As it happens, any real number whose decimal representation has a portion which repeats indefinitely or terminates* can be expressed as a ratio of integers, and thus is rational.
We can find the fraction using a bit of algebra.
So, in our given example, we have
Note that this process can be generalized to handle cases in which the repeating portion starts after nonrepeating digits as well.
*A terminating decimal can be converted to a fraction by multiplying and dividing by an appropriate power of
very difficult to demonstrate this without algebra
This is a rational number as it may be written as the fraction
If the use of the
Divide both sides by 9