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Real Numbers

From counting mangoes to Euclid's algorithm, irrational numbers, and decimal expansions, this page gives you the full Class 10 Real Numbers chapter in one place with examples, proofs, and practice.

Natural NumbersIntegersRational NumbersEuclid's LemmaHCFIrrationalityDecimal Expansions

Complete Chapter Roadmap - 16 Topics

01Natural Numbers 02Whole Numbers 03Integers 04Rational Numbers 05Irrational Numbers 06Real Numbers 07What Divides Means 08Euclid's Division Lemma 09Remainder Rule 10Why Learn This?11Euclid's Algorithm 12Applications of HCF 13Fundamental Theorem of Arithmetic 14HCF and LCM 15Proving Irrationality 16Decimal Expansions

Natural Numbers

The counting numbers - the oldest mathematics on Earth

NCERT Connection

Every natural number has a successor obtained by adding 1. Every natural number except 1 has a predecessor obtained by subtracting 1.

Imagine your teacher asks, "How many students are present today?" You begin with 1, 2, 3, 4... and never start from zero. These are the most natural counting numbers, which is exactly why they are called Natural Numbers.

Natural numbers were born out of daily human need: counting sheep, fruits, coins, people, and days. They are the first number system every learner meets.

The formal set is

\mathbb{N} = \{1, 2, 3, 4, 5, \ldots\}

. They begin at 1, extend forever, and do not include zero, negatives, or fractions.

Quick Think

Which of these are Natural Numbers? 7, -3, 0, 45, \frac{1}{2}, 1000

Answer: 7, 45, and 1000 only.

Whole Numbers

Adding zero changes the story

Key Relationship

Every Natural Number is also a Whole Number, but not every Whole Number is Natural because 0 belongs to Whole Numbers.

\mathbb{N} \subset \mathbb{W}

Suppose a box of laddoos becomes empty after everyone has eaten. Natural numbers cannot describe an empty box, but Whole Numbers can.

Whole Numbers are the natural numbers together with zero:

\mathbb{W} = \{0, 1, 2, 3, 4, 5, \ldots\}

This single addition of zero is historically one of the most important ideas in mathematics, and Indian mathematics played a huge role in shaping it.

Quick Think

A batsman gets out on a duck and scores 0. Is 0 a Natural Number or a Whole Number?

Answer: Whole Number only.

Integers

Going below zero - temperature, depth, and debt

Hierarchy

Natural Numbers are contained in Whole Numbers, and Whole Numbers are contained in Integers.

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z}

Temperatures like -5°C, basement floors like B2, and overdrafts in a bank account all demand numbers below zero.

Integers are the full family of negative whole numbers, zero, and positive whole numbers:

\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

They stretch in both directions on the number line. Positive integers go right, negative integers go left, and zero sits in the middle.

Quick Think

Arrange in ascending order: 5, -7, 0, -2, 3, -10

Answer: -10, -7, -2, 0, 3, 5

Rational Numbers

The sharing numbers - fractions, ratios, and parts

When one pizza is shared equally among four friends, each person gets

\frac{1}{4}

. Fractions like this require a wider number system than integers.

A rational number is any number that can be written in the form

\frac{p}{q}

where

p, q \in \mathbb{Z}

and

q \neq 0

Every integer is also rational because

5 = \frac{5}{1}

-7 = \frac{-7}{1}

, and

0 = \frac{0}{1}

The decimal expansion of a rational number is either terminating or non-terminating repeating.

Examples

Check which numbers are rational

$5 = \frac{5}{1}$ is rational.
$0.5 = \frac{1}{2}$ is rational.
$0.\overline{3} = \frac{1}{3}$ is rational.
$\sqrt{2}$ is not rational.

Irrational Numbers

Infinite decimals with no repeating pattern

Common Mistakes

$\frac{22}{7}$ is rational. It is only an approximation of $\pi$ .
$\sqrt{4} = 2$ is rational because 4 is a perfect square.
$\sqrt{5}$ is irrational because 5 is not a perfect square.

The diagonal of a square tile of side 1 metre is

\sqrt{2}

. Its decimal expansion goes on forever without repeating: 1.4142135...

An irrational number cannot be written as

\frac{p}{q}

with integers

p

and

q\neq0

Numbers like

\sqrt{2}

\sqrt{3}

\pi

, and

e

are irrational. Non-perfect square roots of natural numbers are a common source of irrational numbers.

Real Numbers

The complete family of numbers on the number line

Real Numbers include every rational number and every irrational number. If a number can be placed on the number line, it is a real number.

The complete inclusion chain is

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

This chapter is called Real Numbers because it brings together the full number system used in everyday arithmetic and higher mathematics.

Quick Think

Is every integer a real number? Is every real number an integer?

Answer: Every integer is real, but not every real number is an integer.

What 'Divides' Actually Means

The precise meaning of divisibility

If 12 toffees are shared equally among 3 children with nothing left over, we say 3 divides 12 and write

3 \mid 12

Formally, a non-zero integer

a

divides an integer

b

if there exists an integer

c

such that

b = a \times c

If there is a remainder, then

a

does not divide

b

Textbook Illustration

State whether each is true or false

$3 \mid 93$ is true because $93 = 3 \times 31$ .
$6 \mid 28$ is false because the division leaves a remainder.
$0 \mid 4$ is false because division by zero is not defined.
$5 \mid 0$ is true because $0 = 5 \times 0$ .

Euclid's Division Lemma

The universal packing rule of division

If 17 mangoes are packed into boxes of 5, then 3 boxes are full and 2 mangoes remain. In symbols:

17 = 5 \times 3 + 2

Euclid's Division Lemma says that for any two positive integers

a

and

b

, there exist unique integers

q

and

r

such that

a = bq + r, \quad 0 \leq r < b

Here

a

is the dividend,

b

is the divisor,

q

is the quotient, and

r

is the remainder.

Solved Example

Express 347 in the form 11q + r

Dividing 347 by 11 gives quotient 31 and remainder 6, so

347 = 11 \times 31 + 6

The Golden Rule of Remainder

Why the remainder must always be smaller than the divisor

If you say 25 = 6 x 3 + 7, the remainder is 7, which is bigger than the divisor 6. That means the division is not complete because you could extract one more group of 6.

This is why Euclid's Lemma always enforces

0 \leq r < b

When the divisor is 2, the remainder can only be 0 or 1. When the divisor is 3, the remainder can only be 0, 1, or 2.

Quick Check

Check whether these are valid: 25 = 6 x 3 + 7, 25 = 6 x 4 + 1, 25 = 6 x 5 - 5

Answer: Only the middle one is valid.

Why Learn This?

How Euclid's Lemma helps classify integers

When a number is divided by 2, the only possible remainders are 0 and 1. That gives two forms:

2q \text{ and } 2q+1

So every positive even integer is of the form

2q

, and every positive odd integer is of the form

2q+1

This logic extends to other divisors as well. Dividing by 3 gives forms

3q

3q+1

, and

3q+2

Standard Proof

Why every positive integer is either 2q or 2q+1

Apply Euclid's Lemma with divisor 2. Since the remainder can only be 0 or 1, every integer must be of the form

2q

2q+1

Euclid's Division Algorithm

Finding the HCF step by step

Euclid's Division Algorithm repeatedly applies Euclid's Lemma until the remainder becomes zero.

The last non-zero remainder is the HCF. This is often faster than listing factors or doing full prime factorisation.

For example:

210 = 55 \times 3 + 45

55 = 45 \times 1 + 10

45 = 10 \times 4 + 5

10 = 5 \times 2 + 0

So the HCF is 5.

Show the full HCF chain for HCF(4052, 12576)

Since 12576 is larger, start there.

$12576 = 4052 \times 3 + 420$
$4052 = 420 \times 9 + 272$
$420 = 272 \times 1 + 148$
$272 = 148 \times 1 + 124$
$148 = 124 \times 1 + 24$
$124 = 24 \times 5 + 4$
$24 = 4 \times 6 + 0$

So the HCF is 4.

Practice

Find the HCF using Euclid's Division Algorithm: (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Applications of HCF

Largest equal grouping problems

Whenever a question asks for the greatest size, largest equal grouping, maximum tile size, or same number of columns, the idea is usually HCF.

If 420 Kaju Barfis and 130 Badam Barfis must be stacked equally with the least tray area, each stack should contain the HCF of 420 and 130, which is 10.

If 616 soldiers and 32 band members march in the same number of columns, the maximum possible number of columns is HCF(616, 32) = 8.

Fundamental Theorem of Arithmetic

Every composite number has unique prime-factor DNA

Every composite number can be expressed as a product of primes, and this factorisation is unique except for the order of the factors.

For example,

1176 = 2^3 \times 3 \times 7^2

and no matter how you factor 1176, you reach the same prime powers.

This theorem is the base for many proofs in this chapter, including results about decimal expansions and the last digit of powers.

Solved Example

Can 4^n ever end with 0?

No. Since

4^n = (2^2)^n = 2^{2n}

its prime factorisation contains only 2. A number ending in 0 must be divisible by 10, and therefore must have both 2 and 5 as prime factors.

HCF and LCM Relationship

The shortcut formula for two numbers

For any two positive integers

\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

This formula is valid for two numbers only. It is a very common exam trap to apply it blindly to three numbers.

Using prime factorisation, HCF takes the common primes with the minimum powers, while LCM takes all primes with the maximum powers.

Worked Example

Find HCF and LCM of 90 and 144

90 = 2 \times 3^2 \times 5, \qquad 144 = 2^4 \times 3^2

\text{HCF} = 2 \times 3^2 = 18, \qquad \text{LCM} = 2^4 \times 3^2 \times 5 = 720

Proving Irrationality

The contradiction method

To prove that

\sqrt{2}

is irrational, assume the opposite:

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

where

a

and

b

are co-prime.

Squaring gives

2b^2 = a^2

so 2 divides

a^2

, and therefore 2 divides

a

. Let

a = 2c

Substituting back shows that 2 also divides

b

. But then

a

and

b

are both even, contradicting the assumption that they are co-prime. Hence

\sqrt{2}

is irrational.

The same contradiction method works for

\sqrt{3}

\sqrt{5}

, and expressions like

3 + 2\sqrt{5}

Template

How to prove expressions like 3 + 2√5 are irrational

Assume

3 + 2\sqrt{5}

is rational. Then isolate

\sqrt{5}

and conclude it would also be rational, which is impossible.

Show the contradiction proof for √3

Assume $\sqrt{3} = \frac{a}{b}$ where $a$ and $b$ are co-prime.

Squaring gives $3b^2 = a^2$ . So 3 divides $a^2$ , hence 3 divides $a$ .

Let $a = 3c$ . Then $a^2 = 9c^2$ , so $3b^2 = 9c^2$ and therefore $b^2 = 3c^2$ .

This means 3 divides $b$ as well, contradicting the fact that $a$ and $b$ are co-prime.

Therefore, $\sqrt{3}$ is irrational.

Decimal Expansions of Rational Numbers

When decimals terminate and when they repeat

If a rational number

\frac{p}{q}

is in lowest terms, then its decimal expansion terminates if and only if the prime factorisation of

q

is of the form

2^m \times 5^n

If the denominator has any prime factor other than 2 or 5, then the decimal expansion is non-terminating repeating.

For example,

\frac{17}{8}

terminates because

8 = 2^3

, while

\frac{64}{455}

repeats because

455 = 5 \times 7 \times 13

contains primes other than 2 and 5.

The number of decimal places in a terminating decimal is

\max(m, n)

when the denominator is

2^m \times 5^n

Practice

State whether each decimal terminates or repeats: 23/8, 125/441, 35/50, 77/210

Complete Chapter Summary

#	Concept	Key Fact / Formula
1	Natural Numbers	N = {1, 2, 3, ...}
2	Whole Numbers	W = {0, 1, 2, 3, ...}
3	Integers	Z includes negatives, zero, positives
4	Rational Numbers	p/q with q != 0
5	Irrational Numbers	Non-terminating, non-repeating decimals
6	Real Numbers	N subset W subset Z subset Q subset R
7	Divisibility	a divides b if b = a x c
8	Euclid's Lemma	a = bq + r, 0 <= r < b
11	Euclid's Algorithm	Last non-zero remainder is the HCF
13	Fundamental Theorem	Unique prime factorisation
14	HCF x LCM	HCF(a,b) x LCM(a,b) = a x b
15	Irrationality	Assume rational, derive contradiction
16	Decimal Expansions	Terminating iff denominator = 2^m x 5^n

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