Chapter – 1

Real Numbers

In this post we have given the detailed notes of class 10 Maths Chapter 1 (Real Numbers) in English. These notes are useful for the students who are going to appear in class 10 board exams.

Chapter 1: Real Numbers

Introduction to Real Numbers

All rational numbers and irrational numbers taken together form the set of real numbers. Real numbers are denoted by R. Every real number can be represented by a unique point on the number line, and every point on the number line represents a unique real number.

Examples: √3, 22/7, √15, 5/9 are all real numbers.

Types of Real Numbers

Real numbers are broadly classified into two types:

• Positive real numbers — numbers whose value is greater than zero, e.g. 3, 7/2, √5.
• Negative real numbers — numbers whose value is less than zero, e.g. −3, −7/2, −√5.

Real numbers can further be divided into rational numbers (numbers that can be written in the form p/q, where p and q are integers and q ≠ 0) and irrational numbers (numbers that cannot be written in this form).

Properties of Real Numbers

Real numbers satisfy four important properties:

• 1. Closure Property: When two real numbers are added or multiplied, the result is always a real number.
• 2. Commutative Property: Adding or multiplying two real numbers in any order gives the same result, i.e. a + b = b + a and a × b = b × a.
• 3. Associative Property: Regrouping three real numbers while adding or multiplying does not change the result, i.e. (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• 4. Distributive Property: Multiplication distributes over addition and subtraction, i.e. a × (b + c) = a × b + a × c and a × (b − c) = a × b − a × c.

Euclid’s Division Algorithm

Euclid’s Division Algorithm is a technique to compute the Highest Common Factor (HCF) of two positive integers. In simple terms, given a positive integer a and another positive integer b, a can be divided by b in such a way that it leaves a remainder r that is smaller than b. This is nothing but the familiar process of long division. Though the statement looks very simple, it has many applications related to the divisibility properties of integers, and it is mainly used to calculate the HCF of two positive integers.

Euclid’s Division Lemma

Given positive integers a and b, there exist unique whole numbers q and r satisfying:

a = bq + r, where 0 ≤ r < b

Finding HCF Using Euclid’s Division Algorithm

To obtain the HCF of two positive integers, say c and d, with c > d, follow the steps below:

Step 1: Apply Euclid’s division lemma to c and d, to find whole numbers q and r such that c = dq + r, 0 ≤ r < d.

Step 2: If r = 0, d is the HCF of c and d. If r ≠ 0, apply the division lemma to d and r.

Step 3: Continue the process till the remainder is zero. The divisor at that stage will be the required HCF.

Example: Find the HCF of 455 and 42 using Euclid’s division algorithm.

Solution: We start with the larger integer, 455. Applying Euclid’s lemma:

455 = 42 × 10 + 35

Now taking divisor 42 and remainder 35, applying the lemma again:

42 = 35 × 1 + 7

Now taking divisor 35 and remainder 7, applying the lemma again:

35 = 7 × 5 + 0

Since the remainder has now become zero, we cannot proceed further. The divisor at this stage, i.e. 7, is the HCF of 455 and 42.

Euclid’s Division Algorithm is not only useful for computing the HCF of large numbers, it is also important because it is one of the earliest algorithms to be used as a computer program.

The 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 prime factors occur.

For example, we regard 2 × 3 × 5 × 7 as the same as 3 × 5 × 7 × 2.

The HCF and LCM of two positive integers can be found using the Fundamental Theorem of Arithmetic — this method is called the prime factorisation method.

Example: Find the HCF and LCM of 6 and 20 by the prime factorisation method.

Solution: Here 6 = 2¹ × 3¹ and 20 = 2 × 2 × 5 = 2² × 5¹.

HCF (6, 20) = 2¹ = 2

LCM (6, 20) = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

Relationship between HCF, LCM and the Product of Two Numbers

For any two positive integers a and b:

HCF (a, b) × LCM (a, b) = a × b

Check: HCF (6, 20) × LCM (6, 20) = 2 × 60 = 120, and 6 × 20 = 120. Hence verified.

Note: This relationship holds only for two numbers, not for three or more numbers.

Revisiting Irrational Numbers

Irrational Number

A number “s” is called irrational if it cannot be written in the form p/q, where p and q are integers and q ≠ 0. Some examples of irrational numbers are:

√2, √3, √15, π, 0.10110111011110… etc.

Theorem 1.3

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Proof that √2 is Irrational

To prove: √2 is irrational.

Proof: Let us assume, to the contrary, that √2 is rational.

So we can find integers r and s (s ≠ 0) such that √2 = r/s.

Suppose r and s have a common factor other than 1. Then we divide by the common factor to get √2 = a/b, where a and b are coprime.

So b√2 = a.

Squaring both sides and rearranging, we get:

2b² = a²

Therefore, 2 divides a².

By Theorem 1.3, it follows that 2 divides a.

So we can write a = 2c for some integer c.

Substituting for a, we get 2b² = 4c², that is, b² = 2c².

This means 2 divides b², and so, by Theorem 1.3, 2 divides b as well.

Therefore, a and b have at least 2 as a common factor. But this contradicts the fact that a and b are coprime.

This contradiction has arisen because of our incorrect assumption that √2 is rational. Hence, we conclude that √2 is irrational. In the same way, it can be proved that √3, √5, √7, etc. are all irrational numbers.

Rational Numbers and Their Decimal Expansions

Rational Number

A number that is of the form p/q, or that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is called a rational number.

Examples: 5/9, −3/7, 12/13, etc. are rational numbers.

Numerator and Denominator

In a rational number written in the form p/q, p is called the numerator and q is called the denominator.

For example, in the rational number 2/3, 2 is the numerator and 3 is the denominator. Similarly, in −5/6, −5 is the numerator and 6 is the denominator, and in 12/−13, 12 is the numerator and −13 is the denominator.

Representation of Rational Numbers on the Number Line

On the number line, positive integers lie to the right of zero and negative integers lie to the left of zero. Similarly, negative rational numbers are represented to the left of zero and positive rational numbers are represented to the right of zero on the number line.

Terminating and Non-Terminating Decimal Expansions

The decimal expansion of a rational number is either terminating or non-terminating repeating (recurring). Let us examine some rational numbers and their decimal expansions:

(i) 0.375 = 375/1000 = 3/8

(ii) 0.104 = 104/1000 = 13/125

(iii) 0.0875 = 875/10000 = 7/80

(iv) 23.3408 = 233408/10000 = 14588/625

Notice that all these denominators, when expressed in their lowest terms, have only 2 and/or 5 as prime factors.

Theorem — Terminating Decimal Expansion

Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p/q, where p and q are coprime, and the prime factorisation of q is of the form 2^m × 5ⁿ, where m and n are non-negative integers.

Converse Theorem

Let x = p/q be a rational number, where p and q are coprime and the prime factorisation of q is of the form 2^m × 5ⁿ, where m and n are non-negative integers. Then x has a decimal expansion which terminates.

Non-Terminating Repeating Decimal Expansion

Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2^m × 5ⁿ, where m and n are non-negative integers. Then x has a decimal expansion which is non-terminating repeating (recurring).

Example: 1/3 = 0.333… (non-terminating repeating, since denominator 3 has a prime factor other than 2 or 5), whereas 7/8 = 0.875 (terminating, since 8 = 2³).

Key Points to Remember

• Real numbers R include all rational and irrational numbers.
• Euclid’s Division Lemma: for positive integers a and b, a = bq + r, where 0 ≤ r < b.
• Euclid’s Division Algorithm is used to find the HCF of two positive integers by repeated application of the division lemma.
• The Fundamental Theorem of Arithmetic: every composite number can be expressed as a product of primes, uniquely, apart from the order of factors.
• HCF (a, b) × LCM (a, b) = a × b — true only for two numbers.
• √2, √3, √5, √7, etc. are all irrational numbers — proved by contradiction using the fact that if a prime p divides a², then p divides a.
• A rational number p/q (in lowest terms) has a terminating decimal expansion if and only if q is of the form 2^m × 5ⁿ.
• If q is not of the form 2^m × 5ⁿ, the decimal expansion of p/q is non-terminating repeating.

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