Question

The following real numbers have decimal expansions as given below. State whether they are rational or irrational. If they are rational, express them in the form `p/q`, where p and q are co-prime integers and q ≠ 0 and then what can you say about the prime factors of q?

1.636363

Answer 1

The number 1.636363 has a terminating decimal expansion.

1. State whether they are rational or irrational: 

The number 1.636363 is rational because all terminating decimals can be expressed as a fraction `p/q`.

2. Express in the form `p/q`:

`1.636363 = 1636363/1000000`

The fraction is already in its simplest form where p and q are coprime.

So, in the form `p/q`, the number is `1636363/1000000`.

3. What can you say about the prime factors of q?:

The denominator q is 1,000,000.

Its prime factorisation is:

1000000 = 10⁶

= (2 × 5)⁶

= 2⁶ × 5⁶

The prime factors of q are 2 and 5. 

A rational number has a terminating decimal expansion if and only if the prime factors of its denominator are only 2 or 5.