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 \(\frac{p}{q}\), what can you say about the prime factors of q?
1) 43.123456789
2) 0.120120012000120000…
3) 43.123456789123456789...

1) 43.123456789
\(\Rightarrow \) \(43.123456789 = {43123456789 \over {10^9}} = {43123456789 \over {2^9 × 5^9}}\)

The above expression has denominator of the form \(2^n × 5^m\) where n and m are non-negative integers.

So, the number is a rational number, a terminating decimal number.


2) .0.120120012000120000…
Since,the given number is a non terminating non repeating number, it is not rational.


3) 43.123456789123456789....
The number is a rational number because decimal expansion is non terminating repeating.

It has denominator of the form \(2^n × 5^m\) where n and m are non-negative integers.