(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.