0.053053053053053 is it rational or irrational why

Question

0.053053053053053 is it a rational or irrational and why

Answer 1

A repeating decimal is rational. Assuming this is repeating 0.053 then we can say:

So x = 53/999, a rational number

Answer 2

It's rational.

A terminating decimal is always rational.

We can move the decimal point 14 places to the right, giving a whole number, then divide that by 10^14 which is a whole number getting back to the original decimal, so we can write the decimal as a ratio of two whole numbers (so it's rational).

Also decimals that have repeating groups are also rational .. so if the '053' repeats for ever that is also rational.

Why ?

0.053053053053 ... = (53/1000) + (53/1000000) + (53/1000000000) + ... = (53/1000) [ 1 + 1/1000 + 1/1000000 ... ] . Inside the parenthesis is a geometric series with a=1, r=1/1000 so it sums to 1/ 1-r = 1 / (999/1000) = 1000 / 999

So the decimal is 53 / 1000 x 1000 / 999 = 53 / 999 which is rational.