Chapter 4: Number System

1. Foundations: Real Numbers & Surds

Real Numbers (R) consist of all Rational (Q) and Irrational (Q') numbers. A Surd is a specific type of irrational number—a root of a rational number that cannot be expressed as a rational fraction.

Pure Surd: Entirely contained (e.g., √5).
Mixed Surd: Contains a rational coefficient (e.g., 2√3).
Like Surds: Identical irrational parts (e.g., 5√2 and 3√2).

2. Simplification (Perfect Square Method)

We simplify surds to their most basic form by extracting the largest perfect square factor.

Example: Simplify √72
1. Prime factorize: 72 = 2 × 2 × 2 × 3 × 3 = 36 × 2
2. Apply property: √(36 × 2) = √36 × √2
3. Result: 6√2

3. Laws of Surds & Indices

Surds obey specific laws derived from the laws of indices:

Product: √a × √b = √(ab)
Quotient: √a / √b = √(a/b)
Index: ⁿ√a = a^(1/n)

4. Rationalization (The Conjugate Method)

To eliminate a root from the denominator, multiply by the conjugate.

Example: Rationalize 5 / (√7 - √2)
1. Conjugate is (√7 + √2).
2. Multiply: [5(√7 + √2)] / [(√7 - √2)(√7 + √2)]
3. Denominator identity (a² - b²): (√7)² - (√2)² = 7 - 2 = 5
4. Simplify: [5(√7 + √2)] / 5 = √7 + √2