. 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?
Sol. i. 43.123456789
Since this number has a terminating decimal expansion, it is a rational number of the form p/q and q is of the form 2m×5n
That is, the prime factor of q will be 2 or 5 or both.
The decimal expansion is neither terminating nor recurring.
Therefore, the given number is an irrational number.
Since the decimal expansion is non-terminating recurring, the given number is a rational number of the form p/q and q is not of the form 2m×5n that is, the prime factors of q will also have a factor other than 2 or 5.