Classify the following numbers as rational or irrational:
(i) $\sqrt{81}$
(ii) $\sqrt{12}$
(iii) $0.33333\ldots$
(iv) $0.123451234512345\ldots$
(v) $1.01001000100001\ldots$ (Notice the pattern: Is it repeating a single block?)
(vi) $23.560185612239874790120$

Find the explicit fractions in case they are rational.

(i) $\sqrt{81}$

⇒ $\sqrt{81} = 9 = \dfrac{9}{1}$

Since 9 can be written as $\dfrac{p}{q}$ with q ≠ 0,

$\sqrt{81}$ is rational.

(ii) $\sqrt{12}$

⇒ $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Since $\sqrt{3}$ is irrational, $2\sqrt{3}$ is also irrational.

So, $\sqrt{12}$ is irrational.

(iii) 0.33333…

This is a repeating decimal with the digit 3 repeating.

Let x = 0.3333…

⇒ 10x = 3.3333…

⇒ 10x - x = 3.3333… - 0.3333… = 3

⇒ 9x = 3

⇒ x = $\dfrac{3}{9} = \dfrac{1}{3}$.

Since 0.33333… can be written as $\dfrac{p}{q}$ with q ≠ 0,

So, 0.33333… is rational and equals $\dfrac{1}{3}$.

(iv) 0.123451234512345…

This is a repeating decimal with the block "12345" repeating.

Let x = 0.12345 12345…

⇒ 100000x = 12345.12345…

⇒ 100000x - x = 12345.12345… - 0.12345… = 12345

⇒ 99999x = 12345

⇒ x = $\dfrac{12345}{99999} = \dfrac{4115}{33333}$.

Since 0.123451234512345… can be written as $\dfrac{p}{q}$ with q ≠ 0,

So, 0.123451234512345… is rational and equals $\dfrac{4115}{33333}$.

(v) 1.01001000100001…

The pattern increases the number of zeros each time (one 0, then two 0's, then three 0's, etc.). This means there is no single repeating block.

So, 1.01001000100001… is non-terminating and non-repeating.

Hence, it is irrational.

(vi) 23.560185612239874790120

This is a terminating decimal with 21 decimal places.

It can be written as :

⇒ $\dfrac{23560185612239874790120}{10^{21}}$.

Since 23.560185612239874790120 can be written as $\dfrac{p}{q}$ with q ≠ 0,

So, 23.560185612239874790120 is rational.