'Adam and Eve' or 'Odd and Even'

The following presents a few simple rules to quickly assess and construct odd timelines.

Algebraic Representation

As a software engineer, the first idea that comes to mind to test the evenness of a number x would be:


odd number:  x % 2 != 0
even number: x % 2 == 0

This approach is inadequate for algebraic transformation, so here's a better representation:


odd number:  2m + 1
even number: 2n

Some Rules


... and their proofs:

1. The sum of even numbers is even.


2a + 2b + 2c + ... = 2(a + b + c + ...) = 2n.

2. The sum/difference of an even and an odd number is itself odd.


2a + 2b + 1 = 2(a + b) + 1 = 2m + 1.
2a - 2b - 1 = 2(a - b) - 1 = 2m - 1. (difference when subtracting odd from even)
2a + 1 - 2b = 2(a - b) + 1 = 2m + 1. (difference when subtracting even from odd: not commutative)

3. The sum of k odd numbers inherits oddness of k.


2a + 1 + 2b + 1 + 2c + 1 + ... = 2(a + b + c + ...) + k = 2m + k.
2m is even, so sum inherits oddness of k.

4. The difference of k odd numbers inherits oddness of k.

(This is a corollary to 3.)


2a + 1 - 2b - 1 = 2(a - b) = 2m.
(2a + 1) - (2b + 1) - (2c + 1) = 2(a - b - c) - 1 = 2n - 1.
(2a + 1) - (2b + 1) - (2c + 1) - (2d + 1) = 2 (a - b - c - d - 1) = 2k.
Generally, the first + 1 and the second - 1 cancel each other out, thus
an even number of 1s is subtracted from k, thus not changing the oddness
of k, thus oddness of k is inherited by the total difference.

5. The difference of two odd numbers is even.

(This is a corollary to 3 and 4.)


2a + 1 - 2b - 1 = 2(a - b) = 2m.
2m is even.

6. The product of only odd multipliers is odd.


By definition, none of the multipliers has 2 as its factor, hence neither has the product, hence the product is odd.