Real Numbers Class 10 Notes: Chapter Overview
Revise real numbers, prime factors, HCF–LCM and irrationality proofs with Class 10 notes, worked examples, a worksheet and practice quiz.
Real numbers include rational numbers and irrational numbers. A rational number can be written as p/q with integers p and q and q ≠ 0. An irrational number cannot be written in that form. In Class 10, prime factorisation provides a way to reason about divisibility and prove that certain square roots are irrational.
What counts as a real number?
| Example | Classification | Reason |
|---|---|---|
| −3 | Integer and rational | It equals −3/1. |
| 0.75 | Rational | It equals 3/4. |
| √2 | Irrational | It has no expression as a ratio of integers. |
| √9 = 3 | Rational | A square-root symbol does not always mean irrational. |
Prime factorisation and the fundamental theorem
Every integer greater than 1 has a prime factorisation that is unique apart from the order of its factors. For example, 90 = 2 × 3² × 5. The factors can be grouped differently while finding them, but the final primes and their counts agree. The number 1 is neither prime nor composite.
HCF and LCM
Find the HCF and LCM of 24 and 36
- 124 = 2³ × 3 and 36 = 2² × 3².
- 2HCF = 2² × 3 = 12; use the smaller powers common to both.
- 3LCM = 2³ × 3² = 72; use the larger powers of all primes present.
- 4Check 12 × 72 = 864 = 24 × 36. This product identity applies to two positive integers.
An irrationality proof
Why √2 is irrational
- 1Assume √2 = a/b in lowest terms, with b ≠ 0. Squaring gives a² = 2b².
- 2a² is even, so a is even; write a = 2k. Then b² = 2k², so b is even too.
- 3Both a and b have a factor 2, contradicting lowest terms. Therefore the assumption was false.
Study sequence
- Learn the prime-factor method and work one HCF–LCM example without looking.
- Use the Exercise 1.1 companion to check textbook work.
- Complete the original worksheet, then attempt the online real-numbers quiz.
- For proofs, identify the assumption, the divisibility step and the contradiction.
Fundamental theorem, HCF and LCM word problems
Important questions and worksheet PDF