CBSE · Class 6-12

Class 10 · Mathematics (Standard)

Unit 1 · Class 10Mathematics (Standard)

Real Numbers

Fundamental Theorem of Arithmetic, HCF and LCM by prime factorisation, and the irrationality of numbers like $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$.

Lessons

5 concepts

1
Fundamental Theorem of Arithmetic
Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique apart from the order in which the primes appear. E.g. $12600 = 2^3 \cdot 3^2 \cdot 5^2 \cdot 7$ . This uniquene…
2
HCF and LCM by Prime Factorisation
Write each number as a product of primes. Then $\text{HCF} =$ product of smallest powers of common primes; $\text{LCM} =$ product of greatest powers of all primes involved. For two positive integers $a,b$ : $\text{HCF}(…
3
Proving Irrationality
To prove $\sqrt{p}$ (with $p$ prime) is irrational, assume $\sqrt{p} = a/b$ with $\gcd(a,b)=1$ . Squaring gives $pb^2 = a^2$ , so $p\mid a^2 \Rightarrow p\mid a$ . Write $a = pk$ : then $b^2 = pk^2$ , so $p\mid b$ . This contr…
4
Decimal Expansions of Rationals
A rational number $x = p/q$ in lowest terms has a terminating decimal expansion iff $q = 2^m 5^n$ . Otherwise the expansion is non-terminating but recurring. E.g. $3/8 = 0.375$ terminates because $8 = 2^3$ ; $1/6 = 0.1\ove…
5
Applications of HCF and LCM
HCF is used when you need the largest common measure (splitting into equal groups, tiling). LCM is used for the smallest common multiple (meeting times, common denominators). Real-life word problems typically reduce to e…

Skills to practise

Skill · medium

Real Numbers — Practice