Rational Numbers, Exponents, Squares & Cubes
1. Rational Numbers
Properties (Summary)
| Property | Addition | Multiplication |
|---|---|---|
| Closure | Sum of rationals = rational | Product of rationals = rational |
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a+b)+c = a+(b+c) | (ab)c = a(bc) |
| Identity | 0 (a + 0 = a) | 1 (a × 1 = a) |
| Inverse | a + (—a) = 0 | a × (1/a) = 1 (a ≠ 0) |
| Distributive | — | a(b+c) = ab + ac |
Representing Rational Numbers on Number Line
Between any TWO rational numbers, there are INFINITELY MANY rational numbers.
Finding Rational Numbers Between Two Rationals
Take the mean: (a + b)/2. Repeat.
2. Exponents and Powers
Laws (Extended to Negative Exponents)
| Law | Formula |
|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of power | (aᵐ)ⁿ = aᵐⁿ |
| Power of product | (ab)ᵐ = aᵐ bᵐ |
| Power of quotient | (a/b)ᵐ = aᵐ/bᵐ |
| Zero exponent | a⁰ = 1 (a≠0) |
| Negative exponent | a⁻ⁿ = 1/aⁿ |
Standard Form (Scientific Notation)
Writing very LARGE or very SMALL numbers compactly. A × 10ⁿ, where 1 ≤ A < 10.
- 150,000,000 km (Earth-Sun distance) = 1.5 × 10⁸ km
- 0.000000001 m (diameter of atom) = 1.0 × 10⁻⁹ m
3. Squares and Square Roots
Properties of Square Numbers
- A number ending in 2, 3, 7, or 8 is NEVER a perfect square
- Square of an EVEN number = EVEN. Square of ODD = ODD.
Finding Square Roots
| Method | When to Use |
|---|---|
| Prime Factorisation | Pair the prime factors. Each pair gives one factor of the root. |
| Long Division Method | Large numbers. A step-by-step algorithm. |
Pythagorean Triplets
Three natural numbers (a, b, c) satisfying a² + b² = c². For any m > 1: (2m, m²—1, m²+1).
m=2 → (4, 3, 5). m=3 → (6, 8, 10).
4. Cubes and Cube Roots
Perfect Cubes
Numbers that are the cube of an integer: 1, 8, 27, 64, 125, 216...
Finding Cube Roots
- Prime Factorisation: Group into TRIPLETS of prime factors.
- Estimation: For numbers that are perfect cubes.
Properties
- Cube of negative = NEGATIVE. (—2)³ = —8.
- Cube of even = EVEN. Cube of odd = ODD.
