NCERT solutions class 9 maths chapter 1 â€“ Number Systems are provided here. These NCERT solutions are created by the BYJUâ€™S expert faculties to help students in the preparation of their board exams. These expert faculties solve and provide the NCERT maths solution so that it should help students to solve the problems comfortably. They also concentrate on making the solutions in such a way that it is easy to understand for the students. They give a detailed and stepwise explanation of each answer to the questions given in the exercises in the NCERT textbook for class 9.

In CBSE class 9 Number Systems chapter, students are introduced to a lot of important topics which will be helpful for those who wish to pursue mathematics as a subject in further classes. Based on these NCERT solutions, students can prepare for their upcoming Board Exams. These solutions are helpful as the syllabus covered here follows NCERT guidelines.

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## Exercise 1.1 Page: 5

** Q1. Is zero a rational number? Can you write it in the form p/q where p and q are integers and q â‰ 0?**

** Q2. Find six rational numbers between 3 and 4.**

** Q3. Find five rational numbers between 3/5 and 4/5 .**

** Q4. State whether the following statements are true or false. Give reasons for your answers.**

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## Exercise 1.2 Page: 8

** Q1. State whether the following statements are true or false. Justify your answers.**

**(i)Every irrational number is a real number.**

**Q2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.**

**Q3. Show how 5 can be represented on the number line. **

** Q4. Classroom activity (Constructing the â€˜square root spiralâ€™) : Take a large sheet of paper and construct the â€˜square root spiralâ€™ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1P2 perpendicular to OP1 of unit length (see Fig. 1.9). Now draw a line segment P2P3 perpendicular to OP2. Then draw a line segment P3P4 perpendicular to OP3. Continuing in Fig. 1.9 : **

**Constructing this manner, you can get the line segment Pnâ€“1Pn by square root spiral drawing a line segment of unit length perpendicular to OPnâ€“1. In this manner, you will have created the points P2, P3,â€¦., Pn,â€¦ ., and joined them to create a beautiful spiral depicting 2, 3, 4, â€¦**

Solution:

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## Exercise 1.3 Page: 14

** Q1. Write the following in decimal form and say what kind of decimal expansion each has :****(i) 36/100 **

**(ii) 1/11 **

**(iii) 4 1/8 **

** (iv) 3/13 **

**(v) 2/11**

**(vi) 329/400**

**You know that = \($\overline{0.142857}$\). Can you predict what the decimal expansions of 2/7,3/7,4/7,5/7,6/7 are, without actually doing the long division? If so, how?**

**[Hint: Study the remainders while finding the value of 1/7 carefully.]**

** Q3. Express the following in the form p/q, where p and q are integers and q â‰ 0.**

**Express 0.99999â€¦. in the form p/q . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.**

** Q5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer.**

Solution:

** Q6. Look at several examples of rational numbers in the form p/q (q â‰ 0), where p and q are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?**

Solution:

We observe that when q isÂ 2, 4, 5, 8, 10â€¦ Then the decimal expansion is terminating. For example:

1/2= 0.Â 5, denominatorÂ qÂ = 2^{1}

7/8= 0.Â 875, denominatorÂ qÂ = 2^{3}

4/5= 0.Â 8, denominatorÂ qÂ = 5^{1}

We can observe that the terminating decimal may be obtained in the situation where prime factorization of the denominator of the given fractions has the power of only 2 or only 5 or both.

**Q7. Write three numbers whose decimal expansions are non-terminating non-recurring.**

Solution:

We know that all irrational numbers are non-terminating non-recurring. three numbers with decimal expansions that are non-terminating non-recurring are:

a) âˆš3 = 1.732050807568..

b) âˆš26 = 5.099019513592..

c) âˆš101 = 10.04987562112..

**Q8. Find three different irrational numbers between the rational numbers 5/7 and 9/11.**

Solution:

5/7 = 0.Â 714285â€¦

9/11 = 0.81â€¦.

Three different irrational numbers are:

a) 0.73073007300073000073â€¦

b) 0.75075007300075000075â€¦

c) 0.76076007600076000076â€¦

**Q9.Â Classify** **the following numbers as rational or irrational according to their type:**

(i) **âˆš23**

Sol:

âˆš23 = 4.79583152331â€¦

Since the number is non-terminating non-recurring therefore, it is an irrational number.

(ii)** âˆš225**

Sol:

âˆš225= 15 = 15/1

Since the number can be represented inÂ p/q form, it is a rational number.

(iii)** 0.3796**

Sol:

Since the number, 0.3796, is terminating, it is a rational number.

(iv)**7.478478 **

Sol:

The number, 7.478478, is non-terminating but recurring, it is a rational number.

(v)**1.101001000100001â€¦**

Since the number, 1.101001000100001â€¦, is non-terminating non-repeating (non-recurring), it is an irrational number.

## Exercise 1.4 Page: 18

**Q1. Visualise 3.765 on the number line, using successive magnification.**

Solution:

**Q2. Visualise \($\overline{4.26}$\) on the number line, up to 4 decimal places.**

Solution:

\(\overline{4.26}\)=4.26262626â€¦.. \(\overline{4.26}\) up to 4 decimal places= 4.2626

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## Exercise 1.5 Page: 24

** Q1. Classify the following numbers as rational or irrational:**

**(i) 2 â€“ âˆš5Â **

** (iv) 1/âˆš2**

**(v) 2Î **

**Q2. Simplify each of the following expressions:**

**(i) (3 +âˆš3 ) (2 +âˆš2 ) **

** Q3. Recall, Ï€ is defined as the ratio of the circumference (say c) of a circle to its diameter, (say d). That is, Ï€ = c/d This seems to contradict the fact that Ï€ is irrational. How will you resolve this contradiction?**

Solution:

There is no contradiction. When we measure a value with a scale, we only obtain anÂ approximate value. We never obtain an exact value. Therefore, we may not realize whether c or d is irrational. The value ofÂ Ï€ is almost equal to 22/7or 3.142857â€¦Â

**Q4. RepresentÂ (âˆš9.3) on the number line.**

**Q5. RationalizeÂ the denominators of the following**

**(iv) 1 / âˆš7 â€“ 2 **

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## Exercise 1.6 Page: 26

**Q1. Find:**** (i) 64 ^{1/2}**

**Q2. Find:**** (i) 9 ^{3/2}**

**Q3. Simplify**:

** (i) 2 ^{2/3}. 2^{1/5}**

## NCERT Solutions for class 9 Maths Chapter 1- Number Systems

As this is one of the important topics in maths, It has a weightage of 6 marks in class 9 (Number system â€“ Number Systems) maths board exams. On an average three questions are asked from this unit.

- One out of three questions in part A (1 marks).
- One out of three questions in part B (2 marks).
- One out of three questions in part C (3 marks).

This chapter talks about

- Introduction of Number Systems
- Irrational Numbers
- Real Numbers and their Decimal Expansions
- Representing Real Numbers on the Number Line.
- Operations on Real Numbers
- Laws of Exponents for Real Numbers
- Summary

List of Exercises

Exercise 1.1 Solutions 4 Questions ( 2 long, 2 short)

Exercise 1.2 Solutions 4 Questions ( 3 long, 1 short)

Exercise 1.3 Solutions 9 Questions ( 9 long)

Exercise 1.4 Solutions 2 Questions ( 2 long)

Exercise 1.5 Solutions 5 Questions ( 4 long 1 short)

Exercise 1.6 Solutions 3 Questions ( 3 long)

## NCERT Solutions for class 9 Maths Chapter 1- Number Systems

NCERT solutions for class 9 maths chapter 1 â€“ Number System is the first chapter of class 9 Maths. Number System is introduced in class 9 and is discussed in detail. The chapter discusses the Number Systems and their applications. The introduction of the chapter includes whole numbers, integers and rational numbers.

The chapter starts with the introduction of Number Systems in section 1.1 followed by two very important topics in section 1.2 and 1.3

- Irrational Numbers â€“ The numbers which canâ€™t be written in the form of p/q.
- Real Numbers and their Decimal Expansions â€“ Here you study the decimal expansions of real numbers and see whether it can help in distinguishing between rational and irrationals.

Next, it discusses the following topics.

- Representing Real Numbers on the Number Line â€“ In this the solutions for 2 problems in Exercise 1.4.
- Operations on Real Numbers â€“ Here you explore some of the operations like addition, subtraction, multiplication and division on the irrational numbers.
- Laws of Exponents for Real Numbers â€“ Use these laws of exponents to solve the questions.

### Key advantages of NCERT Solutions for class 9 Maths Chapter 1- Number Systems

- These NCERT Solutions for class 9 Maths helps you solve and revise the whole syllabus of class 9.
- After going through the stepwise solutions given by our subject expert teachers, you will be able to score more marks.
- It follows NCERT guidelines which help in preparing the students accordingly.
- It contains all the important questions from the examination point of view.
- It helps in scoring well in maths in board exams.