Real Numbers – WBBSE Class 10 Mathematics Notes & Guide

Real numbers form the backbone of many problems in Class 10 mathematics. Understanding how every integer can be broken down into prime factors, how primes interact with products, and how to handle surds and irrational numbers will help you solve a wide range of questions confidently.

Key concepts

Fundamental Theorem of Arithmetic

Every natural number greater than 1 can be written uniquely as a product of prime numbers, ignoring the order of the factors. For example, 84 = 2 × 2 × 3 × 7 = 2²·3·7. This representation is called the prime factorisation of the number.

Euclid’s Lemma

If a prime number p divides the product ab of two integers, then p must divide at least one of a or b. Symbolically, p|ab ⇒ p|a or p|b. This lemma is essential for proving the uniqueness part of the Fundamental Theorem of Arithmetic.

Rationalisation of surds

Rationalisation removes surds (square‑root terms) from the denominator of a fraction. For a simple denominator √a, multiply numerator and denominator by √a. For a binomial denominator like √a+√b, multiply by its conjugate √a-√b.

Representation of real numbers on the number line

All real numbers can be placed on a straight line. Rational numbers appear at points that can be expressed as p/q (with integer p,q, q≠0). Irrational numbers fill the gaps between rationals; they cannot be written as a terminating or repeating decimal.

Properties of irrational numbers

• The sum or difference of a rational number and an irrational number is irrational.
• The product of a non‑zero rational number and an irrational number is irrational.
• The square of an irrational number may be rational (e.g., (√2)² = 2) or irrational (e.g., (√2+√3)² is irrational).

Important definitions and formulas

• Prime number: a natural number greater than 1 having exactly two distinct positive divisors – 1 and itself.
• Composite number: a natural number greater than 1 that is not prime.
• Prime factorisation: n = p₁^(e₁) × p₂^(e₂) × … × pₖ^(eₖ).
• Euclid’s Lemma: p|ab ⇒ p|a or p|b for prime p.
• Rationalising a simple surd: frac1√a = frac√aa.
• Rationalising a binomial surd: frac1√a+√b = frac√a-√ba-b.
• Irrationality test: If √n cannot be expressed as a fraction in lowest terms, it is irrational (true for non‑perfect squares).

Solved examples

Example 1 – Prime factorisation and HCF/LCM

Find the HCF and LCM of 180 and 210 using their prime factorisations.

1. Factorise each number:

• 180 = 2²·3²·5
• 210 = 2·3·5·7

1. Identify common primes with the smallest powers for HCF:

• HCF = 2¹·3¹·5¹ = 30

1. Identify all primes with the highest powers for LCM:

• LCM = 2²·3²·5·7 = 2²×3²×5×7 = 4×9×5×7 = 1260

Answer: HCF = 30, LCM = 1260.

Example 2 – Rationalising a binomial surd

Rationalise frac5√2+√3.

1. Multiply numerator and denominator by the conjugate √2-√3: frac5√2+√3 × frac√2-√3√2-√3 = frac5(√2-√3)(√2)²-(√3)²
2. Simplify denominator: (√2)²-(√3)² = 2-3 = -1.
3. Final expression: frac5(√2-√3)-1 = -5√2+5√3.

Answer: -5√2+5√3.

Example 3 – Proving irrationality using Euclid’s Lemma

Prove that √15 is irrational.

1. Assume the contrary: √15=p/q where p,q are co‑prime integers.
2. Square both sides: 15 = (p²)/(q²) ⇒ p² = 15q².
3. Hence p² is divisible by 3 and 5, so p is divisible by 3 and 5 (by Euclid’s Lemma). Let p = 3·5·k = 15k.
4. Substitute back: (15k)² = 15q² ⇒ 225k² = 15q² ⇒ q² = 15k².
5. This shows q is also divisible by 3 and 5, contradicting the assumption that p and q are co‑prime.

Conclusion: √15 cannot be expressed as a fraction; it is irrational.

Common mistakes to avoid

• Skipping the order of primes: While the order of prime factors does not matter, forgetting to write the highest power for LCM leads to a smaller value.
• Wrong conjugate: When rationalising √a+√b, the conjugate is √a-√b, not √a+√b.
• Assuming all surds are irrational: √4=2 is rational. Check whether the radicand is a perfect square.
• Mixing up HCF and LCM formulas: Remember HCF uses minimum exponents, LCM uses maximum exponents.

Exam tips

• Write the prime factorisation clearly; it saves time for HCF/LCM questions.
• Always multiply by the conjugate in one step; write both numerator and denominator to avoid sign errors.
• For irrationality proofs, start with the assumption of rationality and use Euclid’s Lemma to reach a contradiction.
• When a question asks for a number line representation, mark the exact point and label it with the decimal (if terminating) or indicate it as an irrational point.
• Keep the definition of rational and irrational numbers handy; many conceptual questions are based on these definitions.

Frequently asked questions

1. Can a composite number have more than one distinct prime factorisation?

No. By the Fundamental Theorem of Arithmetic, the prime factorisation of a natural number is unique, apart from the order of the factors.

2. Why do we multiply by the conjugate when rationalising a binomial surd?

Multiplying by the conjugate converts the denominator into a difference of squares, which eliminates the surd because (√a+√b)(√a-√b) = a-b, a rational number.

3. Is the sum of two irrational numbers always irrational?

Not necessarily. For example, √2+ (2-√2) = 2 which is rational. However, the sum of a rational and an irrational number is always irrational.