At school I asked the question and I kept wondering "Can fractions or decimals be odd or even?"
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There is a natural way to define "oddness" for fractions. For integer $x$, let $\nu_{2}(x)$ denote the number of $2$'s that divide $x$. For example, $\nu_2(6)=1, \nu_2(4)=2, \nu_2(12)=2, \nu_2(1)=0, \nu_2(7)=0$. We can leave $\nu_2(0)$ undefined, or set it equal to $\infty$, as you like.
We can extend this to fractions via $$\nu_2\left(\frac{m}{n}\right)=\nu_2(m)-\nu_2(n)$$
This satisfies the lovely relation $$\nu_2(xy)=\nu_2(x)+\nu_2(y)$$ which holds even when $x,y$ are fractions.
With this tool in hand, we can define a number $x$ as "odd" if $\nu_2(x)=0$. The product of two odd numbers is odd, while the product of an odd number and a non-odd number is non-odd. However there is no natural definition of "even" numbers. We could take $\nu_2(x)\neq 0$ (but then the product of two even numbers might be odd), or $\nu_2(x)>0$ (but then we need a third term for $\nu_2(x)<0$).
See also a more comprehensive answer here.
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Correct. I have some questions that I hope you can answer. – Shaelynn McCabe Sep 26 '14 at 16:04
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Why is this relation lovely? Is there any particular application for this generalization? I am just curious, thanks :) – Aditya P Apr 04 '17 at 16:21
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1@Aditya, (1) The relation is lovely because it relates multiplication with addition. (2) You are responding to a post from 2.5 years ago; usually this will not get you an answer. – vadim123 Apr 04 '17 at 19:15
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No, odd-ness and even-ness is defined only for Integers.
For more info: Parity
taninamdar
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Still, no. Unless you want to define it yourself. But if you ask about the standard definition, then no. – taninamdar Sep 19 '14 at 15:50
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You are right, I am just in a corny mood today, I was thinking about 0.99999999..... – imranfat Sep 19 '14 at 15:51
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$0.\overline{9}$ is same as $1$. So except for integers which have alternate non-terminating decimal representation, parity is not defined for other decimals. – taninamdar Sep 19 '14 at 15:52
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Parity does not apply to non-integer numbers.
A non-integer number is neither even nor odd.
Parity applies to integers and also functions. So I wouldn't say that parity only applies to integers.
For instance, if $f(x)=x^n$ and $n$ is an integer, then the parity of $n$ is the parity of the function.
k170
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