Real Numbers — Mathematics Class 10 Notes (CBSE & HBSE)
Free NCERT Mathematics notes for Real Numbers (Class 10) on Siksha Sarovar, aligned to CBSE and Haryana Board (HBSE). This chapter is broken into 3 topics with clear explanations, formulas, solved examples and board-pattern practice — free to read, no sign-up required.
Board exam focus — Real Numbers (CBSE & HBSE)
Foundation of number theory. CBSE includes case-study on HCF/LCM applications. HBSE focuses on theorem-based proofs and prime factorization.
Fundamental Theorem of Arithmetic
Fundamental Theorem of Arithmetic
Statement: Every composite number can be expressed as a product of primes, and this factorisation is unique, apart from the order.
HCF and LCM by Prime Factorisation
| HCF | LCM | |
|---|---|---|
| Rule | Smallest power of common prime factors | Greatest power of all prime factors |
KEY FORMULA: HCF(a,b) × LCM(a,b) = a × b (two numbers only)
Example: 180 = 2² × 3² × 5 and 252 = 2² × 3² × 7
- HCF = 2² × 3² = 36
- LCM = 2² × 3² × 5 × 7 = 1260
- Verify: 36 × 1260 = 45360 = 180 × 252 ✓
When to Use HCF vs LCM
| Situation | Use |
|---|---|
| Largest number that divides both | HCF |
| Smallest number divisible by both | LCM |
| Maximum equal groups | HCF |
| Simultaneous events (bells ringing together) | LCM |
CBSE Tip: Prime factorisation must be written in index form (2³ × 3 × 5). Always verify HCF × LCM = product of the two numbers!
Irrational Numbers and Proofs
Rational vs Irrational
| Type | Form | Decimal |
|---|---|---|
| Rational | p/q (q≠0) | Terminating or recurring |
| Irrational | Cannot be p/q | Non-terminating, non-recurring |
Proof: √2 is Irrational (Contradiction)
- Assume √2 = p/q, HCF(p,q) = 1
- Square: p² = 2q² → 2 | p² → 2 | p (by prime lemma)
- Let p = 2m → q² = 2m² → 2 | q
- 2 divides both p and q → contradicts HCF = 1
- ∴ √2 is irrational ∎
Same proof works for √3, √5, √7, √11.
Operations on Irrationals
- Rational + Irrational = always irrational
- Rational × Irrational (≠0) = always irrational
- Irrational ± Irrational = may be rational OR irrational
Terminating Decimals
Rule: p/q (lowest terms) terminates ⟺ q = 2ⁿ × 5ᵐ
| q | Type |
|---|---|
| 8 = 2³ | Terminating |
| 25 = 5² | Terminating |
| 6 = 2 × 3 | Non-terminating |
| 15 = 3 × 5 | Non-terminating |
Always SIMPLIFY the fraction first before checking!
Euclid's Division Algorithm
Euclid's Division Lemma
For positive integers a, b, ∃ unique q, r such that: a = bq + r, where 0 ≤ r < b
Algorithm for HCF
- Apply lemma: a = bq + r
- If r = 0 → HCF = b
- If r ≠ 0 → replace a ← b, b ← r, repeat
Example: HCF(455, 42)
- 455 = 42 × 10 + 35
- 42 = 35 × 1 + 7
- 35 = 7 × 5 + 0 → HCF = 7
Applications
| Problem | Use |
|---|---|
| Largest tile for a floor | HCF of floor dimensions |
| Equal groups from two sets | HCF of group sizes |
| Minimum time for events to coincide | LCM of time intervals |
Euclid's algorithm is more efficient than prime factorisation for large numbers.
Frequently asked questions
Are these Real Numbers notes free?
Yes — the Real Numbers notes for Mathematics (Class 10) on Siksha Sarovar are completely free to read, with no account required.
Do these notes follow CBSE and HBSE?
Yes. The Real Numbers notes are NCERT-aligned and include guidance for both CBSE and Haryana Board (HBSE), with important questions and MCQs for revision.
What does the Real Numbers chapter cover?
Concept explanations, key formulas and definitions, fully solved examples and board-pattern practice questions for Real Numbers.