9th Class Math – Chapter 2: Real Numbers

Chapter 2: Real Numbers

Real numbers include all rational and irrational numbers. They can be represented on a number line.

**Natural Numbers (N)**: {1, 2, 3, 4, ...}

**Whole Numbers (W)**: {0, 1, 2, 3, 4, ...}

**Integers (Z)**: {..., -3, -2, -1, 0, 1, 2, 3, ...}

**Rational Numbers (Q)**: Numbers that can be expressed as p/q where p and q are integers and q ≠ 0

**Irrational Numbers**: Numbers that cannot be expressed as p/q (e.g., √2, π)

#### Commutative Property

Addition: a + b = b + a

Multiplication: a × b = b × a

#### Associative Property

Addition: (a + b) + c = a + (b + c)

Multiplication: (a × b) × c = a × (b × c)

#### Distributive Property

a × (b + c) = a × b + a × c

#### Identity Property

Additive Identity: a + 0 = a

Multiplicative Identity: a × 1 = a

#### Inverse Property

Additive Inverse: a + (-a) = 0

Multiplicative Inverse: a × (1/a) = 1 (where a ≠ 0)

Real numbers can be represented on a number line. Every point on the number line corresponds to a unique real number.

**Rational Numbers**: Can be written as terminating or repeating decimals

Examples: 1/2 = 0.5, 1/3 = 0.333..., 5 = 5.0

**Irrational Numbers**: Non-terminating, non-repeating decimals

Examples: √2 ≈ 1.414..., π ≈ 3.14159..., e ≈ 2.71828...

**Example 1**: Identify whether √16 is rational or irrational.

Solution: √16 = 4, which is a rational number.

**Example 2**: Find the additive inverse of -7.

Solution: The additive inverse of -7 is 7, because -7 + 7 = 0.

Classify the following as rational or irrational: √25, √3, 22/7, 0.1010010001...

Find the multiplicative inverse of -3/4

Verify the distributive property for a = 2, b = 3, c = 4