#### (i)Insert a rational number and an irrational number between the following. 2 and 3 (ii)Insert a rational number and an irrational number between the following.0.1 and 0.1 (iii)Insert a rational number and an irrational number between the following.$\frac{1}{3}\, and\, \frac{1}{2}$ (iv)Insert a rational number and an irrational number between the following.$\frac{-2}{5}\, and\, \frac{1}{2}$ (v)Insert a rational number and an irrational number between the following.0.15 and 0.16 (vi)Insert a rational number and an irrational number between the following.$\sqrt{2}$ and $\sqrt{3}$ (vii)Insert a rational number and an irrational number between the following.2.357 and 3.121 (viii)Insert a rational number and an irrational number between the following..0001 and .001 (ix)Insert a rational number and an irrational number between the following.3.623623 and 0.484848 (x) Insert a rational number and an irrational number between the following.6.375289 and 6.375738

(i) Answer.  Rational number: $\frac{5}{2}$

Irrational number: 2.040040004 ……….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.

Between 2 and 3
Rational number: 2.5 = $\frac{25}{10}$$= \frac{5}{2}$

and irrational number : 2.040040004

(ii) Answer. Rational number: $\frac{19}{1000}$

Irrational number 0.0105000500005 ……..
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0 and 0.1:
0.1 can be written as 0.10
Rational number: 0.019 = $\frac{19}{1000}$
and irrational number 0.0105000500005

(iii)Answer. Rational number $\frac{21}{60}$
Irrational number : 0.414114111 ……
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions. They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
LCM of 3 and 2 is 6.
We can write $\frac{1}{3}$ as $\frac{1\times 20}{3\times 20}= \frac{20}{60}$

and $\frac{1}{2}$ as $\frac{1\times 30}{3\times 30}= \frac{30}{60}$
Also, $\frac{1}{3}$ = 0.333333….
And  $\frac{1}{2}= 0\cdot 5$
So, rational number between $\frac{1}{3}$  and $\frac{1}{2}$ is $\frac{21}{60}$
and irrational number : 0.414114111 ……

Irrational number: 0.151551555 …….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.

$\frac{-2}{5}= -0\cdot 4$ and $\frac{1}{2}= -0\cdot 5$
Rational number between -0.4 and 0.5 is 0
And irrational number: 0.151551555 …….

(v) Answer. Rational number: $\frac{151}{1000}$
Irrational number: 0.151551555 ……….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0.15 and 0.16
Rational number : 0.151 = $\frac{151}{1000}$
and irrational number 0.151551555

(vi) Answer.  Rational number: $\frac{3}{2}$
Irrational number: 1.585585558 ………
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions. They are non-terminating non-recurring in nature.
Between $\sqrt{2}\, and\, \sqrt{3}$

$\sqrt{2}= 1\cdot 414213562373$
$\sqrt{3}= 1\cdot 732050807568$
Rational number: 1.5 = $\frac{3}{2}$
and irrational number: 1.585585558

Irrational number: 3.101101110………
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 2.357 and 3.121
Rational number: 3
Irrational number: 3.101101110………
(viii) Answer.  Rational number: $\frac{2}{10000}$
Irrational number: 0.000113133133 ……….
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 0.0001 and 0.001
Rational number: 0.0002 = $\frac{2}{10000}$
Irrational number: 0.000113133133

Irrational number: 1.909009000 ……
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 3.623623 and 0.484848
A rational number between 3.623623 and 0.484848 is 1.
An irrational number between 3.623623 and 0.484848 is 1.909009000 ……

(x) Answer. A rational number is $\frac{63753}{10000}$
An irrational number is 6.375414114111……..
Solution.
Any number which can be represented in the form of p/q where q is not equal to zero is a rational number. Also, both p and q should be rational when the fraction is expressed in the simplest form.
Irrational numbers are real numbers which cannot be represented as simple fractions.
They are non-terminating non-recurring in nature. It means they keep on continuing after decimal point and do not have a particular pattern/sequence after the decimal point.
Between 6.375289 and 6.375738:
A rational number is 6.3753 = $\frac{63753}{10000}$
An irrational number is 6.375414114111……..