Question

Make a table with columns for rational numbers, irrational numbers and real numbers and write the following numbers in their proper places in the table.

(1) $1.\overline{57}$
(2) $\sqrt{5}$
(3) 4.10547194
(4) 4.8
(5) $0.\overline{735}$
(6) $\sqrt{25}$
(7) $\sqrt{10}$
(8) $\sqrt{196}$
(9) 3.819023...
(10) 6.10203040...

Answer 1

A number in the non-terminating recurring decimal form is a rational number and a number whose decimal form is non- terminating but is not recurring is called an irrational number.

(1) $1.\overline{57}$
$1.\overline{57}$ has 57 as the recurring term. Hence, it is a rational number.

(2) $\sqrt{5}$
$\sqrt{5}$ = 2.2360……, is a non-terminating and non-repeating decimal number. Hence, it is irrational.

(3) 4.10547194….
In 4.10547194…is a non-terminating and non-repeating decimal number. Hence, it is irrational.

(4) 4.8
4.8 can be written as $\frac{48}{10}=\frac{24}{5}$
This number is of the form $\frac{p}{q}$, where p = 24 and q = 5 are integers and q ≠ 0.
So, $\frac{24}{5}$ is a rational number, i.e. 4.8 is a rational number.

(5) $0.\overline{735}$
$0.\overline{735}$has 735 as a recurring term. Hence, it is a rational number.

(6) $\sqrt{25}$
$\sqrt{25}$ can be expressed as 5.
This number is of the form $\frac{p}{q}$, where p = 5 and q = 1 are integers and q ≠ 0.
Hence, $\sqrt{25}$ is a rational number.

(7) $\sqrt{10}$
$\sqrt{10}$= 3.16227……, is a non-terminating and non-repeating decimal number. Hence, it is irrational.

(8) $\sqrt{196}$
$\sqrt{196}$can be expressed as 14.
This number is of the form $\frac{p}{q}$, where p = 14 and q = 1 are integers and q ≠ 0.
Hence, $\sqrt{196}$ is a rational number.

(9) 3.819023…
In 3.819023…is a non-terminating and non-repeating decimal number. Hence, it is irrational.

(10) 6.10203040…
In 6.10203040…is a non-terminating and non-repeating decimal number.Hence, it is irrational.
All the numbers can be classified as

Rational	Irrational	Real
$1.\overline{57}$	$\sqrt{5}$	$1.\overline{57}$
4.8	4.10547194	$\sqrt{5}$
$0.\overline{735}$	$\sqrt{10}$	4.10547194
$\sqrt{25}$	3.819023...	4.8
$\sqrt{196}$	6.10203040...	$0.\overline{735}$
		$\sqrt{25}$
		$\sqrt{10}$
		$\sqrt{196}$
		3.819023...
		6.10203040...