not be odd or even. But I want a rigorous math proof for this. Can someone help me with this proof? Also, can I do this proof by contradiction?
Definition. 


  Let $n\in \mathbb{Z}$ be an integer. $n$ is called even if ...


According to the definition only integers are even or odd. It is not something that you have to prove.
Closely related to other answers but might be good for the purpose. By contradiction:

For a given irrational a, assume it is odd.
a/2 is also irrational; call it b
2b is obviously even since odd + odd = even
so a is even.
So all irrationals are even, 
which shows they cannot be split into two categories at least.
All rationals can be expressed as the product of two irrationals 
(which is  just saying that a rational divided by an irrational is irrational) 
so all rationals are even too.


∴ 1 is even