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?

1.585585558...

Answer 1

The given decimal number is 1.585585558...

We want to determine if it is rational or irrational.

Observing the decimals:

The digits do not show a clear repeating block pattern.

The sequence goes 1.585585558... which looks like a non-terminating, non-repeating decimal expansion.

Decimals that are non-terminating and non-recurring are irrational numbers.

The text provides this number exactly as example (iii) in one of the exercises and states:

1.585585558... is a non-terminating non-recurring decimal, so it is an irrational number.

Therefore:

The number 1.585585558... is irrational.

It cannot be expressed as a fraction `p/q` with integer co-prime p and q.

Hence, no discussion about prime factors of q applies since q does not exist in this case.