Concept Notes 1: The Mathematics of Parity — Even & Odd Numbers
Parity is the mathematical property of an integer being either even or odd. Understanding parity rules provides powerful short-cuts for solving complex numerical problems.
1. Formal Algebraic Definitions of Parity
- Even Number ($2k$): Any integer divisible by $2$ without a remainder. Expressible as $2k$ (where $k \in \mathbb{Z}$).
Set: $\{\dots, -4, -2, 0, 2, 4, 6, 8, \dots\}$. (Note: $0$ is an even number!)
- Odd Number ($2k + 1$): Any integer that leaves a remainder of $1$ when divided by $2$. Expressible as $2k + 1$ (where $k \in \mathbb{Z}$).
Set: $\{\dots, -3, -1, 1, 3, 5, 7, \dots\}$.
2. Parity Arithmetic Rules
| Operation | Rule | Algebraic Proof | Example |
|---|---|---|---|
| Even + Even | Even | $2a + 2b = 2(a + b)$ | $4 + 6 = 10$ |
| Odd + Odd | Even | $(2a + 1) + (2b + 1) = 2(a + b + 1)$ | $3 + 5 = 8$ |
| Even + Odd | Odd | $2a + (2b + 1) = 2(a + b) + 1$ | $4 + 5 = 9$ |
| Even $\times$ Even | Even | $(2a)(2b) = 4ab = 2(2ab)$ | $4 \times 6 = 24$ |
| Odd $\times$ Odd | Odd | $(2a + 1)(2b + 1) = 2(2ab + a + b) + 1$ | $3 \times 5 = 15$ |
| Even $\times$ Odd | Even | $(2a)(2b + 1) = 2(2ab + a)$ | $4 \times 5 = 20$ |
Key Takeaway on Products: A product of integers is odd if and only if EVERY factor is odd. If even a single factor in a product is even, the entire product is even!