Number System

Real Numbers, Rational/Irrational, Surds and Indices

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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.

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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).

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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

### Surd Operations

• **√a × √b = √(ab)**
• **√a ÷ √b = √(a/b)**
• **(√a)² = a**
• **√(a²) = |a|**

### Rationalising Factors

• For 1/√a → multiply by √a/√a
• For 1/(a + √b) → multiply by (a − √b)/(a − √b)
• For 1/(√a + √b) → multiply by (√a − √b)/(√a − √b)

### Key Identity for Conjugates

• (√a + √b)(√a − √b) = a − b

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Worked Examples

### Example 1: Classifying Numbers **Question:** Classify the following as rational or irrational: (a) 0.272727... (b) 0.101001000100001...

**Solution:** (a) 0.272727... = 0.27̅ — The digits "27" repeat indefinitely. → Non-terminating but repeating → **Rational** (equals 27/99 = 3/11)

(b) 0.101001000100001... — The pattern changes (increasing zeros), never repeats. → Non-terminating, non-repeating → **Irrational**

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### Example 2: Simplifying Using Laws of Indices **Question:** Simplify: (2³ × 2⁵) ÷ 2⁴

**Solution:** Step 1: Apply multiplication rule in numerator 2³ × 2⁵ = 2³⁺⁵ = 2⁸

Step 2: Apply division rule 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴ = **16**

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### Example 3: Rationalising the Denominator **Question:** Rationalise: 5/(√7 − √2)

**Solution:** Step 1: Identify the conjugate of denominator Conjugate of (√7 − √2) is (√7 + √2)

Step 2: Multiply numerator and denominator by conjugate = 5(√7 + √2) / [(√7 − √2)(√7 + √2)]

Step 3: Apply identity (a − b)(a + b) = a² − b² Denominator = (√7)² − (√2)² = 7 − 2 = 5

Step 4: Simplify = 5(√7 + √2) / 5 = **√7 + √2**

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### Example 4: Simplifying Surds **Question:** Simplify: √72 + √50 − √18

**Solution:** Step 1: Break each into prime factors under root √72 = √(36 × 2) = 6√2 √50 = √(25 × 2) = 5√2 √18 = √(9 × 2) = 3√2

Step 2: Combine like surds = 6√2 + 5√2 − 3√2 = (6 + 5 − 3)√2 = **8√2**

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Common Mistakes

• **Mistake:** Thinking π = 22/7 exactly, so π is rational.

→ **Fix:** 22/7 is only an approximation. π is irrational — its decimal never terminates or repeats.

• **Mistake:** Adding unlike surds directly: √2 + √3 = √5.

→ **Fix:** √2 + √3 cannot be simplified further. Only like surds (same radicand) can be combined.

• **Mistake:** Applying aᵐ × bⁿ = (ab)ᵐ⁺ⁿ — mixing different bases.

→ **Fix:** The rule aᵐ × aⁿ = aᵐ⁺ⁿ works only for the **same base**.

• **Mistake:** Writing √(a + b) = √a + √b.

→ **Fix:** The square root does not distribute over addition. √(9 + 16) = √25 = 5, not √9 + √16 = 3 + 4 = 7.

• **Mistake:** Forgetting that √(a²) = |a|, not just a.

→ **Fix:** √((-3)²) = √9 = 3, not −3. The principal square root is always non-negative.

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Quick Reference

• **Rational:** p/q form, decimal terminates or repeats
• **Irrational:** Cannot be p/q, decimal never terminates or repeats
• **Same base multiplication:** Add exponents (aᵐ × aⁿ = aᵐ⁺ⁿ)
• **Same base division:** Subtract exponents (aᵐ ÷ aⁿ = aᵐ⁻ⁿ)
• **Rationalise a + √b:** Multiply by conjugate (a − √b)
• **Simplify surds:** Factor out perfect squares first, then combine like surds