Question 1:
Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
Answer:
(i)
The denominator is of the form 5m.
Hence, the decimal expansion ofis terminating.
(ii)
The denominator is of the form 2m.
Hence, the decimal expansion of is terminating.
(iii)
455 = 5 × 7 × 13
Since the denominator is not in the form 2m × 5n, and it also contains 7 and 13 as its factors, its decimal expansion will be non-terminating repeating.
(iv)
1600 = 26 × 52
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of is terminating.
(v)
Since the denominator is not in the form 2m × 5n, and it has 7 as its factor, the decimal expansion of is non-terminating repeating.
(vi)
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of is terminating.
(vii)
Since the denominator is not of the form 2m × 5n, and it also has 7 as its factor, the decimal expansion of is non-terminating repeating.
(viii)
The denominator is of the form 5n.
Hence, the decimal expansion of is terminating.
(ix)
The denominator is of the form 2m × 5n.
Hence, the decimal expansion of is terminating.
(x)
Since the denominator is not of the form 2m × 5n, and it also has 3 as its factors, the decimal expansion of is non-terminating repeating.
Question 2:
Write down the decimal expansions of those rational numbers in Question 1 above which have terminating decimal expansions.
Answer:
(viii)
Question 3:
The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form , what can you say about the prime factor of q?
(i) 43.123456789 (ii) 0.120120012000120000… (iii)
Answer:
(i) 43.123456789
Since this number has a terminating decimal expansion, it is a rational number of the form and q is of the form
i.e., the prime factors of q will be either 2 or 5 or both.
(ii) 0.120120012000120000 …
The decimal expansion is neither terminating nor recurring. Therefore, the given number is an irrational number.
(iii)
Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form and q is not of the form i.e., the prime factors of q will also have a factor other than 2 or 5.