Q1. 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.
Q2. 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 0. \(\overline{4}\) both are rational numbers
Q3. Find, whether the following numbers are rational and irrational
(i) \(\sqrt{7}\)
(ii) \(\sqrt{4}\)
(iii) \(2 + \sqrt{3}\)
(iv) \(\sqrt{3} + \sqrt{2}\)
(v) \(\sqrt{3} + \sqrt{5}\)
(vi) \((\sqrt{2} – 2)^{2}\)
(vii) \((2 – \sqrt{2})(2 + \sqrt{2})\)
(viii) \((\sqrt{2}+\sqrt{3})^{2}\)
(ix) \(\sqrt{5}\) – 2
(x) \(\sqrt{23}\)
(xi) \(\sqrt{225}\)
(xii) 0.3796
(xiii) 7.478478……
(xiv) 1.101001000100001……
Solution:
(i)\(\sqrt{7}\) is not a perfect square root so it is an Irrational
number.
(ii) \(\sqrt{4}\) is a perfect square root so it is an rational number.
We have,
\(\sqrt{4}\) can be expressed in the form of
\(\frac{a}{b}\), so it is a rational number. The decimal
representation of \(\sqrt{9}\) is 3.0. 3 is a rational number.
(iii) \(2 + \sqrt{3}\)
Here, 2 is a rational number and \(\sqrt{3}\) is an irrational number
So, the sum of a rational and an irrational number is an irrational number.
(iv) \(\sqrt{3} + \sqrt{2}\)
\(\sqrt{3}\) is not a perfect square and it is an irrational number and \(\sqrt{2}\) 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 \(\sqrt{3} + \sqrt{2}\) is an irrational number.
(v) \(\sqrt{3} + \sqrt{5}\)
\(\sqrt{3}\) is not a perfect square and it is an irrational number and \(\sqrt{5}\) 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 \(\sqrt{3} + \sqrt{5}\) is an irrational number.
(vi) \((\sqrt{2} – 2)^{2}\)
We have, \((\sqrt{2} – 2)^{2}\)
= 2 + 4 – 4 \(\sqrt{2}\)
= 6 + 4 \(\sqrt{2}\)
6 is a rational number but 4\(\sqrt{2}\) is an irrational number.
The sum of a rational number and an irrational number is an irrational number, so \((\sqrt{2} + \sqrt{4})^{2}\) is an irrational number.
(vii) \((2 – \sqrt{2})(2 + \sqrt{2})\)
We have,
\((2 – \sqrt{2})(2 + \sqrt{2}) = (2)^{2}-(\sqrt{2})^{2}\) [Since, (a + b)(a – b) = a2 – b2]
4 – 2 = \(\frac{2}{1}\)
Since, 2 is a rational number.
\((2 – \sqrt{2})(2 + \sqrt{2})\) is a rational number.
(viii) \((\sqrt{2}+\sqrt{3})^{2}\)
We have,
\((\sqrt{2}+\sqrt{3})^{2}\) = 2 + 2 \(\sqrt{6}\) + 3 = 5 + \(\sqrt{6}\) [Since, (a+b)2 = a2 + 2ab + b2
The sum of a rational number and an irrational number is an irrational number, so \((\sqrt{2}+\sqrt{3})^{2}\) is an irrational number.
(ix) \(\sqrt{5}\) – 2
The difference of an irrational number and a rational number is an irrational number.
(\(\sqrt{5}\) – 2 ) is an irrational number.
(x) \(\sqrt{23}\)
\(\sqrt{23}\) = 4.795831352331…
As decimal expansion of this number is non-terminating, non-recurring so it is an irrational number.
(xi) \(\sqrt{225}\)
\(\sqrt{225}\) = 15 = \(\frac{15}{1}\)
\(\sqrt{225}\) is rational number as it can be represented in \(\frac{p}{q}\) 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
Q4. Identify the following as irrational numbers. Give the decimal representation of rational numbers:
(i) \(\sqrt{4}\)
(ii) 3 \(\times \sqrt{18}\)
(iii) \(sqrt{1.44}\)
(iv) \(\sqrt{{\frac{9}{27}}}\)
(v) – \(\sqrt{64}\)
(vi) \(\sqrt{100}\)
Solution:
(i) We have,
\(\sqrt{4}\) can be written in the form of
\(\frac{p}{q}\). So, it is a rational number. Its decimal
representation is 2.0
(ii). We have,
3 \(\times \sqrt{18}\)
= 3 \(\times \sqrt{2 x 3 x 3}\)
= 9 \(\times \sqrt{2}\)
Since, the product of a ratios and an irrational is an irrational number.
9 \(\times \sqrt{2}\)is an irrational.
3\(\times \sqrt{18}\) is an irrational number.
(iii) We have,
\(sqrt{1.44}\)
= \(\sqrt{\frac{144}{100}}\)
= \(\frac{12}{10}\)
= 1.2
Every terminating decimal is a rational number, so 1.2 is a rational number.
Its decimal representation is 1.2.
(iv) \(\sqrt{{\frac{9}{27}}}\)
We have,
\(\sqrt{{\frac{9}{27}}}\)
= \(\frac{3}{\sqrt{27}}\)
= \(\frac{1}{\sqrt{3}}\)
Quotient of a rational and an irrational number is irrational numbers so
\(\frac{1}{\sqrt{3}}\) is an irrational number.
\(\sqrt{{\frac{9}{27}}}\) is an irrational number.
(v) We have,
– \(\sqrt{64}\)
= – 8
= – \(\frac{8}{1}\)
= – \(\frac{8}{1}\) can be expressed in the form of \(\frac{a}{b}\),
so – \(\sqrt{64}\) is a rational number.
Its decimal representation is – 8.0.
(vi) We have,
\(\sqrt{100}\)
= 10 can be expressed in the form of \(\frac{a}{b}\),
so \(\sqrt{100}\) is a rational number
Its decimal representation is 10.0.
Q5. In the following equations, find which variables x, y and z etc. represent rational or irrational numbers:
(i) \(x^{2}\) = 5
(ii) \(y^{2}\) = 9
(iii) \(z^{2}\) = 0.04
(iv) \(u^{2}\) = \(\frac{17}{4}\)
(v) \(v^{2}\) = 3
(vi) \(w^{2}\) = 27
(vii) \(t^{2}\) = 0.4
Solution:
(i) We have,
\(x^{2}\) = 5
Taking square root on both the sides, we get
x = \(\sqrt{5}\)
\(\sqrt{5}\) is not a perfect square root, so it is an irrational number.
(ii) We have,
=\(y^{2}\) = 9
= 3
= \(\frac{3}{1}\) can be expressed in the form of \(\frac{a}{b}\), so it a rational number.
(iii) We have,
\(z^{2}\) = 0.04
Taking square root on the both sides, we get
z = 0.2
\(\frac{2}{10}\) can be expressed in the form of \(\frac{a}{b}\), so it is a rational number.
(iv) We have,
\(u^{2}\) = \(\frac{17}{4}\)
Taking square root on both sides, we get,
u = \(\sqrt{\frac{17}{4}}\)
u = \(\frac{\sqrt{17}}{2}\)
Quotient of an irrational and a rational number is irrational, so u is an Irrational number.
(v) We have,
\(v^{2}\) = 3
Taking square root on both sides, we get,
v = \(\sqrt{3}\)
\(\sqrt{3}\) is not a perfect square root, so v is irrational number.
(vi) We have,
\(w^{2}\) = 27
Taking square root on both the sides, we get,
w = \(3\sqrt{3}\)
Product of a irrational and an irrational is an irrational number. So w is an irrational number.
(vii) We have,
\(t^{2}\) = 0. 4
Taking square root on both sides, we get,
t = \(\sqrt{\frac{4}{10}}\)
t = \(\frac{2}{\sqrt{10}}\)
Since, quotient of a rational and an Irrational number is irrational
number. \(t^{2}\) = 0.4 is an irrational number.
Q6. 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) \(\sqrt{2}\) is an irrational number.
Now, \(\sqrt{2}\) – \(\sqrt{2}\) = 0.
0 is the rational number.
(ii) Let two irrational numbers are \(3\sqrt{2}\) and \(\sqrt{2}\).
\(3\sqrt{2}\) – \(\sqrt{2}\) = \(2\sqrt{2}\)
\(5\sqrt{6}\) is the rational number.
(iii) \(\sqrt{11}\) is an irrational number.
Now, \(\sqrt{11}\) + (- \(\sqrt{11}\)) = 0.
0 is the rational number.
(iv) Let two irrational numbers are \(4\sqrt{6}\) and \(\sqrt{6}\)
\(4\sqrt{6}\) + \(\sqrt{6}\)
\(5\sqrt{6}\) is the rational number.
(iv) Let two Irrational numbers are \(7\sqrt{5}\) and
\(\sqrt{5}\)
Now, \(7\sqrt{5}\) \(\times \sqrt{5}\)
= 7 \(\times\) 5
= 35 is the rational number.
(v) Let two irrational numbers are \(\sqrt{8}\) and \(\sqrt{8}\).
Now, \(\sqrt{8}\) \(\times \sqrt{8}\)
8 is the rational number.
(vi) Let two irrational numbers are \(4\sqrt{6}\) and \(\sqrt{6}\)
Now, \(\frac{4\sqrt{6}}{\sqrt{6}}\)
= 4 is the rational number
(vii) Let two irrational numbers are \(3\sqrt{7}\) and \(\sqrt{7}\)
Now, 3 is the rational number.
(viii) Let two irrational numbers are \(\sqrt{8}\) and \(\sqrt{2}\)
Now \({\sqrt{2}}\) is an rational number.
Q7. 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.
Q8. 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.
Q9. Find one irrational number between 0.2101 and 0.2222 … = 0. \(\overline{2}\)
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.
Q10. 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.
Q11. 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.
Q12. 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.
Q13. Prove that \(\sqrt{3} + \sqrt{5}\) is an irrational number.
If possible, let \(\sqrt{3} + \sqrt{5}\) be a rational number equal to x.
Then,
x= \(\sqrt{3} + \sqrt{5}\)
\(x^{2} = (\sqrt{3} + \sqrt{5})^{2}\)
\(x^{2}\) = 8 + \(2\sqrt{15}\)
\(\frac{x^{2}-8}{2}\) = \(\sqrt{15}\)
Now, \(\sqrt{\frac{x^{2}-8}{2}}\) is rational
\(\sqrt{15}\) is rational
Thus, we arrive at a contradiction.
Hence, \(\sqrt{3} + \sqrt{5}\) is an irrational number.