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.overline(123456789)`

Explanation

To determine whether the given real numbers are rational or not, we need to analyze their decimal expansions. A number is rational if it can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

### Step-by-Step Solution:

**(i) For the number \( 43.123456789 \):**

1. **Identify the type of decimal:** The decimal \( 43.123456789 \) is terminating because it has a finite number of digits after the decimal point.

...