Parity for non integers.

Question

First thing, the set of non integers: $\mathbb{R-Z=L}$

So could we have a parity for the numbers in set $\mathbb{L}$?

I thought of it and here is my work:

Let $c$ be in $\mathbb{L}$, then $c$ is even if $\lfloor c\rfloor=2k$ where $k$ is in $\mathbb{N}$ and $c$ is odd if $\lfloor c\rfloor=2k+1$ where $k\in\mathbb{N}$.

Is this correct?

And then? This parity does not behave well with respect to sum and product, for instance. — Andreas Caranti, Dec 31 '13 at 16:04
But since this is your first post on MSE, I have forgotten to welcome you. Also, thanks for having formatted your question properly in $\LaTeX$. — Andreas Caranti, Dec 31 '13 at 16:07
And thank you, it took me a while to understand that thing. Happy new year! — InverseSquareRoot, Dec 31 '13 at 16:08
I would side with http://en.wikipedia.org/wiki/Parity_(mathematics) when it says "Parity does not apply to non-integer numbers." Maybe it's just me, but I don't see anything useful coming out of it. — Andreas Caranti, Dec 31 '13 at 16:09
There is a way to make sense of parity of rational numbers; call a rational $a/b$ in lowest terms even or odd according to whether $a$ is even or odd. The product of two odds will be odd, the product of two evens, even, but the product of odd and even could be either. You can work out how it goes for sums. — Gerry Myerson, Dec 31 '13 at 16:20

score 3 · Answer 1 · edited Apr 13 '17 at 12:21

Parity arguments will not work in rings like $\,\Bbb R\,$ where $\,2\,$ is invertible. Indeed suppose that $\,2y=1.\,$ Then we deduce $\ x = (2y)x = 2(yx),\,$ so every element is even.

However, parity arguments will work in any ring $R$ which has a modular image that is the same as the integers modulo $2\,$ (i.e. $\,R/I \cong \Bbb Z/2).$ This is true for many familiar rings, e.g. the subring of rationals with odd denominator, various rings of algebraic integers, e.g. the Gaussian integers $\,m + n\, i\,$ for $\, m,n\in\Bbb Z.\,$ See this answer for further discussion and links (which also shows how to define parity to certain types of rings with infinite elements).

Parity for non integers.

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