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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) \(5. \overline{234}\)
(iii) 23.245789
(iv) \(7.\overline{3427}\)
(v) 0.120120012000120000…
(vi) 23.142857
(vii) 2.313313313331…
(viii) 0.02002000220002…
(ix) 3.300030000300003…
(x) 1.7320508…
(xi) 2.645713
(xii) 2.8284271…

1 Answer

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(i) 7.2345

Here, 7.2345 has terminating decimal expansion.

So, it represents a rational number.

i.e. 7.2345 = 7.2345/10000 = p/q

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

(ii) \(5. \overline{234}\)

\(5. \overline{234}\) is non-terminating but repeating.

So, it would be a rational number.

In a non-terminating repeating expansion of p/q ,

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 = 23.245789/1000000 = p/q

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

In a terminating expansion of p/q , q is of the form 2n5m

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

(iv) \(7.\overline{3427}\)

\(7.\overline{3427}\) is non-terminating but repeating.

So, it would be a rational number.

In a non-terminating repeating expansion of p/q ,

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 = 23.142857/1000000 = p/q

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

In a terminating expansion of p/q , q is of the form 2n5m

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 = 2.645713/1000000 = p/q

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

In a terminating expansion of p/q , q is of the form 2n5m

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.

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