Number System

Real Numbers, Rational/Irrational, Surds and Indices

---

Overview

The Number System forms the foundation of all mathematics tested in TN TET Paper II. Questions from this topic appear consistently, testing both conceptual understanding and computational skills. You must be able to classify numbers, perform operations on surds, and apply laws of indices fluently.

This topic connects directly to algebra, geometry (when dealing with irrational lengths like √2), and even data handling. For TN TET, expect 2–4 questions that test definitions, properties, simplification of surds, and index manipulation. Mastery here builds confidence for the entire quantitative section.

The key challenge is distinguishing between number types and applying the correct rules during simplification. Students who memorise definitions without understanding relationships often make careless errors.

---

Key Concepts

• **Real numbers** = All numbers on the number line = Rational numbers ∪ Irrational numbers. Every point on the number line corresponds to a real number.

• **Rational numbers** = Numbers expressible as p/q where p, q are integers and q ≠ 0. Their decimal expansion either terminates or repeats. Examples: 3/4 = 0.75, 1/3 = 0.333...

• **Irrational numbers** = Numbers that cannot be written as p/q. Their decimal expansion is non-terminating and non-repeating. Examples: √2, √3, π, e.

• **Natural numbers ⊂ Whole numbers ⊂ Integers ⊂ Rational numbers ⊂ Real numbers.** This hierarchy is frequently tested.

• **Surds** = Irrational roots that cannot be simplified to remove the root sign. √5 is a surd; √4 = 2 is not a surd.

• **Like surds** share the same radicand (number under root): 3√5 and 7√5 are like surds. Only like surds can be added or subtracted directly.

• **Indices (exponents)** represent repeated multiplication. The laws of indices allow simplification of complex expressions involving powers.

• **Rationalisation** = Removing the surd from the denominator by multiplying by an appropriate factor (conjugate or the surd itself).

---

Formulas / Key Facts

### Classification Quick Test | If decimal is... | Number type | |------------------|-------------| | Terminating | Rational | | Non-terminating, repeating | Rational | | Non-terminating, non-repeating | Irrational |

### Laws of Indices 1. **aᵐ × aⁿ = aᵐ⁺ⁿ** — Same base, add powers when multiplying 2. **aᵐ ÷ aⁿ = aᵐ⁻ⁿ** — Same base, subtract powers when dividing 3. **(aᵐ)ⁿ = aᵐⁿ** — Power of a power, multiply exponents 4. **(ab)ⁿ = aⁿbⁿ** — Power distributes over multiplication 5. **(a/b)ⁿ = aⁿ/bⁿ** — Power distributes over division 6. **a⁰ = 1** (where a ≠ 0) 7. **a⁻ⁿ = 1/aⁿ** — Negative exponent means reciprocal 8. **a^(1/n) = ⁿ√a** — Fractional exponent means root