# NCERT Solutions Class 9 Maths Chapter 1 Exercise 1.3

One of the most significant and well-respected educational organizations in the nation, the Central Board of Secondary Education (abbreviated as CBSE) answers to the Union Government of India. Numerous public and private schools that are part of the CBSE board use the NCERT curriculum.

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## CBSE Class 9 NCERT Solutions Maths Chapter 1 Exercise 1.3

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### Access NCERT Solutions for Class 9 Maths Chapter 1 – Number System

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### Exercise 1.3 Class 9 Maths answers for their perusal online

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### NCERT Solutions for Class 9 Maths Chapter 1 Other Exercises

The first chapter of Mathematics for Class 9 is titled Number System. In this chapter, the Number System is covered in great detail. The number systems and their uses are covered in this chapter. Whole, integer, and rational numbers are all included in the chapter’s introduction.

The chapter begins with an overview of number systems in section 1.1, followed by sections 1.2 and 1.3 on two crucial subjects.

Irrational numbers are those that cannot be expressed as p/q.

Real Numbers and their Decimal Expansions – In this section, students examine real number decimal expansions to determine if they may be used to discern between logical and irrational behaviour. All the definitions and simplified terms are provided in the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3.

It then talks about the following subjects.

Real Numbers on the Number Line – These are the answers to the two problems in Exercise 1.4.

Operations on Real Numbers – In this section, students will learn about various irrational number operations, such as addition, subtraction, multiplication, and division.

Exponentiation Laws for Real Numbers – To answer the problems, use these exponentiation laws.

The NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3 can help students to learn more about Number Systems and how to resolve all types of issues. One of the greatest academic tools for studying for their CBSE exams is this one.

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The NCERT Solutions for Class 9 Mathematics Chapters are as follows. There are 15 chapters in total

Chapter 1 Number System

Chapter 2 Polynomials

Chapter 3 Coordinate Geometry

Chapter 4 Linear Equations in Two Variables

Chapter 5 Introduction to Euclid’s Geometry

Chapter 6 Lines and Angles

Chapter 7 Triangles

Chapter 9 Areas of Parallelograms and Triangles

Chapter 10 Circles

Chapter 11 Constructions

Chapter 12 Heron’s Formula

Chapter 13 Surface Areas and Volumes

Chapter 14 Statistics

Chapter 15 Probability

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### NCERT Solutions for Class 9

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### CBSE Study Materials for Class 9

The foundation for Class 10 and beyond is laid by the CBSE Study Materials for Class 9. The mathematics, science, and social studies curricula for Class 9 are vast and difficult. It covers every foundational idea that’s crucial for competitive exams like the IIT JEE, Olympiads, NEET, and more. As a result, students must prepare for and pass their Class 9 exams. In order to help students with their studies, Extramarks has provided the CBSE Study Materials for Class 9.

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### CBSE Study Materials

NCERT, or the National Council of Educational Research and Training, is an autonomous organisation. The Societies Registration Act allowed for the establishment of this organisation in 1961. The Government of India controls it. Its goal is to advance the nation’s literary, artistic, and humanitarian growth. The NCERT textbooks and NCERT curriculum are published for students in classes 1 to 12. The NCERT textbooks and curricula are regarded as reliable resources for the CBSE Board Exam. These textbooks and curricula serve as the foundation for competitive exams like the National Eligibility Entrance Test (NEET), Union Public Service Commission (UPSC), Indian Institute of Technology (IIT), Provincial Civil Service (PCS), etc. in addition to being applicable for annual exams or CBSE Board Examinations.

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Q.1

$\begin{array}{l}\text{Write the following in decimal form and say what kind of}\\ \text{decimal expansion each has:}\\ \left(\text{i}\right)\text{ }\frac{\text{36}}{\text{100}}\text{ }\left(\text{ii}\right)\text{ }\frac{\text{1}}{\text{11}}\text{ }\left(\text{iii}\right)\text{ 4}\frac{\text{1}}{\text{8}}\text{ }\left(\text{iv}\right)\text{ }\frac{\text{3}}{\text{13}}\text{ }\left(\text{v}\right)\text{ }\frac{\text{2}}{\text{11}}\text{ }\left(\text{vi}\right)\text{ }\frac{\text{329}}{\text{400}}\end{array}$

Ans

$\begin{array}{l}\left(\mathrm{i}\right)\frac{36}{100}=0.36\\ \text{It is a terminating decimal because remainder is zero.}\\ \left(\mathrm{ii}\right)\frac{1}{11}=0.0909090909\dots \\ \text{It is a non-terminating decimal because remainder is not zero.}\\ \left(\mathrm{iii}\right)4\frac{1}{8}=4.125\\ \text{It is a terminating decimal because remainder is zero.}\\ \left(\mathrm{iv}\right)\frac{3}{13}=0.230769230769\dots \\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=0.\overline{230769}\\ \text{It is a non terminating decimal because it is a repeating}\\ \\ \left(\mathrm{v}\right)\frac{2}{11}=0.181818181818\dots \\ \text{\hspace{0.17em}\hspace{0.17em}}=0.\overline{18}\\ \text{It is a terminating decimal because it is a repeating}\\ \text{\hspace{0.17em}}\\ \left(\mathrm{vi}\right)\frac{329}{400}=0.8225\\ \text{It is a terminating decimal because remainder is zero.}\end{array}$

Q.2

$\begin{array}{l}\text{You know that }\frac{\text{1}}{\text{7}}\text{ = 0}\overline{\text{.142857}}\text{. Can you predict what the}\\ \text{decimal expansions of }\frac{\text{2}}{\text{7}}\text{ ,}\frac{\text{3}}{\text{7}}\text{ ,}\frac{\text{4}}{\text{7}}\text{, }\frac{\text{5}}{\text{7}}\text{, }\frac{\text{6}}{\text{7}}\text{ are, without actually}\\ \text{doing the long division? If so, how?}\end{array}$

Ans

$\begin{array}{l}\mathrm{Given}:\text{}\frac{1}{7}=0.\overline{142857}\\ \therefore \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{2}{7}=2×0.\overline{142857}\\ =0.\overline{285714}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{3}{7}=3×0.\overline{142857}\\ =0.\overline{428571}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{4}{7}=4×0.\overline{142857}\\ =0.\overline{571428}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{5}{7}=5×0.\overline{142857}\\ =0.\overline{714285}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{6}{7}=6×0.\overline{142857}\\ =0.\overline{857142}\end{array}$

Q.3

$\begin{array}{l}\text{Express the following in the form }\frac{\text{p}}{\text{q}}\text{, where p and q are}\\ \text{integers and q¹0.}\\ \left(\text{i}\right)\text{ 0 }\left(\text{ii}\right)\text{ 0.4}\overline{\text{7}}\text{ }\left(\text{iii}\right)\text{ 0.}\overline{\text{001}}\end{array}$

Ans

$\begin{array}{l}\left(\mathrm{i}\right)\text{\hspace{0.17em}}\mathrm{Let}\text{x}=0.\overline{6}...\left(\mathrm{i}\right)\\ \text{Multiplying both sides by 10, we get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}10\mathrm{x}=10×0.\overline{6}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}10\mathrm{x}=6.\overline{6}...\left(\mathrm{ii}\right)\\ \text{Subtracting equation}\left(\mathrm{i}\right)\text{from equation}\left(\mathrm{ii}\right),\text{we get}\\ 10\mathrm{x}-\mathrm{x}=6.\overline{6}-0.\overline{6}\\ 9\mathrm{x}=6\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{x}=\frac{6}{9}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{x}=\frac{2}{3}\\ \therefore \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}0.\overline{6}=\frac{2}{3}\\ \left(\mathrm{ii}\right)\\ \text{\hspace{0.17em}}\mathrm{Let}\text{x}=0.4\overline{7}...\left(\mathrm{i}\right)\\ \text{Multiplying both sides of equation}\left(\mathrm{i}\right)\text{by 10, we get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}10\mathrm{x}=10×0.4\overline{7}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}10\mathrm{x}=4.\overline{7}...\left(\mathrm{ii}\right)\\ \mathrm{Multiplying}\text{both sides of equation}\left(\mathrm{ii}\right)\text{by 10, we get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}100\mathrm{x}=10×4.\overline{7}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}100\mathrm{x}=47.\overline{7}...\left(\mathrm{iii}\right)\\ \text{Subtracting equation}\left(\mathrm{ii}\right)\text{from equation}\left(\mathrm{iii}\right),\text{we get}\\ 100\mathrm{x}-10\mathrm{x}=47.\overline{7}-4.\overline{7}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}90\mathrm{x}=43\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{x}=\frac{43}{90}\\ \therefore \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}0.4\overline{7}=\frac{43}{90}\\ \left(\mathrm{iii}\right)\mathrm{Let}\text{x}=0.\overline{001}...\left(\mathrm{i}\right)\\ \text{Multiplying both sides of equation}\left(\mathrm{i}\right)\text{by 1000, we get}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1000x}=1000×0.\overline{001}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=1.\overline{001}...\left(\mathrm{ii}\right)\\ \text{Subtracting equation}\left(\mathrm{i}\right)\text{from equation}\left(\mathrm{ii}\right),\text{we get}\\ \text{1000x\hspace{0.17em}}-\mathrm{x}=1.\overline{001}-0.\overline{001}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}999\mathrm{x}=1\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{x}=\frac{1}{999}\\ \therefore \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}0.\overline{001}=\frac{1}{999}\end{array}$

Q.4

$\begin{array}{l}\text{Express 0.99999, in the form }\frac{\text{p}}{\text{q}}\text{. Are you surprised by your answer?}\\ \text{With your teacher and classmates discussway the answer makes sense.}\end{array}$

Ans

$\begin{array}{l}\text{\hspace{0.17em}Let x}=0.99999\dots \\ =0.\overline{9}\dots \left(i\right)\\ \text{Multiplying both sides by 10, we get}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}10x=10×0.\overline{9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}10x=9.\overline{9}\dots \left(ii\right)\\ \text{Subtracting equation}\left(i\right)\text{from equation}\left(ii\right),\text{we get}\\ 10x-x=9.\overline{9}-0.\overline{9}\\ 9x=9\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=\frac{9}{9}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=1\\ \therefore \text{\hspace{0.17em}}\text{\hspace{0.17em}}0.99999\dots =1\end{array}$

Q.5

$\begin{array}{l}\mathrm{What}\mathrm{can}\mathrm{the}\mathrm{maximum}\mathrm{number}\mathrm{of}\mathrm{digits}\mathrm{be}\mathrm{in}\mathrm{the}\mathrm{repeating}\\ \mathrm{block}\mathrm{of}\mathrm{digits}\mathrm{in}\mathrm{the}\mathrm{decimal}\mathrm{expansion}\mathrm{of}\frac{1}{17}?\mathrm{Perform}\mathrm{the}\\ \mathrm{division}\mathrm{to}\mathrm{check}\mathrm{your}\mathrm{answer}.\end{array}$

Ans

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\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}140\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{136}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}40\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{34}\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}60\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{51}\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}90\\ \text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{85}\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}50\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{34}\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}160\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{153}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}70\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{68}\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}20\\ \text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{17}\\ \text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}30\\ \text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{17}\\ \text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}130\\ \text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{119}\\ \text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}110\\ \text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}\underset{¯}{102}\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}80\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{68}\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}120\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\underset{¯}{119}\\ \text{}\text{}\text{}\text{}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ Thus,\text{\hspace{0.17em}}\frac{1}{17}=0.\overline{0588235294117647}\end{array}$

Q.6

$\begin{array}{l}\mathrm{Look}\mathrm{at}\mathrm{several}\mathrm{examples}\mathrm{of}\mathrm{rational}\mathrm{numbers}\mathrm{in}\mathrm{the}\mathrm{form}\\ \frac{\mathrm{p}}{\mathrm{q}}\left(\mathrm{q}\ne 0\right),\mathrm{where}\mathrm{p}\mathrm{and}\mathrm{q}\mathrm{are}\mathrm{integers}\mathrm{with}\mathrm{no}\mathrm{common}\\ \mathrm{factors}\mathrm{other}\mathrm{than}1\mathrm{and}\mathrm{having}\mathrm{terminating}\mathrm{decimal}\\ \mathrm{representations}\left(\mathrm{expansions}\right).\mathrm{Can}\mathrm{you}\mathrm{guess}\mathrm{what}\mathrm{property}\\ \mathrm{q}\mathrm{must}\mathrm{satisfy}?\end{array}$

Ans

$\begin{array}{l}\text{Since},\text{the rational number}\frac{p}{q}\text{will be terminating decimal if denominator q}\\ \text{is either 2, 4, 5, 8, 10 and so on}\dots \\ \frac{9}{2}=4.5\\ \frac{11}{4}=2.75\\ \frac{17}{8}=2.125\\ \frac{13}{5}=2.6\\ \frac{32}{10}=3.2\\ \text{We see that rational number}\frac{\text{p}}{q}\text{will be terminating if prime factors of q}\\ \text{are either 2 only or multiple of 2 and 5 only or both}\text{.}\end{array}$

Q.7  Write three numbers whose decimal expansions are non-terminating non-recurring.

Ans Three numbers whose decimal expansions are non- terminating non-recurring are as follows: 0.030030012003000050004123000… 0.01200012500003500050010008879000102003… 1.5200050040060080010030010040038001…

Q.8

$\begin{array}{l}\mathrm{Find}\mathrm{three}\mathrm{different}\mathrm{irrational}\mathrm{numbers}\mathrm{between}\mathrm{the}\\ \mathrm{rational}\mathrm{numbers}\frac{5}{7}\mathrm{and}\frac{9}{11}.\end{array}$

Ans

$\begin{array}{l}\text{Since},\text{}\frac{5}{7}=0.\overline{714285}\text{and}\frac{9}{11}=0.\overline{81}\\ \text{Three irrational numbers between}\frac{5}{7}\text{and}\frac{9}{11}\text{are:}\\ 0.72005006004000202005004\dots \\ 0.75005006004000202005004\dots \\ 0.80005006004000202005004\dots \end{array}$

Q.9

$\begin{array}{l}\text{Classify the following numbers as rational or irrational:}\\ \left(\text{i}\right)\text{ }\sqrt{\text{23}}\text{ }\left(\text{ii}\right)\text{ }\sqrt{\text{225}}\text{ }\left(\text{iii}\right)\text{ 0.3796 }\\ \left(\text{iv}\right)\text{ 7.478478}...\text{ }\left(\text{v}\right)\text{ 1.101001000100001}...\end{array}$

Ans

$\begin{array}{l}\left(\mathrm{i}\right)\sqrt{23}=4.7983152331272\dots \\ \text{Since},\text{this number is non-terminating and non-repeating, therefore}\\ \text{it is irrational number.}\\ \left(\mathrm{ii}\right)\sqrt{225}=15,\text{It is rational number.}\\ \left(\mathrm{iii}\right)0.3796,\text{It is rational number because this number is terminating.}\\ \left(\mathrm{iv}\right)7.478478\dots =7.\overline{478},\text{this is rational number as it is non-terminating}\\ \text{and repeating.}\\ \left(\mathrm{v}\right)1.101001000100001\dots ,\text{​}\\ \text{It is irrational number because it is non-terminating non repeating.}\end{array}$

## FAQs (Frequently Asked Questions)

### 1. How many exercises are there in Class 9 Mathematics Chapter 1?

There are toilet 6 exercises in the Class 9 Mathematics Chapter 1 textbook. Listwise is given as follows

Exercise 1.1 4 Questions ( 2 long, 2 short)

Exercise 1.2 4 Questions ( 3 long, 1 short)

Exercise 1.3 9 Questions ( 9 long)

Exercise 1.4 2 Questions ( 2 long)

Exercise 1.5 5 Questions ( 4 long 1 short)

Exercise 1.6 3 Questions ( 3 long)

The Extramarks provide all exercise solutions. Students can check the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3.

### 2. How can the Extramarks the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3 assist students?

The Mathematics specialists have prepared the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3 to the questions in accordance with the most recent recommendations made by the CBSE and NCERT. To make their preparation quick and simple to grasp, they have broken down the solutions into steps. All the pertinent examinations and illustrations for the questions with real-world examples are provided.

Students can develop a strong foundation for all the topics by using these answers, which are organized in a methodical fashion. Extramarks makes sure that all the themes and subtopics from each chapter are addressed, and they have also created the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3 so that studying will be more enjoyable, engaging, and effective for students.

### 3. What are the major features of the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3?

Students can use the NCERT Solutions For Class 9 Maths Chapter 1 Exercise 1.3 to complete and review the Ch 1 Maths Class 9 Ex 1.3.

They will be able to earn higher scores after carefully practising the step-by-step answers provided by subject-matter experts.

It adheres to NCERT standards. From the standpoint, the test comprises all the crucial questions. It aids in achieving high Mathematics scores on standardized tests.