# '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 even odd)
2a + 1 - 2b = 2(a - b) + 1 = 2m + 1. (difference odd even: not ``commutative)
`

### 3. The sum of *k* odd numbers is odd if *k* is odd.

`
2a + 1 + 2b + 1 + 2c + 1 + ... = 2(a + b + c + ...) + k = 2m + k.
2m is even, so solely evenness of `*k* determines evenness of the sum.

### 4. The difference of two odd numbers is even.

(This is a corollary to 3.)
`
2a + 1 - 2b - 1 = 2(a - b).
2m is even, so solely evenness of `*k* determines evenness of the sum.

### 5. 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.
`

© Bernhard Wagner, May 27th, 2012.