It's sometimes useful and interesting to generalize mathematical concepts - e.g., turning the familiar notion of number of things in a given set into the concept of cardinality of a set, etc. Lately I stumbled upon this Math.SE question about possible ways to generalize the concept of parity of integers to noninteger rationals and it got me thinking: why can't we just say that a rational is even iff its last digit represents an even number (of course we have to have some fixed base in mind, say the decimal one)? I.e., take a look at the following statement:
An integer X is even iff it can be represented as 2Y, where Y is an integer as well.
It is the definition of parity, and it makes conceptual sense. We can go further and say:
An integer is even iff its last digit represents an even integer.
(I.e. in decimal an integer is even iff its last digit is either '0', '2', '4', '6' or '8'. Also, to determine whether a given digit represents an even integer we have to use the definition, obviously.)
Now, the last statement makes technical sense - it's a perfectly valid mathematical theorem.
Here's the crux: to generalize the concept of parity, to me it seems that the only thing we should do is modify the last statement a bit, like so:
A rational number is even iff its last digit represents an even integer.
Here's what we just did: we came up with another definition under which the notion of parity for integers still holds and a new notion of parity for noninteger rationals emerges.
So we can say that, for example, the number 0.5 is odd, while 0.24 is even.
My question is: is this way of generalizing a mathematical concept valid at all?
P.S.It's also quite neat that if you replace "rational number" in the last definition by "real number" you get that any irrational number is both even and odd (vacuous truth, because no irrational has a last digit). So, we can say that any irrational is evodd (or maybe odden?) :)
EDIT: We're talking about the positive integer positional numeral systems here! Thanks to @Mauro ALLEGRANZA for making me state this explicitly.
EDIT 2: Never mind the infinite invisible zeros, e.g. even though 0.5=0.50=0.500=..., we still say that 0.5 is odd. Thanks to @Hunan Rostomyan for noticing this :)