The following real numbers have decimal expansions as given below. In each case examine whether they are rational or not. If they are a rational number of the form p/q, what can be said about q?

(i) 7.2345

(ii)

(iii) 23.245789

(iv)

(v) 0.120120012000120000…

(vi) 23.142857

(vii) 2.313313313331…

(viii) 0.02002000220002…

(ix) 3.300030000300003…

(x) 1.7320508…

(xi) 2.645713

(xii) 2.8284271…

(i) 7.2345

Here, 7.2345 has terminating decimal expansion.

So, it represents a rational number.

i.e. 7.2345 = =

Thus, q = 10⁴, those factors are 2³ × 5³

(ii)

is non-terminating but repeating.

So, it would be a rational number.

In a non-terminating repeating expansion of ,

q will have factors other than 2 or 5.

(iii) 23.245789

23.245789 is terminating decimal expansion

So, it would be a rational number.

i.e. 23.245789 = =

Thus, q = 10⁶, those factors are 2⁵ × 5⁵

In a terminating expansion of , q is of the form 2ⁿ5^m

So, prime factors of q will be either 2 or 5 or both.

(iv)

is non-terminating but repeating.

So, it would be a rational number.

In a non-terminating repeating expansion of ,

q will have factors other than 2 or 5.

(v) 0.120120012000120000…

0.120120012000120000… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(vi) 23.142857

23.142857 is terminating expansion.

So, it would be a rational number.

i.e. 23.142857 = =

Thus, q = 10⁶, whose factors are 2⁵ × 5⁵

In a terminating expansion of , q is of the form 2ⁿ5^m

So, prime factors of q will be either 2 or 5 or both.

(vii) 2.313313313331…

2.313313313331… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(viii) 0.02002000220002…

0.02002000220002… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(ix) 3.300030000300003…

3.300030000300003… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(x) 1.7320508…

1.7320508… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.

(xi) 2.645713

2.645713 is terminating expansion

So, it would be a rational number.

i.e. 2.645713 = =

Thus, q = 10⁶, those factors are 2⁵ × 5⁵

In a terminating expansion of , q is of the form 2ⁿ5^m

So, prime factors of q will be either 2 or 5 or both.

(xii) 2.8284271…

2.8284271… is non-terminating and non-repeating.

So, it is not a rational number as we see in the chart.