Is 0. rational or irrational? what is an example of a repeater decimal?

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