Class 10 Mathematics Ch 1 Short Notes: Real Numbers

Contents hide

1 The Chapter at a Glance

2 Definitions and Theorems

3 HCF and LCM by Prime Factorisation

4 The 2\sqrt{2}2 Proof in Four Lines

5 Rational and Irrational Together

The whole of Chapter 1 on two pages for last-minute revision: every definition, theorem and rule, and the key proof at a glance. The full detailed notes are available separately.

1 The Chapter at a Glance

Real Numbers

1 · Fundamental Theorem of Arithmetic (FTA)Every composite number splits into primes in only one way, apart from the order of the factors

2 · HCF (Highest Common Factor) and LCM (Lowest Common Multiple)Smallest and greatest powers of primes

3 · Theorem 1.2if a prime

p

divides

a^2

, it divides

a

4 · Irrationality proofs

\sqrt{2}

\sqrt{3}

\sqrt{5}

by contradiction

2 Definitions and Theorems

Term / theorem	Meaning
Prime number	Exactly two factors, 1 and itself (2, 3, 5, 7, …)
Composite number	More than two factors (4, 6, 9, …); 1 is neither prime nor composite
Coprime numbers	No common factor other than 1 (HCF $=1$ )
Theorem 1.1: Fundamental Theorem of Arithmetic (FTA)	“Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.”
Theorem 1.2	If a prime $p$ divides $a^2$ , then $p$ divides $a$ ( $a$ a positive integer). Written $p\mid a^2\Rightarrow p\mid a$ , where $\mid$ means “divides” and $\Rightarrow$ means “implies”.
Irrational number	Cannot be written as $\dfrac{p}{q}$ with $p,q$ integers, $q\ne0$
Theorem 1.3	$\sqrt{2}$ is irrational

3 HCF and LCM by Prime Factorisation

Rules

HCF = product of the smallest power of each common prime factor

LCM = product of the greatest power of each prime factor involved

\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b\quad\text{(for two numbers)}

Numbers	Factorised	HCF	LCM
6, 20	$2\times3$ ; $2^2\times5$	2	60
96, 404	$2^5\times3$ ; $2^2\times101$	4	$\dfrac{96\times404}{4}=9696$
6, 72, 120	$2\times3$ ; $2^3\times3^2$ ; $2^3\times3\times5$	6	360

Common Mistake

For three numbers, HCF × LCM is not in general equal to the product: for 6, 72, 120, $6\times360=2160$ but $6\times72\times120=51840$ .

Three numbers (NCERT’s Note to the Reader)

\text{LCM}(p,q,r)=\dfrac{p\times q\times r\times\text{HCF}(p,q,r)}{\text{HCF}(p,q)\times\text{HCF}(q,r)\times\text{HCF}(p,r)}

\text{HCF}(p,q,r)=\dfrac{p\times q\times r\times\text{LCM}(p,q,r)}{\text{LCM}(p,q)\times\text{LCM}(q,r)\times\text{LCM}(p,r)}

for positive integers

p,q,r

. Check with 6, 72, 120: LCM

=360

, HCF

=6

Ends in 0?

A number ends in 0 only if its prime factors include both 2 and 5. $4^n=2^{2n}$ and $6^n=2^n\times3^n$ have no 5, so they never end in 0 (a prime factorisation has only one form, so no 5 can appear).

4 The $\sqrt{2}$ Proof in Four Lines

Assume

\sqrt{2}

is rational:

\sqrt{2}=\dfrac{a}{b}

a,b

coprime,

b\ne0

proof by contradiction

2b^2=a^2\Rightarrow2\mid a

; put

a=2c

(

c

an integer)square, Theorem 1.2

b^2=2c^2\Rightarrow2\mid b

Theorem 1.2 again

2 divides both

a

and

b

, but they were coprime: contradictionso

\sqrt{2}

is irrational

Replace 2 by any prime $p$ (3, 5, 7, …) and the same four lines prove $\sqrt{p}$ irrational.

5 Rational and Irrational Together

Rule	Example	Assume the number $=\dfrac{a}{b}$ ; rearranging gives
rational ± irrational = irrational	$5-\sqrt{3}$	$\sqrt{3}=\dfrac{5b-a}{b}$
non-zero rational × or ÷ irrational = irrational	$3\sqrt{2}$	$\sqrt{2}=\dfrac{a}{3b}$

Two irrationals can give a rational: $\sqrt{2}\times\sqrt{2}=2$ , $\sqrt{2}+(-\sqrt{2})=0$ .

Last-minute checklist

State FTA word for word, including “apart from the order”
Factorise, then smallest powers for HCF, greatest powers for LCM
Product rule for two numbers; it does not hold in general for three
Proofs: assume rational, coprime $a,b$ , square, Theorem 1.2 twice, contradiction
Not in the CBSE 2026-27 syllabus, and no section in the current NCERT reprint: Euclid’s division algorithm, decimal expansions of rational numbers

More for this chapter

Notes Important Questions MCQ / Objective NCERT Solutions

1 The Chapter at a Glance

2 Definitions and Theorems

3 HCF and LCM by Prime Factorisation

4 The 2\sqrt{2}2​ Proof in Four Lines

5 Rational and Irrational Together

4 The $\sqrt{2}$ Proof in Four Lines