Prove that a periodic decimal is rational.?

Question

List steps and explain what you're doing please, thanks.

Answer 1

periodic decimal can always be written as fractions with integer numerators and denominators, the definition of a rational number.

eg: 1.34444... = x

10x = 13.4444...

subtract the two: 9x = 12.1
x = 12.1/9 = 121/90.

Answer 2

Let us say that the devcimal representation of a fraction is:

0.abcdeabcdeabcde.....
cycle is 5

we look for P/Q (rational fraction) such that

P/Q = 0.abcdeabcdeabcde.....

multiply both sides by 10^5 = 100,000

100,000*(P/Q) = abcde.abcdeabcdeabcde....
subtract the original value of P/Q)

100,000*(P/Q) = abcde.abcdeabcdeabcde....
minus 1*(P/Q) = 0.abcdeabcdeabcdeabcde....
gives
99,999*(P/Q) = abcde (the fraction part cancels out)

P/Q = abcde/99,999


Example (with a cyle of 6)

P/Q = 0.142857142857142857.....
1,000,000*(P/Q) = 142857.142857142857142857...

subtract the first from the second:

999,999*(P/Q) = 142857
P/Q = 142857/999,999 = 1/7

(there is a bit of work between 142857/999,999 and 1/7, it is tedious but it is not complicated.