Question

0.123123123... = 41/333 if I want both a/b to be integers. But how did this come about? How does one find out that 0.123123123 = 41/333?

Answer 1

The usual trick is to multiply by 10^{number of repeating digits}

1) N = 0.123123123123123...

123 is the repeating pattern. That's 3 digits.
10^3 = 1000

2) 1000N = 123.123123123123...

Subtract 1) from 2)

1000N = 123.123123123123...
N = 0.123123123123123...

999N = 123

N = 123/999

Then you can spend some quality time trying to reduce the fraction.

Answer 2

Thank you kindly.