## RD Sharma Class 9 Solutions Chapter 1 Number Systems

### RD Sharma Class 9 Solutions Chapter 1 Number System Ex 1.1

Question 1.
Is zero a rational number? Can you write it P in the form $$\frac { p }{ q }$$ , where p and q are integers and q â‰  0? [NCERT]
Solution:
Yes, zero is a rational number e.g.

Question 2.
Find five rational numbers between 1 and 2. [NCERT]
Solution:
We know that one rational number between two numbers a and b = $$\frac { a+b }{ 2 }$$
Therefore one rational number between 1 and 2

Question 3.
Find six rational numbers between 3 and 4. [NCERT]
Solution:
One rational number between 3 and 4

Question 4.
Find five rational numbers between $$\frac { 3 }{ 5 }$$ and $$\frac { 4 }{ 5 }$$
Solution:

Question 5.
Are the following statements true or false?
(i) Every whole number is a natural number. [NCERT]
(ii) Every integer is a rational number.
(iii) Every rational number is an integer.
(iv) Every natural number is a whole number,
(v) Every integer is a whole number.
(vi) Every rational number is a whole number.
Solution:
(i) False, as 0 is not a natural number.
(ii) True.
(iii) False, as $$\frac { 1 }{ 2 }$$, $$\frac { 1 }{ 3 }$$ etc. are not integers.
(iv) True.
(v) False, âˆµ negative natural numbers are not whole numbers.
(vi) False, âˆµ proper fraction are not whole numbers

### RD Sharma Class 9 Chapter 1 Number System Ex 1.2

Question 1.
Express the following rational numbers as decimals:

Solution:

Question 2.

Solution:

Question 3.
Look at several examples of rational numbers in the form $$\frac { p }{ q }$$ (q â‰  0), where pÂ and q are integers with no common factors other than 1 and having terminating decimal representations. Can you guess what property q must satisfy?
Solution:
The property is if the denominators have factors 2 or 5 or both, the decimal representation will be terminating e.g.

### RD Sharma Class 9 Solutions Chapter 1 Number System Ex 1.3

Question 1.
Express each of the following decimals in the form $$\frac { p }{ q }$$:
(i) 0.39
(ii) 0.750
(iii) 2.15
(iv) 7.010
(v) 9.90
(vi) 1.0001
Solution:

Question 2.
Express each of the following decimals in the form $$\frac { p }{ q }$$:

Solution:

### RD Sharma Solutions Class 9 Chapter 1 Number System Ex 1.4

Question 1.
Define an irrational number.
Solution:
A number which cannot be expressed in the form of $$\frac { p }{ q }$$ where p and q are integers and q â‰  0 is called an irrational number.

Question 2.
Explain, how irrational numbers differ from rational numbers?
Solution:
A rational number can be expressed in either terminating decimal or non-terminating recurring decimals but an irrational number is expressed in non-terminating non-recurring decimals.

Question 3.
Examine, whether the following numbers are rational or irrational:

Solution:

Question 4.
Identify the following as rational or irrational numbers. Give the decimal representation of rational numbers:

Solution:

Question 5.
InÂ  the following equation, find which variables x, y, z etc. represent rational or irrational numbers:

Solution:

Question 6.
Given two rational numbers lying between 0.232332333233332… and 0.212112111211112.
Solution:
Two rational numbers lying between 0.232332333233332… and 0.212112111211112… will be 0.232 and 0.212

Question 7.
Give two rational numbers lying between 0.515115111511115… and 0.5353353335…
Solution:
Two rational numbers lying between 0.515115111511115… and 0.535335333533335… will be 0.515, 0.535

Question 8.
Find one irrational numbers between 0.2101 and 0.2222… = 0.$$\overline { 2 }$$.
Solution:
One irrational number lying between 0.2101 and 0.2222… = 0.$$\overline { 2 }$$ will be 2201.0010001…

Question 9.
Find a rational number and also an irrational number lying between the numbers, 0.3030030003… and 0.3010010001…
Solution:
Between two numbers 0.3030030003… and 0.3010010001…, a rational will be 0.301 and irrational number will be 0.3020020002…

Question 10.
Find three different irrational numbers between the rational numbers $$\frac { 5 }{ 7 }$$ and $$\frac { 9 }{ 11 }$$. [NCERT]
Solution:

Question 11.
Give an example of each, of two irrational numbers whose:
(i) difference is a rational number.
(ii) difference is an irrational number.
(iii) sum is a rational number.
(iv) sum is an irrational number.
(v) product is a rational number.
(vi) product is an irrational number.
(vii) quotient is a rational number.
(viii) quotient is an irrational number.
Solution:
(i) Two numbers whose difference is also a rational number, e.g. $$\sqrt { 2 }$$
, $$\sqrt { 2 }$$Â  which are irrational numbers.
âˆ´ Difference = $$\sqrt { 2 }$$Â –Â $$\sqrt { 2 }$$ =Â 0 which is also a rational number.
(ii) Two numbers whose difference is an irrational number.
e.g. $$\sqrt { 3 }$$Â  and $$\sqrt { 2 }$$Â  which are irrational numbers.
Now difference = $$\sqrt { 3 }$$Â  –$$\sqrt { 2 }$$Â  which is also an irrational number.
(iii) Let two irrational numbers be $$\sqrt { 3 }$$Â  and –$$\sqrt { 2 }$$Â  which are irrational numbers.
Now sum = $$\sqrt { 3 }$$Â  + (-$$\sqrt { 3 }$$) = $$\sqrt { 3 }$$
– $$\sqrt { 3 }$$Â  = 0 Which is a rational number.
(iv) Let two numbers be $$\sqrt { 5 }$$Â  , $$\sqrt { 3 }$$ which are irrational numbers.
Now sum = $$\sqrt { 5 }$$ + $$\sqrt { 3 }$$ Â which is an irrational number.
(v) Let numbers be $$\sqrt { 3 }$$Â  +$$\sqrt { 2 }$$andÂ  $$\sqrt { 3 }$$Â  –$$\sqrt { 2 }$$which are irrational numbers.
Now product = ($$\sqrt { 3}$$Â  +$$\sqrt { 2 }$$ ) ($$\sqrt { 3 }$$ –$$\sqrt { 2 }$$)
= 3-2 = 1 which is a rational number.
(vi) Let numbers be $$\sqrt { 3 }$$ and $$\sqrt { 5 }$$ , which are irrational number.
Now product = $$\sqrt { 3 }$$ xÂ $$\sqrt { 5 }$$ Â =Â $$\sqrt { 3×5 }$$
= $$\sqrt { 15 }$$
which is an irrational number.
(vii) Let numbers be 6 $$\sqrt { 2 }$$Â  and 2 $$\sqrt { 2 }$$ which are irrational numbers.
Quotient =$$\frac { 6\sqrt { 2 } }{ 2\sqrt { 2 } }$$ = 3 which is a rational number.
(viii)Â Let numbers be $$\sqrt { 3 }$$andÂ $$\sqrt { 5 }$$ which are irrational numbers.
Now quotient =$$\frac { \sqrt { 3 } }{ \sqrt { 5 } }$$ =Â $$\sqrt { \frac { 3 }{ 5 } }$$ which is anÂ  irrational number.

Question 12.
Find two irrational numbers between 0.5 and 0.55.
Solution:
Two irrational numbers between 0.5 and 0.55 will be 0.51010010001… and 52020020002…

Question 13.
Find two irrational numbers lying betwee 0.1 and 0.12.
Solution:
Two irrational numbers lying between 0.1 and 0.12 will be 0.1010010001… and 0.1020020002…

Question 14.
Prove that $$\sqrt { 3 }$$+$$\sqrt { 5 }$$ is an irrational number.
Solution:

### RD Sharma Class 9 PDF Chapter 1 Number System Ex 1.5

Question 1.
Complete the following sentences:
(i) Every point on the number line corresponds to a … number which many be either … or
(ii) The decimal form of an irrational number is neither … nor …
(iii) The decimal representation of a rational number is either … or …
(iv) Every real number is either … number or … number.
Solution:
(i) Every point on the number line corresponds to a real number which many be either rational or irrational.
(ii) The decimal form of an irrational number is neither terminating nor repeating.
(iii) The decimal representation of a rational number is either terminating or nonÂ­terminating, recurring.
(iv) Every real number is either rational number or an irrational number.

Question 2.
Find whether the following statements are true or false:
(i)Â  Every real number is either rational or irrational.
(ii) Ï€ is an irrational number.
(iii) Irrational numbers cannot be represented by points on the number line.
Solution:
(i) True. (Value of Ï€ = 3.14)
(ii) False : we can represent irrational number also.

Question 3.
Represent $$\sqrt { 6 }$$, $$\sqrt { 7 }$$, $$\sqrt { 8 }$$ on the number line.
Solution:

Question 4.
Represent $$\sqrt { 3.5 }$$ , $$\sqrt { 9.4 }$$and $$\sqrt { 10.5 }$$ on the real number line.
Solution:

#### Number System Class 9 RD Sharma Class 9 Solutions Ex 1.6

Question 1.
Visualise 2.665 on the number line, using successive magnification.
Solution:
2.665
âˆµÂ It lies between 2 and 3

Question 2.
Visualise the representation of 5.3$$\overline { 7 }$$ on theÂ number line upto 5 decimal places, that is upto 5.37777.Â Â Â Â Â Â Â Â  [NCERT]
Solution:
5.37777.

### RD Sharma Class 9 Solution Chapter 1 Number System MCQS

Question 1.
Visualise 2.665 on the number line, using successive magnification.
Solution:
2.665
âˆµÂ It lies between 2 and 3

Question 2.
Visualise the representation of 5.3$$\overline { 7 }$$ on theÂ number line upto 5 decimal places, that is upto 5.37777.Â Â Â Â Â Â Â Â  [NCERT]
Solution:
5.37777.

Class 9 RD Sharma Solutions Chapter 1 Number System Ex 1.6

the correct alternative in each of the following:
Question 1.
Which one of the following is a correct statement?
(a) Decimal expansion of a rational number is terminating
(b) Decimal expansion of a rational number is non-terminating
(c) Decimal expansion of an irrational number is terminating
(d) Decimal expansion of an irrational number is non-terminating and non-repeating
Solution:
Decimal expansion of an irrational number is non-terminating and non-repeating . (d)

Question 2.
Which one of the following statements is true?
(a) The sum of two irrational numbers is always an irrational-number
(b) The sum of two irrational numbers is always a rational number
(c) The sum of two irrational numbers may be a rational number or an irrational number
(d) The sum of two irrational numbers is always an integer
Solution:
The sum of two irrational numbers may be a rational number or an irrational number (c)

Question 3.
Which of the following is a correct statement?
(a) Sum of two irrational numbers is always irrational
(b) Sum of a rational and irrational number is always an irrational number
(c) Square of an irrational number is always a rational number
(d) Sum of two rational numbers can never be an integer
Solution:
Sum of a rational and irrational number is always an irrational numberÂ Â Â Â Â Â Â Â  (b)

Question 4.
Which of the following statements is true?
(a) Product of two irrational numbers is always irrational
(b) Product of a rational and an irrational number is always irrational
(c) Sum of two irrational numbers can never be irrational
(d) Sum of an integer and a rational number can never be an integer
Solution:
Product of a rational and an irrational number is always irrationalÂ Â Â  (b)

Question 5.
Which of the following is irrational?

Solution:

Question 6.
Which of the following is irrational?
(a) 14
(b)Â  0.14$$\overline { 16 }$$
(c)Â  Â 0.$$\overline { 1416 }$$
(d)Â  0.1014001400014
Solution:
0.1014001400014…….. is irrational as it is non-terminating nor repeating decimal, (d)

Question 7.
Which of the following is rational?

Solution:

Question 8.
The number 0.318564318564318564… is:
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number
Solution:
The number = 0.318564318564318564…………
= 0.$$\overline { 318564 }$$
âˆµ The decimal is non-terminating and recurring
âˆ´ It is rational number.Â Â  (c)

Question 9.
If n is a natural number, then $$\sqrt { n }$$is
(a) always a natural number
(b) always a rational number
(c) always an irrational number
(d) sometimes a natural number and sometimes an irrational number
Solution:
If n is a natural number then $$\sqrt { n }$$ may sometimes a natural number and sometime an irrational number e.g.
If n = 2 then $$\sqrt { n }$$ =$$\sqrt { 2 }$$ which is are irrational and if n = 4, then $$\sqrt { n }$$= $$\sqrt { 4 }$$ =Â  2 which is a rational number.Â Â Â Â Â Â  (d)

Question 10.
Which of the following numbers can be represented as non-terminating, repeating decimals?

Solution:

Question 11.
Every point on a number line represents
(a) a unique real number
(b) a natural number
(c) a rational number
(d) an irrational number
Solution:
Every point on a number line represents a unique real number.Â Â Â Â Â Â Â Â  (a)

Question 12.
Which of the following is irrational?
(a) 0.15
(b) 0.01516
(c) 0.$$\overline { 1516 }$$
(d) 0.5015001500015..
Solution:
As it is non-terminating non-repeating decimals while others are terminating or non-terminating repeating decimals. (d)

Question 13.
The number 1.$$\overline { 27 }$$ in the formÂ $$\frac { p }{ q }$$Â  , where pÂ and q are integers and q â‰  0, is

Solution:

Question 14.

Solution:

Question 15.

Solution:

Question 16.

Solution:

Question 17.

Solution:

Question 18.

Solution:

Question 19.

Solution:
An irrational number between 2 and 2.5 is $$\sqrt { 5 }$$ as it has approximate value 2.236… (b)

Question 20.
The number of consecutive zeros in 23 x 34 x 54 x 7, is
(a) 3
(b) 2
(c) 4
(d) 5
Solution:
In 23 x 34 x 54 x 7, number of consecutive zero will be 3 as 23 x 54 = 2 x 2 x 2 x 5x 5 x 5 x 5 = 5000Â Â Â Â Â  (a)

Question 21.
The smallest rational number by which $$\frac { 1 }{ 3 }$$ should be multiplied so that its decimal expansion terminates after one place of decimal, is

Solution:

RD Sharma Class 9 Solutions Chapter 1 Number Systems Exercise 1.4

RD Sharma Class 9 Solutions Chapter 1 Number Systems Exercise 1.6