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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...
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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.
