CBSE Class 10 Mathematics Real Numbers Notes

Real Numbers form the foundation of Class 10 Mathematics. This chapter revisits Euclid’s Division Algorithm, explores the Fundamental Theorem of Arithmetic, and deepens our understanding of rational and irrational numbers. A thorough grasp of this chapter is essential for board examinations and higher mathematics.

Key Topics Covered

• Euclid’s Division Lemma and Algorithm

• Fundamental Theorem of Arithmetic

• Revisiting Irrational Numbers

• Rational Numbers and Their Decimal Expansions

1. Euclid’s Division Lemma

Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder) such that:

a = bq + r , where 0 ≤ r < b

‘a’ is the dividend, ‘b’ is the divisor, ‘q’ is the quotient, and ‘r’ is the remainder. This relationship holds for every pair of positive integers.

Euclid’s Division Algorithm

The Euclidean Division Algorithm finds the HCF (Highest Common Factor) of two positive integers. Steps:

1. Step 1: Apply Euclid’s Division Lemma to a and b (a > b) to find q and r such that a = bq + r.

2. Step 2: If r = 0, the HCF is b. If r ≠ 0, apply the lemma again to b and r.

3. Step 3: Continue until the remainder is zero. The divisor at that stage is the HCF.

Example: Find the HCF of 135 and 225.

225 = 135 × 1 + 90

135 = 90 × 1 + 45

90 = 45 × 2 + 0 ➤ HCF = 45

2. Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is unique — apart from the order in which the prime factors occur.

Every composite number has a unique prime factorisation.

HCF and LCM using Prime Factorisation

Using prime factorisation, HCF and LCM are computed as:

• HCF = Product of the smallest power of each common prime factor.

• LCM = Product of the greatest power of each prime factor.

• Important Relation: HCF(a, b) × LCM(a, b) = a × b

3. Irrational Numbers

A number is called irrational if it cannot be written in the form p / q, where p and q are integers and q ≠ 0. Numbers like √2, √3, √5 and combinations involving them are irrational.

Proof of Irrationality of √2

Proof by contradiction: Assume √2 is rational, so √2 = p/q where p, q are coprime (HCF = 1). Squaring gives 2q² = p² ⇒ p² is even ⇒ p is even. Let p = 2m. Then q² = 2m² ⇒ q is also even. This contradicts HCF(p, q) = 1. ∴ √2 is irrational.

Key Properties of Irrational Numbers

• Rational + Irrational = Irrational

• Non-zero Rational × Irrational = Irrational

• Examples: √2, √3, √5, √7, 2+√3, 3√5 are all irrational.

4. Rational Numbers and Decimal Expansions

The decimal expansion of a rational number p/q is either terminating or non-terminating repeating. The type depends on the prime factorisation of denominator q.

Terminating Decimal	Non-Terminating Repeating
Condition: q = 2ⁿ × 5ᵐ (n, m ≥ 0) Only 2 and 5 as prime factors. e.g. 7/8 = 0.875, 3/20 = 0.15	Condition: q has factors other than 2 or 5 Includes 3, 7, 11, etc. e.g. 1/3 = 0.333…, 5/6 = 0.833…

5. Quick Formula Summary

These are the most important formulas frequently tested in CBSE board examinations:

Euclid’s Lemma	a = bq + r, where 0 ≤ r < b
HCF × LCM Relation	HCF(a,b) × LCM(a,b) = a × b
Terminating Decimal	p/q terminates if q = 2ⁿ × 5ᵐ
Irrational Numbers	√2, √3, √p (prime p) are irrational
Rational + Irrational	= Irrational
Non-zero Rational × Irrational	= Irrational