Irrational Numbers : Show that $0.1248163264...$ is irrational

Question

I was working through some basic Number Theory Problems in Rosen and came across the following problem :

Show that the real number $0.1248163264...$ represented in base 10 is an irrational number

I am slightly stumped. Can someone help me out? A hint would be great.

Answer 1

HINT: I expect that you know that a number is rational if and only if its decimal expansion is eventually periodic. Suppose that the expansion eventually repeats with period $p$.

• Show that there are two consecutive powers of $2$ after the initial aperiodic segment whose lengths (when written in the usual way in base ten) are the same multiple of $p$.

I’ve left a further hint in the spoiler-protected block below; click on the block to see it.

Show that on the one hand these two powers of $2$ must end in the same digit, and on the other hand that they cannot end in the same digit.

Answer 2

A number is rational if and only if it has an eventually repeating decimal expansion.

So you should show that this decimal expansion does not have an eventually repeating expansion.

Answer 3

Starting from the first digit after the decimal point, the digit gets multiplied by two every single time without stopping, also expanding the numbers of digits going on without an end. This has no repetition, proving it's irrational.