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# RD Sharma solutions for Class 9 Mathematics chapter 1 - Number Systems

## Chapter 1: Number Systems

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.50Ex. 1.60Others

#### Chapter 1: Number Systems Exercise 1.10 solutions [Page 5]

Ex. 1.10 | Q 1 | Page 5

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

Ex. 1.10 | Q 2 | Page 5

Find five rational numbers between 1 and 2.

Ex. 1.10 | Q 3 | Page 5

Find six rational numbers between 3 and 4.

Ex. 1.10 | Q 4 | Page 5

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

Ex. 1.10 | Q 5.1 | Page 5

State whether the following statement is true or false. Give reasons for your answers.

Every whole number is a natural number.

Ex. 1.10 | Q 5.2 | Page 5

State whether the following statement is true or false. Give reasons for your answers.

Every integer is a rational number.

Ex. 1.10 | Q 5.3 | Page 5

State whether the following statement is true or false. Give reasons for your answers.

Every rational number is an integer.

Ex. 1.10 | Q 5.4 | Page 5

State whether the following statement is true or false. Give reasons for your answers.

Every natural number is a whole number.

Ex. 1.10 | Q 5.5 | Page 5

State whether the following statement is true or false. Give reasons for your answers.

Every integer is a whole number.

Ex. 1.10 | Q 5.6 | Page 5

State whether the following statement is true or false. Give reasons for your answers.

Every rational number is a whole number.

#### Chapter 1: Number Systems Exercise 1.20 solutions [Page 13]

Ex. 1.20 | Q 1.1 | Page 13

Express the following rational number as decimal:

42/100

Ex. 1.20 | Q 1.2 | Page 13

Express the following rational number as decimal:

327/500

Ex. 1.20 | Q 1.3 | Page 13

Express the following rational number as decimal:

15/4

Ex. 1.20 | Q 2.1 | Page 13

Express the following rational number as decimal:

2/3

Ex. 1.20 | Q 2.2 | Page 13

Express the following rational number as decimal:

-4/9

Ex. 1.20 | Q 2.3 | Page 13

Express the following rational number as decimal:

-2/15

Ex. 1.20 | Q 2.4 | Page 13

Express the following rational number as decimal:

-22/13

Ex. 1.20 | Q 2.5 | Page 13

Express the following rational number as decimal:

437/999

Ex. 1.20 | Q 2.6 | Page 13

Express the following rational number as decimal:

33/26

Ex. 1.20 | Q 3 | Page 13

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 than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?

#### Chapter 1: Number Systems Exercise 1.30 solutions [Page 22]

Ex. 1.30 | Q 1.1 | Page 22

Express the following decimal in the form p/q : 0.39

Ex. 1.30 | Q 1.2 | Page 22

Express the following decimal in the form p/q : 0.750

Ex. 1.30 | Q 1.3 | Page 22

Express the following decimal in the form p/q : 2.15

Ex. 1.30 | Q 1.4 | Page 22

Express the following decimal in the form p/q:

7.010

Ex. 1.30 | Q 1.5 | Page 22

Express the following decimal in the form p/q: 9.90

Ex. 1.30 | Q 1.6 | Page 22

Express the following decimal in the form p/q: 1.0001

Ex. 1.30 | Q 2.1 | Page 22

Express the following decimal in the form p/q: 0.bar4

Ex. 1.30 | Q 2.2 | Page 22

Express the following decimal in the form p/q: 0.bar37

Ex. 1.30 | Q 2.3 | Page 22

Express the following decimal in the form p/q: 0.bar54

Ex. 1.30 | Q 2.4 | Page 22

Express the following decimal in the form p/q: 0.bar621

Ex. 1.30 | Q 2.5 | Page 22

Express the following decimal in the form p/q: 125.bar3

Ex. 1.30 | Q 2.6 | Page 22

Express the following decimal in the form p/q: 4.bar7

Ex. 1.30 | Q 2.7 | Page 22

Express the following decimal in the form p/q: 0.4bar7

#### Chapter 1: Number Systems Exercise 1.40 solutions [Pages 31 - 32]

Ex. 1.40 | Q 1 | Page 31

Define an irrational number ?

Ex. 1.40 | Q 2 | Page 31

Explain, how irrational numbers differ from rational numbers ?

Ex. 1.40 | Q 3.01 | Page 31

Examine, whether the following number are rational or irrational:

sqrt7

Ex. 1.40 | Q 3.02 | Page 31

Examine, whether the following number are rational or irrational:

sqrt4

Ex. 1.40 | Q 3.03 | Page 31

Examine, whether the following number are rational or irrational:

2+sqrt3

Ex. 1.40 | Q 3.04 | Page 31

Examine, whether the following number are rational or irrational:

sqrt3+sqrt2

Ex. 1.40 | Q 3.05 | Page 31

Examine, whether the following number are rational or irrational:

sqrt3+sqrt5

Ex. 1.40 | Q 3.06 | Page 31

Examine, whether the following number are rational or irrational:

(sqrt2-2)^2

Ex. 1.40 | Q 3.07 | Page 31

Examine, whether the following number are rational or irrational:

(2-sqrt2)(2+sqrt2)

Ex. 1.40 | Q 3.08 | Page 31

Examine, whether the following number are rational or irrational:

(sqrt2+sqrt3)^2

Ex. 1.40 | Q 3.09 | Page 31

Examine, whether the following number are rational or irrational:

sqrt5-2

Ex. 1.40 | Q 3.1 | Page 31

Classify the following number as rational or irrational :-

sqrt23

Ex. 1.40 | Q 3.11 | Page 31

Classify the following number as rational or irrational :-

sqrt225

Ex. 1.40 | Q 3.12 | Page 31

Classify the following number as rational or irrational :-

0.3796

Ex. 1.40 | Q 3.13 | Page 31

Classify the following number as rational or irrational :-

7.478478...

Ex. 1.40 | Q 3.14 | Page 31

Classify the following number as rational or irrational :-

1.1010010001...

Ex. 1.40 | Q 4.1 | Page 31

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt4

Ex. 1.40 | Q 4.2 | Page 31

Identify the following as rational or irrational number. Give the decimal representation of rational number:

3sqrt18

Ex. 1.40 | Q 4.3 | Page 31

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt1.44

Ex. 1.40 | Q 4.4 | Page 31

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt(9/27)

Ex. 1.40 | Q 4.5 | Page 31

Identify the following as rational or irrational number. Give the decimal representation of rational number:

-sqrt64

Ex. 1.40 | Q 4.6 | Page 31

Identify the following as rational or irrational number. Give the decimal representation of rational number:

sqrt100

Ex. 1.40 | Q 5.1 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

x2 = 5

Ex. 1.40 | Q 5.2 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

y2 = 9

Ex. 1.40 | Q 5.3 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

z2 = 0.04

Ex. 1.40 | Q 5.4 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

u^2=17/4

Ex. 1.40 | Q 5.5 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

v2 = 3

Ex. 1.40 | Q 5.6 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

w2 = 27

Ex. 1.40 | Q 5.7 | Page 31

In the following equation, find which variables x, y, z etc. represent rational or irrational number:

t2 = 0.4

Ex. 1.40 | Q 6 | Page 31

Give two rational numbers lying between 0.232332333233332... and 0.212112111211112.

Ex. 1.40 | Q 7 | Page 31

Give two rational numbers lying between 0.515115111511115... and 0.535335333533335...

Ex. 1.40 | Q 8 | Page 32

Find one irrational number between 0.2101 and 0.222... = 0.bar2

Ex. 1.40 | Q 9 | Page 32

Find a rational number and also an irrational number lying between the numbers 0.3030030003... and 0.3010010001...

Ex. 1.40 | Q 10 | Page 32

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

Ex. 1.40 | Q 11.1 | Page 32

Give an example of two irrational numbers whose:

difference is a rational number.

Ex. 1.40 | Q 11.2 | Page 32

Give an example of two irrational numbers whose:

difference is an irrational number.

Ex. 1.40 | Q 11.3 | Page 32

Give an example of two irrational numbers whose:

sum is a rational number.

Ex. 1.40 | Q 11.4 | Page 32

Give an example of two irrational numbers whose:

sum is an irrational number.

Ex. 1.40 | Q 11.5 | Page 32

Give an example of two irrational numbers whose:

product is an rational number.

Ex. 1.40 | Q 11.6 | Page 32

Give an example of two irrational numbers whose:

product is an irrational number.

Ex. 1.40 | Q 11.7 | Page 32

Give an example of two irrational numbers whose:

quotient is a rational number.

Ex. 1.40 | Q 11.8 | Page 32

Give an example of two irrational numbers whose:

quotient is an irrational number.

Ex. 1.40 | Q 12 | Page 32

Find two irrational numbers between 0.5 and 0.55.

Ex. 1.40 | Q 13 | Page 32

Find two irrational numbers lying between 0.1 and 0.12.

Ex. 1.40 | Q 14 | Page 32

Prove that sqrt3+sqrt5 is an irrational number.

#### Chapter 1: Number Systems Exercise 1.50 solutions [Page 36]

Ex. 1.50 | Q 1.1 | Page 36

Complete the following sentence:

Every point on the number line corresponds to a _________ number which many be either _______ or ________.

Ex. 1.50 | Q 1.2 | Page 36

Complete the following sentence:

The decimal form of an irrational number is neither ________ nor _________

Ex. 1.50 | Q 1.3 | Page 36

Complete the following sentence:

The decimal representation of a rational number is either ______ or _________.

Ex. 1.50 | Q 1.4 | Page 36

Complete the following sentence:

Every real number is either ______ number or _______ number.

Ex. 1.50 | Q 2.1 | Page 36

Find whether the following statement is true or false.

Every real number is either rational or irrational.

Ex. 1.50 | Q 2.2 | Page 36

Find whether the following statement is true or false.

π is an irrational number.

Ex. 1.50 | Q 2.3 | Page 36

Find whether the following statement is true or false.

Irrational numbers cannot be represented by points on the number line.

Ex. 1.50 | Q 3 | Page 36

Represent sqrt6, sqrt7, sqrt8 on the number line.

Ex. 1.50 | Q 4 | Page 36

Represent sqrt3.5, sqrt9.4, sqrt10.5 on the real number line.

#### Chapter 1: Number Systems Exercise 1.60 solutions [Page 40]

Ex. 1.60 | Q 1 | Page 40

Visualise 2.665 on the number line, using successive magnification.

Ex. 1.60 | Q 2 | Page 40

Visualise the representation of 5.3bar7 on the number line upto 5 decimal places, that is upto 5.37777.

#### Chapter 1: Number Systems Exercise 1.60 solutions [Pages 40 - 42]

Ex. 1.60 | Q 1 | Page 40

Mark the correct alternative in the following:

Which one of the following is a correct statement?

• Decimal expansion of a rational number is terminating

• Decimal expansion of a rational number is non-terminating

• Decimal expansion of an irrational number is terminating

• Decimal expansion of an irrational number is non-terminating and non-repeating

Ex. 1.60 | Q 2 | Page 40

Which one of the following statements is true?

• The sum of two irrational numbers is always an irrational number

• The sum of two irrational numbers is always a rational number

• The sum of two irrational numbers may be a rational number or an irrational number

• The sum of two irrational numbers is always an integer

Ex. 1.60 | Q 3 | Page 40

Which of the following is a correct statement?

•  Sum of two irrational numbers is always irrational

• Sum of a rational and irrational number is always an irrational number

• Square of an irrational number is always a rational number

• Sum of two rational numbers can never be an integer

Ex. 1.60 | Q 4 | Page 40

Which of the following statements is true?

•  Product of two irrational numbers is always irrational

• Product of a rational and an irrational number is always irrational

• Sum of two irrational numbers can never be irrational

•  Sum of an integer and a rational number can never be an integer

Ex. 1.60 | Q 5 | Page 40

Which of the following is irrational?

• $\sqrt{\frac{4}{9}}$

• $\sqrt{\frac{4}{5}}$

• $\sqrt{7}$

• $\sqrt{81}$

Ex. 1.60 | Q 6 | Page 40

Which of the following is irrational?

• 0.14

• 0.14overline16

• 0.overline1416

• 0.1014001400014...

Ex. 1.60 | Q 7 | Page 40

Which of the following is rational?

• $\sqrt{3}$

• $\pi$

• $\frac{4}{0}$

• $\frac{0}{4}$

Ex. 1.60 | Q 8 | Page 40

The number 0.318564318564318564 ........ is:

•  a natural number

• an integer

• a rational number

• an irrational number 0.318564318564318564..... = 0overline318564 is repeating, so it is rational number because rational number is always either terminating or repeating.

Ex. 1.60 | Q 9 | Page 40

If n is a natural number, then  $\sqrt{n}$ is

• always a natural number

• always an irrational number

• always an irrational number

• sometimes a natural number and sometimes an irrational number

Q 10 | Page 41

Which of the following numbers can be represented as non-terminating, repeating decimals?

• $\frac{39}{24}$

• $\frac{3}{16}$

• $\frac{3}{11}$

• $\frac{137}{25}$

Q 11 | Page 41

Every point on a number line represents

•  a unique real number

•  a natural number

•  a rational number

• an irrational number

Q 12 | Page 41

Which of the following is irrational?

• 0.15

•  0.01516

• 0.overline1516

• 0.5015001500015.

Q 13 | Page 41

The number $1 . \bar{{27}}$ in the form $\frac{p}{q}$ , where p and q are integers and q ≠ 0, is

• $\frac{14}{9}$

• $\frac{14}{11}$

• $\frac{14}{13}$

• $\frac{14}{15}$

Q 14 | Page 41

The number $0 . \bar{3}$ in the form $\frac{p}{q}$,where p and q are integers and q ≠ 0, is

• $\frac{33}{100}$

• $\frac{3}{10}$

• $\frac{1}{3}$

• $\frac{3}{100}$

Q 15 | Page 41

$0 . 3 \bar{2}$ when expressed in the form $\frac{p}{q}$ (p, q are integers q ≠ 0), is

• $\frac{8}{25}$

• $\frac{29}{90}$

• $\frac{32}{99}$

• $\frac{32}{199}$

Q 16 | Page 41

$23 . \bar{{43}}$ when expressed in the form $\frac{p}{q}$ (p, q are integers q ≠ 0), is

• $\frac{2320}{99}$

• $\frac{2343}{100}$

• $\frac{2343}{999}$

• $\frac{2320}{199}$

Q 17 | Page 41

$0 . \bar{{001}}$ when expressed in the form $\frac{p}{q}$  (p, q are integers, q ≠ 0), is

• $\frac{1}{1000}$

• $\frac{1}{100}$

• $\frac{1}{1999}$

• $\frac{1}{999}$

Q 18 | Page 41

"The value of "0.overline23  0.overline22  "is"

• 0.overline45

• 0.overline43

• 0.overline45

• 0.45

Q 19 | Page 41

An irrational number between 2 and 2.5 is

• $\sqrt{11}$

• $\sqrt{5}$

• $\sqrt{22 . 5}$

• $\sqrt{12 . 5}$

Q 20 | Page 41

The number of consecutive zeros in $2^3 \times 3^4 \times 5^4 \times 7$ is

• 3

• 2

• 4

• 5

Q 21 | Page 42

The smallest rational number by which1/3should be multiplied so that its decimal expansion terminates after one place of decimal, is

• $\frac{1}{10}$

• $\frac{3}{10}$

• 3

• 30

#### Chapter 1: Number Systems solutions [Page 15]

Q 2 | Page 15

Simplify (3sqrt2 - 2sqrt3)/(3sqrt2 + 2sqrt3) + sqrt12/(sqrt3 - sqrt2)

## Chapter 1: Number Systems

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.50Ex. 1.60Others

## RD Sharma solutions for Class 9 Mathematics chapter 1 - Number Systems

RD Sharma solutions for Class 9 Maths chapter 1 (Number Systems) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 by R D Sharma (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 9 Mathematics chapter 1 Number Systems are Introduction of Real Number, 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.

Using RD Sharma Class 9 solutions Number Systems exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 9 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 1 Number Systems Class 9 extra questions for Maths and can use shaalaa.com to keep it handy for your exam preparation

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