CBSE Class 9

RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems 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 non­terminating, 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)  Every real number is either rational or irrational.
(ii) π is an irrational number.
(iii) Irrational numbers cannot be represented by points on the number line.
Solution:
(i) True. (Value of π = 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:

Hope given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.5 are helpful to complete your math homework.

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems 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 ≠ 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.
In  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. \(\sqrt { 2 } \)
, \(\sqrt { 2 } \)  which are irrational numbers.
∴ Difference = \(\sqrt { 2 } \) – \(\sqrt { 2 } \) = 0 which is also a rational number.
(ii) Two numbers whose difference is an irrational number.
e.g. \(\sqrt { 3 } \)  and \(\sqrt { 2 } \)  which are irrational numbers.
Now difference = \(\sqrt { 3 } \)  –\(\sqrt { 2 } \)  which is also an irrational number.
(iii) Let two irrational numbers be \(\sqrt { 3 } \)  and –\(\sqrt { 2 } \)  which are irrational numbers.
Now sum = \(\sqrt { 3 } \)  + (-\(\sqrt { 3 } \)) = \(\sqrt { 3 } \)
– \(\sqrt { 3 } \)  = 0 Which is a rational number.
(iv) Let two numbers be \(\sqrt { 5 } \)  , \(\sqrt { 3 } \) which are irrational numbers.
Now sum = \(\sqrt { 5 } \) + \(\sqrt { 3 } \)  which is an irrational number.
(v) Let numbers be \(\sqrt { 3 } \)  +\(\sqrt { 2 } \)and  \(\sqrt { 3 } \)  –\(\sqrt { 2 } \)which are irrational numbers.
Now product = (\(\sqrt { 3} \)  +\(\sqrt { 2 } \) ) (\(\sqrt { 3 } \) –\(\sqrt { 2 } \))
= 3-2 = 1 which is a rational number.
(vi) Let numbers be \(\sqrt { 3 } \) and \(\sqrt { 5 } \) , which are irrational number.
Now product = \(\sqrt { 3 } \) x \(\sqrt { 5 } \)  = \(\sqrt { 3×5 } \)
⁼ \(\sqrt { 15 } \)
which is an irrational number.
(vii) Let numbers be 6 \(\sqrt { 2 } \)  and 2 \(\sqrt { 2 } \) which are irrational numbers.
Quotient =\(\frac { 6\sqrt { 2 } }{ 2\sqrt { 2 } }\) = 3 which is a rational number.
(viii) Let numbers be \(\sqrt { 3 } \)and \(\sqrt { 5 } \) which are irrational numbers.
Now quotient =\(\frac { \sqrt { 3 } }{ \sqrt { 5 } }\) = \(\sqrt { \frac { 3 }{ 5 } }\) which is an  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:

Hope given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.4 are helpful to complete your math homework.

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems 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:

Hope given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.3 are helpful to complete your math homework.

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RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.1

These Solutions are part of RD Sharma Class 9 Solutions. Here we have given RD Sharma Class 9 Solutions Chapter 1 Number Systems 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 ≠ 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?
Give reason for your answer.
(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, ∵ negative natural numbers are not whole numbers.
(vi) False, ∵ proper fraction are not whole numbers

 

Hope given RD Sharma Class 9 Solutions Chapter 1 Number Systems Ex 1.1 are helpful to complete your math homework.

If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you.