Quick Summary
Real Numbers expands the number system from integers and fractions to include irrational numbers. Key topics are the classification of real numbers, surds, laws of exponents, and rationalisation. Numerical and proof questions from this chapter appear in all three terms.
Key Definitions
- Natural Numbers (ℕ): Counting numbers 1, 2, 3, ...
- Whole Numbers (W): 0, 1, 2, 3, ...
- Integers (ℤ): ..., −2, −1, 0, 1, 2, ...
- Rational Numbers (ℚ): Numbers of the form p/q, q ≠ 0.
- Irrational Numbers: Non-terminating, non-repeating decimals. Cannot be written as p/q.
- Real Numbers (ℝ): ℚ ∪ Irrational numbers.
- Surd: Irrational n-th root of a positive rational number (e.g., √5, ∛7).
Number System Hierarchy
Surds
- Order of a surd: The index of the radical. √5 is order 2; ∛7 is order 3.
- Like surds: Same order and same radicand. 3√2 and 5√2 are like surds.
- Unlike surds: Different radicand or order.
Laws of Surds
| Law | Example |
|---|---|
| √a × √b = √(ab) | √3 × √5 = √15 |
| √a / √b = √(a/b) | √12 / √3 = √4 = 2 |
| (√a)² = a | (√7)² = 7 |
| √(a²b) = a√b | √12 = 2√3 |
Laws of Exponents
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ / aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1/aⁿ
- a^(1/n) = ⁿ√a
Representation on Number Line
To represent √n on a number line: 1. Mark point A at 0 and B at n on the number line. 2. Mark M as midpoint of OB where O is at 0. 3. Draw semicircle with centre M and radius MB. 4. Draw perpendicular from 1 to meet the semicircle at P. 5. OP = √n.
Important Questions
One Mark
1. Classify 0.1234567... as rational or irrational. 2. Simplify: 3√2 + 5√2. 3. Find the order of the surd ⁴√11. 4. Write the decimal expansion of 1/7 and classify it. 5. Simplify: (√3)⁴.
Short Answer — 3 marks
1. Rationalise the denominator: 5/(2 + √3). 2. Simplify: (√5 + √3)(√5 − √3) and identify the type of number. 3. Express 0.̄3̄2̄ (0.323232...) as a fraction in lowest terms.
Long Answer — 5 marks
1. Represent √7 on the number line using geometric construction. Explain each step. 2. Simplify: [2^(3/4) × 2^(1/4)] / 2^(−1/2) and express the result as a simple power of 2. Also prove that √2 is irrational.
Common Mistakes
- Assuming all square roots are irrational — √9 = 3 is rational.
- Adding unlike surds — 2√3 + 3√5 cannot be simplified further.
- Forgetting that (√a + √b)² ≠ a + b; it equals a + 2√(ab) + b.
- Sign errors when multiplying conjugates — (a + b)(a − b) = a² − b².
Revision Checklist
- [ ] Know the classification of real numbers with examples
- [ ] Can identify surds and their orders
- [ ] Can simplify surds using laws
- [ ] Can rationalise denominators (simple and conjugate forms)
- [ ] Know all six laws of exponents with examples
- [ ] Can represent √n on a number line (construction method)