The following real numbers have decimal expansions as given below. In

1.	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
Answer» 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 2^m 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 2^m x 5ⁿ. The prime factors of q will also have a factor other than 2 or 5.

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

Answer»

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 2^m 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 2^m x 5ⁿ.
The prime factors of q will also have a factor other than 2 or 5.

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