Integers, Rational Numbers, Fractions & Exponents
1. Integers — Deeper Operations
Review
Integers: ...-3, -2, -1, 0, 1, 2, 3...
Properties of Integer Operations
| Property | Addition | Multiplication |
|---|---|---|
| Closure | a + b is ALWAYS an integer | a × b is ALWAYS an integer |
| Commutative | a + b = b + a | a × b = b × a |
| Associative | (a + b) + c = a + (b + c) | (a × b) × c = a × (b × c) |
| Identity | a + 0 = a | a × 1 = a |
| Distributive | — | a × (b + c) = a×b + a×c |
Rules for Multiplication and Division of Signed Numbers
- (+) × (+) = (+) | (+) ÷ (+) = (+)
- (+) × (—) = (—) | (+) ÷ (—) = (—)
- (—) × (+) = (—) | (—) ÷ (+) = (—)
- (—) × (—) = (+) | (—) ÷ (—) = (+)
2. Rational Numbers
Definition
A RATIONAL NUMBER is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.
Examples: 3/4, -5/2, 7 (= 7/1), 0.5 (= 1/2), 0 (= 0/1).
Standard Form
A rational number is in STANDARD FORM when the denominator is POSITIVE and p and q have NO COMMON FACTOR other than 1. Example: -15/20 = -3/4.
Operations on Rational Numbers
- Addition: Find LCM of denominators. Convert to equivalent fractions. Add numerators.
- Subtraction: Same as addition.
- Multiplication: (a/b) × (c/d) = (a×c)/(b×d)
- Division: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c)
Comparison
To compare two rational numbers: convert to equivalent fractions with SAME denominator. Compare numerators. Like integers: use the number line — numbers increase to the RIGHT.
3. Fractions and Decimals — Advanced
Types of Decimals
- Terminating: Division ENDS. 3/8 = 0.375.
- Non-terminating recurring: Digits REPEAT. 1/3 = 0.333... = 0.3̄.
- Non-terminating non-recurring = IRRATIONAL numbers (e.g., π, √2).
Converting Between Forms
- Fraction → Decimal: Divide numerator by denominator.
- Decimal → Fraction (terminating): Write as fraction with 10, 100, 1000... Simplify.
- Decimal → Fraction (recurring): Use algebraic method.
4. Exponents (Powers)
Definition
aⁿ = a × a × a × ... (n times). a = BASE. n = EXPONENT.
Laws of Exponents (for Same Base)
| Law | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n) | 2⁵ ÷ 2² = 2³ |
| Power of Power | (aᵐ)ⁿ = aᵐˣⁿ | (2³)² = 2⁶ |
| Power of Product | (ab)ᵐ = aᵐ bᵐ | (2×3)² = 2²×3² |
| Zero Exponent | a⁰ = 1 (a ≠ 0) | 5⁰ = 1 |
