NCERT Solutions Class 10 Maths Chapter 1 Real Numbers Exercise 1.4

NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.4 explains the decimal forms of rational numbers. Students will understand through this exercise whether a rational number has a terminating or non-terminating repeating decimal. This makes it easier for students to grasp how a denominator of a rational number impacts its decimal expansion. The concepts have been kept easy to grasp and follow in keeping with the syllabus prescribed by CBSE to help students learn this fundamental subject with clarity.

1.0Download NCERT Solutions Class 10 Maths Chapter 1 Real Numbers Exercise 1.4 : Free PDF

2.0Introduction to Rational Numbers and Their Decimal Expansions

Rational numbers are those numbers that may be written in the form of a fraction, $\frac{p}{q}$, with p and q being integers and $q\ne 0$. Moreover, p and q have a greatest common factor (GCF) of 1. For instance, $\frac{1}{3}$ is a rational number while $\frac{3}{6}$ is not. A rational number may have a terminating decimal expansion or a non-terminating repeating decimal expansion. Some examples of rational numbers include $–1,1,\frac{2}{3},2.5$, etc.  Let’s explore these concepts a bit more deeply. 

Terminating Decimal Expansions

A rational number has a terminating decimal representation if, when it is written as a decimal, the digits terminate. This rational number may be written as $\frac{p}{q}$, with the denominator q (in lowest form) having only powers of 2 and 5 as prime factors. Such as 0.475 or $\frac{19}{40}$

Non-terminating Repeating Decimal Expansion

A non-terminating repeating decimal expansion is a form of decimal expression of a rational number where the decimal digits never terminate but recur in a recurring cycle. It is where, after some point, the digits repeat in a cycle. For instance 13 or 0.333….

3.0Class 10 Maths Chapter 1 Real Numbers Exercise 1.4 Overview: Key Concepts

Identifying Terminating and Repeating Decimals

The distinction between terminating and non-terminating repeating decimals simplifies rational numbers and makes solving problems more efficient. In this exercise, we have some methods that help in identifying rational numbers that terminate or repeat after division. These methods include: 

1. For Terminating Rational Numbers: If given a rational number, say, $x=\frac{p}{q}$, such that the prime factorisation of denominator “q” is of the form 2ⁿ5^m, where n & m are positive integers. Then, x has a terminating decimal expansion. This is due to the simple fact that a multiple of 2 and 5 equals 10, which can terminate any number when divided. 
2. For Non-terminating Repeating Rational Numbers: To determine whether a rational number is a non-terminating repeating, consider a rational number, say $x=\frac{p}{q}$, such that the prime factorisation of q is not of the form 2ⁿ5^m, where n & m are non-negative integers. Then, x has a decimal expansion that is non-terminating, repeating (recurring).

Rational Numbers to Decimal Form Conversion:

Converting rational numbers to decimal form aids in identifying whether the expansion is terminating or repeating, thus facilitating simpler calculations. In the exercise, rational numbers can be simply converted into decimal form by understanding one fact. That is, if a number $x=\frac{p}{q}$ is given as a rational number such that its decimal expansion terminates. Then, the prime factorisation of q (in its simplest form) is of the form 2ⁿ×5^m, where n and m are non-negative integers.

This simply means that to convert Rational Numbers to Decimal form, we simply need to write the denominator of the number in the form of 2ⁿ×5^m. Then, simply divide the numerator using basic algebra. 

Start preparing for your exam with this important topic of real numbers. Do it by mastering NCERT Solutions Class 10 Maths Chapter 1 - Exercise 1.4.

4.0NCERT Solutions Class 10 Maths Chapter 1 Exercise 1.4: Detailed Solutions

Q1. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

(i) 13/3125

(ii) 17/8

(iii) 64/455

(iv) 15/1600

(v) 29/343

(vi) 23/(2³5²)

(vii) 129/(2²5⁷7⁵)

(viii) 6/15

(ix) 35/50

(x) 77/210

Solution:

(i) 13/3125 = 13/5⁵

Hence, q = 5⁵, which is of the form 2ⁿ5ᵐ (n = 0, m = 5). So, the rational number 13/3125 has a terminating decimal expansion.

(ii) 17/8 = 17/2³

Hence, q = 2³, which is of the form 2ⁿ5ᵐ (n = 3, m = 0). So, the rational number 17/8 has a terminating decimal expansion.

(iii) 64/455 = 64/(5 × 7 × 13)

Hence, q = 5 × 7 × 13, which is not of the form 2ⁿ5ᵐ. So, the rational number 64/455 has a non-terminating repeating decimal expansion.

(iv) 15/1600 = 15/(2⁶ × 5²) = (3 × 5)/(2⁶ × 5²) = 3/(2⁶ × 5)

Hence, q = 2⁶ × 5, which is of the form 2ⁿ × 5ᵐ (n = 6, m = 1). So, the rational number 15/1600 has a terminating decimal expansion.

(v) 29/343 = 29/7³

Hence, q = 7³, which is not of the form 2ᵐ × 5ⁿ. So, the rational number 29/343 has a non-terminating repeating decimal expansion.

(vi) 23/(2³ × 5²)

Hence, q = 2³ × 5², which is of the form 2ⁿ × 5ᵐ (n = 3, m = 2). So, the rational number has a terminating decimal expansion.

(vii) 129/(2² × 5⁷ × 7⁵)

Hence, q = 2² × 5⁷ × 7⁵, which is not of the form 2ᵐ × 5ⁿ. So, the rational number 129/(2² × 5⁷ × 7⁵) has a non-terminating repeating decimal expansion.

(viii) 6/15 = (2 × 3)/(3 × 5) = 2/5

Hence, q = 5, which is of the form 2ᵐ × 5ⁿ (m = 0, n = 1). So, the rational number 2/5 has a terminating decimal expansion.

(ix) 35/50 = (5 × 7)/(2 × 5²) = 7/(2 × 5)

Hence, q = 2¹ × 5¹, which is of the form 2ᵐ × 5ⁿ (m = 1, n = 1). So, the rational number 35/50 has a terminating decimal expansion.

(x) 77/210 = (7 × 11)/(2 × 3 × 5 × 7) = 11/(2 × 3 × 5)

 Hence, q = 2 × 3 × 5, which is not of the form 2ᵐ × 5ⁿ. So, the rational number 77/210 has a non-terminating repeating decimal expansion.

Q2. Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.

Solution:

(i) 13/3125 = 13/5⁵ = (13 × 2⁵)/(5⁵ × 2⁵) = 416/10⁵ = 0.00416

(ii) 17/8 = 17/2³ = (17 × 5³)/(2³ × 5³) = 2125/10³ = 2.125

(iv) 15/1600 = 3/(2⁶ × 5) = (3 × 5⁵)/(2⁶ × 5⁶) = 9375/10⁶ = 0.009375

(vi) 23/(2³ × 5²) = (23 × 5)/(2³ × 5³) = 115/10³ = 0.115

(viii) 6/15 = 2/5 = 4/10 = 0.4

(ix) 35/50 = 7/10 = 0.7

Q3. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational, or not. If they are rational, and of the form p/q, what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000…...

(iii) 43.123456789 (with a bar over 123456789 indicating repetition)

Solution:

(i) 43.123456789

Since the decimal expansion terminates, the given real number is rational.

It can be expressed in the form p/q as:

43.123456789 = 43123456789 / 1000000000

43.123456789 = 43123456789 / 10⁹

43.123456789 = 43123456789 / (2 × 5)⁹

43.123456789 = 43123456789 / (2⁹ × 5⁹)

Therefore, q = 2⁹ × 5⁹.

The prime factorization of q is of the form 2ⁿ ⋅ 5ᵐ, where n = 9 and m = 9.

(ii) 0.120120012000120000....

Since the decimal expansion is neither terminating nor non-terminating, the given real number is irrational.

(iii) 43.123456789 (with a bar over 123456789)

Since the decimal expansion is non-terminating and repeating, the given real number is rational.

However, since the decimal is non-terminating repeating, q is not of the form 2ᵐ × 5ⁿ.

The prime factors of q will contain factors other than 2 and 5.

5.0Benefits of Studying Class 10 Maths Chapter 1 Real NUmbers Exercise 1.4

• Helps students learn the concept of proving irrational numbers effectively.
• Regular practice helps students understand and solve similar problems with ease.
• Builds a strong base for advanced algebra and number theory in higher classes.
• NCERT solutions are expert-reviewed, ensuring accuracy in understanding concepts.