NCERT Solutions for Class 9th Mathematics

Chapter 1: Number Systems

On this page, you will find all solutions of the chapter Number Systems from all the exercises so, study the NCERT Solutions (Class 9^th Mathematics) .

Number Systems Exercise 1.1

Question 1:

 Is zero a rational number? Can you write it in the form pq, where p and q are integers and q ≠0?

Solution: 

Yes, zero is a rational number it can be written in the form pq.
0 = 01 = 02 = 03 etc. denominator q can also be taken as negative integer.

Question 2:

Find six rational numbers between 3 and 4.

Solution:

Let q_i be the rational number between 3
and 4, where j = 1 to 6.
∴ Six rational numbers are as follows:

Question 3: Find five Rational Numbers between 35 and 45.

Solution: We need to find five rational numbers, therefore, multiply numerator and denominator by 6 we get,

Question 4: State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.

Solution:
(i) True
∵ The collection of all natural numbers and 0 is called whole numbers.
(ii) False
∵ Negative integers are not whole numbers.
(iii) False
∵ Rational numbers are of the form p/q, q ≠ 0 and q does not divide p completely that are not whole numbers.  

NCERT Solutions for Class 9^th Mathematics

Chapter 1: Number Systems, Exercise: 1.2

Question 1:

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

1. Every irrational number is a real number.
2. Every point on the number line is of the form √m, where m is a natural number.
3. Every real number is an irrational number. 

Solution:

1. True; (Because all rational numbers and all irrational numbers form the group of real numbers.
2. False; (Because negative numbers cannot be the square root of any natural number).
3. False; (Because rational numbers are also a part of real numbers).

Question 2:

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. 

Solution:

No, if we take a positive integer i.e. 9 then its square root is 3, which is a rational number. 

Question 3:

Show how √5 can be represented on the number line?

Solution:

Draw a number line and take point O and A on it such that OA = 1 unit. 

Draw BA ⊥ OA as BA = 1 unit. Join OB = √2 units.
Now draw BB₁ ⊥ OB such that BB₁ =1 unit. Join OB₁ = √3 units.
Next, draw B₁B₂⊥ OB₁ such that B₁B₂ = 1 unit.
Join OB₂ = units.
Again draw B₂B₃ ⊥OB₂ such that B₂B₃ = 1 unit.
Join OB₃ = √5 units.

Take O as centre and OB₃ as radius, 

Now, draw an arc which cuts the number line at D.
Point D
represents √5 on the number line.

Question 4: Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP₁, of unit lengths Draw a line segment P₁, P₂ perpendicular to OP₁ of unit length (see figure). Now, draw a line segment P₂P₃ perpendicular to OP₂. Then draw a line segment P₃P₄ perpendicular to OP₃. Continuing in this manner, you can get the line segment P_n-1 P_n by drawing a line segment of unit length perpendicular to OP_{n – 1}. In this manner, you will have created the points P₂, P₃,…… P_n,….. and joined them to create a beautiful spiral depicting √2,√3,√4,……

Solution:

Do the Solution of this Question

NCERT Solutions for Class 9^th Mathematics

Chapter 1: Number Systems, Exercise: 1.3

Question 1: Write the following in decimal form and say what kind of decimal expansion each has.

Solution :

1. We have, 36/100 = 0.36

Hence, the decimal expansion of 36/100 is terminating.

2. On dividing, 1/11, we get,

Hence, the decimal expansion of 1/11 is non-terminating repeating. 

1. We have, 

             That means 33/8

On dividing, 33/8, we get, 

Hence, the decimal expansion of is terminating. 

2. 3/13

On dividing, 3/13, we get,

Here, the repeating block of digits is 230769.

Therefore, 3/13 = 0.23076923

Hence, the decimal expansion of 3/13 is non-terminating repeating. 

3. 2/11

On dividing, 2/11, we get,

Here, the repeating block of digits is 18.

Therefore, 2/11 = 0.1818… =

Hence, the decimal expansion of 2/11 is non-terminating repeating. 

4. 329/400

On dividing, 329/400, we get, 

Therefore, 329/400 = 0.8225

Hence, the decimal expansion of 329/400 is terminating.

Question 2: You know that Can you predict what the decimal expansions of are, without actually doing the long division? If so, how?

Solution:

Thus, without actually doing the long division we can predict the decimal expansions of the given rational numbers.

Question 3: Express the following in the form of where p and q are integers and q ≠ 0.

Solutions:

Question 4: Express 0.99999…in the form p/q Are you surprised by your answer? With you teacher and classmates discuss why the answer makes sense.

Solution:

Let x = 0.99999….. …. (i)
As there is only one repeating digit,
multiplying (i) by 10 on both sides, we get
10x = 9.9999 … (ii)
Subtracting (i) from (ii), we get
10x – x = (99999 ) — (0.9999 )
⇒ 9x = 9 ⇒ x = 9/9 = 1
Thus, 0.9999 =1
As 0.9999… goes on forever, there is no such a big difference between 1 and 0.9999
Hence, both are equal.

Question 5: 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:

In 1/17, in the divisor is 17.
Since, the number of entries in the repeating block of digits is less than the divisor, then the maximum number of digits in the repeating block is 16.
Dividing 1 by 17, we have,

 Question 6 : Look at several examples of rational numbers in the form p/q (q ≠ 0). Where, p and q are integers with no common factors other that 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

Solution: Let us look decimal expansion of the following terminating rational numbers:

We observe that the prime factorisation of q (i.e. denominator) has only powers of 2 or powers of 5 or powers of both.

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

Solution:

√2 = 1.414213562 ………..
√3 = 1.732050808 …….
√5 = 2.23606797 …….

Question 8: Find three different irrational numbers between the rational numbers 5/7 and 9/11.

Solution:

Question 9:

NCERT Solutions for Class 9^th Mathematics

Chapter 1: Number Systems, Exercise: 1.4

Question 1: Visualise 3.765 on the number line, using successive magnification.

Solution:

Question 2:

Solution:

NCERT Solutions for Class 9^th Mathematics

Chapter 1: Number Systems, Exercise: 1.5

Question 1: Classify the following numbers as rational or irrational.

Solution:

Question 2:

Question 3: Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is π = c/d. This seems to contradict the fact that n is irrational. How will you resolve this contradiction?
Solution: When we measure the length of a line with a scale or with any other device, we only get an approximate ational value, i.e. c and d both are irrational.
∴ c/d is irrational and hence π is irrational.
Thus, there is no contradiction in saying that it is irrational.

Question 4:

Question 5:

Solution:

NCERT Solutions for Class 9^th Mathematics

Chapter 1: Number Systems, Exercise: 1.6

Question 1:

Question 2:

Question 3:

NCERT Solutions (Class 9^th Mathematics)

Chapter 1 Number Systems

Chapter 2 Polynomials

Chapter 3 Coordinate Geometry

Chapter 4 Linear Equations in two variables

Chapter 5 Introduction to Euclid Geometry

Chapter 6 Lines and Angles

Chapter 7 Triangles

Chapter 8 Quadrilaterals

Chapter 9 Circles

Chapter 10 Heron’s Formula

Chapter 11 Surface Areas and Volume

Chapter 12 Statistics

Class 9th Mathematics (Download PDF)

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