What you typed is 0.12122... I am assuming that this means 0.1212222222222
You can rationalize any repeating decimal as follows...
multiply the number until only the repeating pattern is after the decimal place...
1000x=121.22222222222222 then multiply the decimal to one more place...
10000x=1212.2222222222222
now if we use subtraction...
10000x=1212.2
1000x=121.2 you get:
9000x=1091 now divide both sides by the multiple of x
x=1091/9000
So x is rational because it is a ratio of integer values.
and 1091/9000=0.1212222222222222222222222222222222222 :P