Classify the following numbers as rational or irrational:

$\begin{matrix} \text{(i)} & \sqrt{23} \\[1.5em] \text{(ii)} & \sqrt{225} \\[1.5em] \text{(iii)} & 0.3796 \\[1.5em] \text{(iv)} & 7.478478… \\[1.5em] \text{(v)} & 1.101001000100001… \\[1.5em] \text{(vi)} & 345.0\overline{456} \\[1.5em] \end{matrix}$

Rational numbers are in the form c, q ≠ 0, and p and q are integers.

(i) $\bold{\sqrt{23}}$ is an irrational number , as it is not a perfect square so it cannot be written in the form $\dfrac{p}{q}$ , q ≠ 0.

(ii) $\sqrt{225}$ = $\sqrt{15×15}$ = 15 = $\dfrac{15}{1}$,

As it can be written in the form $\dfrac{p}{q}$ , q ≠ 0.

∴ $\bold{\sqrt{225}}$ is a rational number .

(iii) 0.3796 = $\dfrac{3796}{1000}$

Since, the decimal expansion is terminating decimal.

∴ 0.3796 is a rational number .

(iv) 7.478478

Let x = 7.478478 $\text{….(i)}$

Since there is three repeating digit after decimal point,

Multiplying both sides by 1000, we get

1000x = 7478.478478… $\text{….(ii)}$

Subtracting (i) from (ii) we get,

999x = 7471

x = $\bold{\dfrac{7471}{999}}$

∴ It is non terminating , repeating rational number .

(v) 1.101001000100001…

Since number of 0's are increasing between two consecutive terms as we move further , So it is non terminating, non repeating decimal.

∴ 1.101001000100001… is an irrational number.

(vi) $345.0\overline{456}$

$345.0\overline{456}$ = 345.0456456…

Let x = 345.0456456…

Multiply both sides by 10, we get

10x = 3450.456456.. $\text{….(i)}$

Since, after decimal there are three repeating digit:

Multiply both sides by 1000, we get

10000x = 3450456.456456… $\text{….(ii)}$

Subtracting (i) from (ii) ,

9990x = 3447006

x = $\bold{\dfrac{3447006}{9990}}$

since, it is non-terminating , repeating decimal.

∴ $\bold{345.0\overline{456}}$ is a Rational number .