The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not.
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 p/q, what can you say about the prime factors of q?
i. 43.123456789
ii. 0.120120012000120000…
iii. 43.123456789
1 Answers
Solution:
i. Since the given number has a terminating decimal expansion, it is a rational number of the form p/q and q is of the form 2m x 5n.
The prime factors of p and q will be either 2 or 5 or both.
ii. Since the decimal expansion is neither terminating nor recurring, the given number is an irrational number.
iii. Since the decimal expansion is nonterminating repeating, the given number is a
rational number of the form p/q and q is not of the form 2m x 5n.
The prime factors of q will also have a factor other than 2 or 5.