In each case, state whether the following numbers are rational or irrational. If they are rational and expressed in the form $\dfrac{p}{q}$, where p and q are coprime integers, then what can you say about the prime factors of q?

$\begin{matrix} \text{(i)} & 279.034 \\[1.5em] \text{(ii)} & 76.\overline{17893} \\[1.5em] \text{(iii)} & 3.010010001… \\[1.5em] \text{(iv)} & 39.546782 \\[1.5em] \text{(v)} & 2.3476817681… \\[1.5em] \text{(vi)} & 59.120120012000… \\[1.5em] \end{matrix}$

(i) 279.034

This can be written as 279.034 = $\dfrac{279034}{1000}$

Since, it is terminating decimal

It is Rational number and the prime factors of its denominator q will be 2 or 5 or both .

(ii) $76.\overline{17893}$

Since it is non-terminating recurring decimal,

$76.\overline{17893}$ = 76.1789317893…

It is a rational number which is non-terminating and repeating. Its denominator q will have prime factors other than 2 or 5.

(iii) 3.010010001…

Since, it is non-terminating non-repeating decimal number

∴ It is an Irrational number.

(iv) 39.546782

This can be written as 39.546782 = $\dfrac{39546782}{1000000}$

Since , it is terminating decimal

It is Rational number and the prime factors of its denominator q will be 2 or 5 or both .

(v) 2.3476817681… = $2.34\overline{7681}$

Since, it is a non-terminating repeating decimal number,

∴ It is a Rational number and its denominator q will have prime factors other than 2 or 5.

(vi) 59.120120012000…

Since, it is non-terminating non-repeating decimal number

∴ It is an Irrational number.