Class 10 Maths Notes — Chapter 1: Real Numbers
Structure of real numbers, prime factorisation, and irrationality — the foundation used throughout the rest of Class 10 algebra.
Detailed NCERT notes
- Real numbers = rational numbers (p/q, q ≠ 0, p, q integers) ∪ irrational numbers (non-terminating, non-recurring decimals).
- Euclid's Division Lemma: for any two positive integers a and b, there exist unique whole numbers q and r such that a = bq + r, where 0 ≤ r < b. This is the base of Euclid's algorithm.
- Euclid's Division Algorithm to find HCF(a, b): apply the lemma repeatedly, taking divisor and remainder as the new pair, until r = 0. The last non-zero divisor is HCF.
- Fundamental Theorem of Arithmetic: every composite number can be expressed as a product of primes, and this factorisation is unique apart from the order of the factors.
- HCF using prime factorisation = product of smallest powers of common prime factors.
- LCM using prime factorisation = product of greatest powers of all prime factors involved.
- Key relation for any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. This does NOT extend directly to three numbers.
- A rational number p/q (in lowest terms) has a terminating decimal expansion iff q can be expressed as 2^m · 5^n where m, n are non-negative integers.
- If q has any prime factor other than 2 or 5, then p/q is non-terminating recurring.
- Irrationality proof by contradiction (standard template): assume √p is rational = a/b (in lowest terms); square both sides ⇒ a² = p·b²; p divides a² ⇒ p divides a (p prime); let a = p·k, substitute, show p divides b too, contradicting 'lowest terms'.
- Sum/difference/product/quotient of a rational (non-zero) and an irrational is always irrational.
- √2, √3, √5, √7, √11 … are all irrational. π and e are irrational (mentioned for context).
- In Board problems, always mention 'p, q are co-prime' explicitly when starting an irrationality proof.
Formulas & key results
- Euclid's Division Lemma: a = bq + r, 0 ≤ r < b
- HCF(a,b) × LCM(a,b) = a × b (only for two numbers)
- Fundamental Theorem of Arithmetic: composite = unique product of primes
Mind map
- Real numbers → Rational + Irrational
- Rational → terminating (den = 2^m·5^n) or non-terminating recurring
- Prove irrationality by contradiction (√2, √3, √5)
- HCF/LCM via prime factorisation or Euclid's algorithm
Tricks & shortcuts
- For HCF word problems: greatest size that fits all quantities.
- For LCM word problems: smallest quantity divisible by all.
- Terminating decimal check: factor denominator; only 2s and 5s → terminating.
- For 3 numbers: LCM × HCF ≠ product; use prime factorisation instead.
Common mistakes to avoid
- Assuming √2 + √3 is rational — it is irrational.
- Forgetting r < b strictly in Euclid's lemma.
- Using HCF × LCM = product for three numbers.
- Skipping 'co-prime' assumption in irrationality proofs.
Competency-based questions & answers
- Q. Prove that 3 + 2√5 is irrational.A. Assume rational = p/q ⇒ √5 = (p − 3q)/(2q). RHS is rational; but √5 is irrational — contradiction. Hence 3 + 2√5 is irrational.
- Q. Find HCF(306, 657) using Euclid's algorithm.A. 657 = 306·2 + 45; 306 = 45·6 + 36; 45 = 36·1 + 9; 36 = 9·4 + 0 ⇒ HCF = 9.
- Q. Without division, decide if 17/8 has a terminating decimal.A. 8 = 2³. Denominator only has 2 ⇒ terminating. 17/8 = 2.125.
- Q. The HCF of two numbers is 9 and their LCM is 360. If one number is 45, find the other.A. Other = (HCF·LCM)/first = 9·360/45 = 72.