Chapter 1: Number System Exercise – 1.4
Question: 1
Define an irrational number.
Solution:
An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers. It cannot be expressed as terminating or repeating decimal.
Question: 2
Explain how an irrational number is differing from rational numbers?
Solution:
An irrational number is a real number which can be written as a decimal but not as a fraction i.e. it cannot be expressed as a ratio of integers. It cannot be expressed as terminating or repeating decimal.
For example, 0.10110100 is an irrational number
A rational number is a real number which can be written as a fraction and as a decimal i.e. it can be expressed as a ratio of integers. . It can be expressed as terminating or repeating decimal.
For examples,
0.10 and
both are rational numbers
Question: 3
Find, whether the following numbers are rational and irrational
(i) √7
(ii) √4
(iii) 2 + √3
(iv) √3 + √2
(v) √3 + √5
(vi) (√2 - 2)2
(vii) (2 - √2) (2 + √2)
(viii) (√2 + √3)2
(ix) √5 - 2
(x) √23
(xi) √225
(xii) 0.3796
(xiii) 7.478478…
(xiv) 1.101001000100001…..
Solution:
(i) √7 is not a perfect square root so it is an Irrational number.
(ii) √4 is a perfect square root so it is an rational number.
We have,
√4 can be expressed in the form of
a/b, so it is a rational number. The decimal representation of √9 is 3.0. 3 is a rational number.
(iii) 2 + √3
Here, 2 is a rational number and √3 is an irrational number
So, the sum of a rational and an irrational number is an irrational number.
(iv) √3 + √2
√3 is not a perfect square and it is an irrational number and √2 is not a perfect square and is an irrational number. The sum of an irrational number and an irrational number is an irrational number, so √3 + √2 is an irrational number.
(v) √3 + √5
√3 is not a perfect square and it is an irrational number and √5 is not a perfect square and is an irrational number. The sum of an irrational number and an irrational number is an irrational number, so √3 + √5 is an irrational number.
(vi) (√2 - 2)2
We have, (√2 - 2)2
= 2 + 4 - 4√2
= 6 + 4√2
6 is a rational number but 4√2 is an irrational number.
The sum of a rational number and an irrational number is an irrational number, so (√2 + √4)2 is an irrational number.
(vii) (2 -√2) (2 + √2)
We have,
(2 - √2) (2 + √2) = (2)2 - (√2)2 [Since, (a + b)(a - b) = a2 - b2]
4 - 2 = 2/1
Since, 2 is a rational number.
(2 - √2)(2 + √2) is a rational number.
(viii) (√2 +√3)2
We have,
(√2 + √3)2 = 2 + 2√6 + 3 = 5+√6 [Since, (a + b)2 = a2 + 2ab + b2
The sum of a rational number and an irrational number is an irrational number, so (√2 + √3)2 is an irrational number.
(ix) √5 - 2
The difference of an irrational number and a rational number is an irrational number. (√5 - 2) is an irrational number.
(x) √23
√23 = 4.795831352331….
As decimal expansion of this number is non-terminating, non-recurring so it is an irrational number.
(xi) √225
√225 = 15 = 15/1
√225 is rational number as it can be represented in p/q form.
(xii) 0.3796
0.3796, as decimal expansion of this number is terminating, so it is a rational number.
(xiii) 7.478478……
7.478478 = 7.478, as decimal expansion of this number is non-terminating recurring so it is a rational number.
(xiv) 1.101001000100001……
1.101001000100001……, as decimal expansion of this number is non-terminating, non-recurring so it is an irrational number
Question: 4
Identify the following as irrational numbers. Give the decimal representation of rational numbers:
(i) √4
(ii) 3 × √18
(iii) √1.44
(iv) √(9/27)
(v) - √64
(vi) √100
Solution:
(i) We have,
√4 can be written in the form of
p/q. So, it is a rational number. Its decimal representation is 2.0
(ii). We have,
3 × √18
= 3 × √2 × 3 × 3
= 9×√2
Since, the product of a ratios and an irrational is an irrational number. 9 ×√2 is an irrational.
3 ×√18 is an irrational number.
(iii) We have,
√1.44
= √(144/100)
= 12/10
= 1.2
Every terminating decimal is a rational number, so 1.2 is a rational number.
Its decimal representation is 1.2.
(iv) √(9/27)
We have,
√(9/27)
=3/√27
= 1/√3
Quotient of a rational and an irrational number is irrational numbers so
1/√3 is an irrational number.
√(9/27) is an irrational number.
(v) We have,
-√64
= - 8
= - (8/1)
= - (8/1) can be expressed in the form of a/b,
so - √64 is a rational number.
Its decimal representation is - 8.0.
(vi) We have,
√100
= 10 can be expressed in the form of a/b,
So √100 is a rational number
Its decimal representation is 10.0.
Question: 5
In the following equations, find which variables x, y and z etc. represent rational or irrational numbers:
(i) x2 = 5
(ii) y2 = 9
(iii) z2 = 0.04
(iv) u2 = 174
(v) v2 = 3
(vi) w2 = 27
(vii) t2 = 0.4
Solution:
(i) We have,
x2 = 5
Taking square root on both the sides, we get
X = √5
√5 is not a perfect square root, so it is an irrational number.
(ii) We have,
= y2 = 9
= 3
= 3/1 can be expressed in the form of a/b, so it a rational number.
(iii) We have,
z2 = 0.04
Taking square root on the both sides, we get
z = 0.2
2/10 can be expressed in the form of a/b, so it is a rational number.
(iv) We have,
u2 = 17/4
Taking square root on both sides, we get,
u = √(17/4)
u = √17/2
Quotient of an irrational and a rational number is irrational, so u is an Irrational number.
(v) We have,
v2 = 3
Taking square root on both sides, we get,
v = √3
√3 is not a perfect square root, so v is irrational number.
(vi) We have,
w2 = 27
Taking square root on both the sides, we get,
w = 3√3
Product of a irrational and an irrational is an irrational number. So w is an irrational number.
(vii) We have,
t2 = 0.4
Taking square root on both sides, we get,
t = √(4/10)
t = 2/√10
Since, quotient of a rational and an Irrational number is irrational number. t2 = 0.4 is an irrational number.
Question: 6
Give an example of each, of two irrational numbers whose:
(i) Difference in a rational number.
(ii) Difference in an irrational number.
(iii) Sum in a rational number.
(iv) Sum is an irrational number.
(v) Product in a rational number.
(vi) Product in an irrational number.
(vii) Quotient in a rational number.
(viii) Quotient in an irrational number.
Solution:
(i) √2 is an irrational number.
Now, √2 -√2 = 0.
0 is the rational number.
(ii) Let two irrational numbers are 3√2 and √2.
3√2 - √2 = 2√2
5√6 is the rational number.
(iii) √11 is an irrational number.
Now, √11 + (-√11) = 0.
0 is the rational number.
(iv) Let two irrational numbers are 4√6 and √6
4√6 + √6
5√6 is the rational number.
(iv) Let two Irrational numbers are 7√5 and √5
Now, 7√5 × √5
= 7 × 5
= 35 is the rational number.
(v) Let two irrational numbers are √8 and √8.
Now, √8 × √8
8 is the rational number.
(vi) Let two irrational numbers are 4√6 and √6
Now, (4√6)/√6
= 4 is the rational number
(vii) Let two irrational numbers are 3√7 and √7
Now, 3 is the rational number.
(viii) Let two irrational numbers are √8 and √2
Now √2 is an rational number.
Question: 7
Give two rational numbers lying between 0.232332333233332 and 0.212112111211112.
Solution:
Let a = 0.212112111211112
And, b = 0.232332333233332...
Clearly, a < b because in the second decimal place a has digit 1 and b has digit 3 If we consider rational numbers in which the second decimal place has the digit 2, then they will lie between a and b.
Let. x = 0.22
y = 0.22112211... Then a < x < y < b
Hence, x, and y are required rational numbers.
Question: 8
Give two rational numbers lying between 0.515115111511115 and 0. 5353353335
Solution:
Let, a = 0.515115111511115...
And, b = 0.5353353335..
We observe that in the second decimal place a has digit 1 and b has digit 3, therefore, a < b.
So If we consider rational numbers
x = 0.52
y = 0.52062062...
We find that,
a < x < y < b
Hence x and y are required rational numbers.
Question: 9
Find one irrational number between 0.2101 and 0.2222 ... =
Solution:
Let, a = 0.2101 and,
b = 0.2222...
We observe that in the second decimal place a has digit 1 and b has digit 2, therefore a < b in the third decimal place a has digit 0.
So, if we consider irrational numbers
x = 0.211011001100011....
We find that a < x < b
Hence x is required irrational number.
Question: 10
Find a rational number and also an irrational number lying between the numbers 0.3030030003... and 0.3010010001...
Solution:
Let,
a = 0.3010010001 and,
b = 0.3030030003...
We observe that in the third decimal place a has digit 1 and b has digit
3, therefore a < b in the third decimal place a has digit 1. So, if we
consider rational and irrational numbers
x = 0.302
y = 0.302002000200002.....
We find that a < x < b and, a < y < b.
Hence, x and y are required rational and irrational numbers respectively.
Question: 11
Find two irrational numbers between 0.5 and 0.55.
Solution:
Let a = 0.5 = 0.50 and b = 0.55
We observe that in the second decimal place a has digit 0 and b has digit
5, therefore a < 0 so, if we consider irrational numbers
x = 0.51051005100051...
y = 0.530535305353530...
We find that a < x < y < b
Hence x and y are required irrational numbers.
Question: 12
Find two irrational numbers lying between 0.1 and 0.12.
Solution:
Let a = 0.1 = 0.10
And b = 0.12
We observe that In the second decimal place a has digit 0 and b has digit 2.
Therefore, a < b.
So, if we consider irrational numbers
x = 0.1101101100011... y = 0.111011110111110... We find that a < x < y < 0
Hence, x and y are required irrational numbers.
Question: 13
Prove that √3 + √5 is an irrational number.
Solution:
If possible, let √3 + √5 be a rational number equal to x.
Then,
Thus, we arrive at a contradiction.
Hence, √3 + √5 is an irrational number.