What are Real Numbers?
Real numbers include all rational numbers (integers, fractions, terminating/recurring decimals) AND irrational numbers (non-repeating, non-terminating decimals like √2, π).
Operations on Real Numbers:
Real numbers are closed under addition, subtraction, multiplication, and division (except by zero). This means when you perform these operations on real numbers, the result is also a real number.
Key Properties:
| Property | Description | Example |
|---|---|---|
| **Associative** | (a + b) + c = a + (b + c) | (2 + √2) + √3 = 2 + (√2 + √3) |
| **Distributive** | a(b + c) = ab + ac | √2(3 + √5) = 3√2 + √10 |
| **Identity** | a + 0 = a; a × 1 = a | √7 + 0 = √7 |
| **Inverse** | a + (-a) = 0; a × (1/a) = 1 (a ≠ 0) | √5 × (1/√5) = 1 |
Rational vs Irrational Results:
- Rational + Rational = Rational
- Rational + Irrational = Irrational
- Irrational + Irrational = Can be Rational or Irrational (e.g., √2 + (-√2) = 0 rational)
- Similar rules apply for multiplication and division