Real Numbers — HCF, LCM & Rationality
The number family tree, the HCF×LCM rule, and how to tell rational from irrational — the Class 10 essentials.
By the PadhoDost Team · 📖 6 min read · Updated 4 August 2026
Part of Class 10 (CBSE) prep🧠 The number family
Real numbers are one big family. Natural numbers (1,2,3…) grow into whole numbers (add 0), then integers (add negatives), then rationals (any p/q fraction). Numbers that REFUSE to be written as p/q — like √2 and π — are the irrationals. Together, rationals + irrationals = all real numbers.
Rational vs Irrational
- ✓Rational: can be written as p/q (q ≠ 0). e.g. 0.5, 2/3, 7
- ✓Its decimal either terminates (0.75) or repeats (0.333…)
- ✓Irrational: cannot be written as p/q. e.g. √2, √3, π
- ✓Its decimal is non-terminating AND non-repeating
HCF and LCM
HCF (Highest Common Factor) is the biggest number that divides two numbers. LCM (Lowest Common Multiple) is the smallest number both divide into. Find them using prime factorisation.
📝 HCF and LCM of 12 and 18
12 = 2 × 2 × 3
18 = 2 × 3 × 3
HCF = common primes = 2 × 3 = 6
LCM = all primes (highest powers) = 2² × 3² = 36
The golden rule
⚡ Quick check
Which of these is an irrational number?
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