Classify the following numbers as rational or irrational:

(A) $\sqrt{81}$

(B) $\sqrt{12}$

(C) 0.33333...

(D) 0.123451234512345...

(E) 1.01001000100001... (Notice the pattern: Is it repeating a single block?)

(F) 23.560185612239874790120

Find the explicit fractions in case they are rational. [Page No. 62]

(A) $\sqrt{81} = 9 = \frac{9}{1}$

Hence, it is a rational.

(B) $\sqrt{12} = 2\sqrt{3}$ (Since $\sqrt{3}$ is irrational.)

Hence, it is an irrational.

(C) $0.33333_{-} = 0.\overline{3} = \frac{1}{3}$

Hence, it is a rational.

(D) $0.123451234512345_{-} = 0.\overline{12345}$

Since, this is a pure repeating decimal.

Let $x = 0.\overline{12345}$ ; ...(i)

$100000x = 12345.\overline{12345}$ ...(ii)

Subtract eq(i) from (ii):

$99999x = 12345$

So, $x = \frac{12345}{99999} = \frac{4115}{33333}$

Hence, it is a rational.

(E) $1.01001000100001_{-}$

Since, this digits follow a pattern (one more zero between 1s each time) but do not form a repeating block.

Hence, it is an irrational.

(F) $23.56018561223987479012$

Since, a terminating decimal (21 digits after point)

i.e., $\frac{23.56018561223987479012}{10^{21}}$

Hence, it is a rational.

Caution
 Students should remember that a non-terminating decimal is not always irrational. If it repeats (like 0.333…) it is rational; only non-terminating, nonrepeating decimals (like 1.01001000100001…) are irrational.