In 1/6=1.6666..., repeating 6 is called repeatend ( or reptend ) . I learn from https://en.wikipedia.org/wiki/Repeating_decimal, the reptend in the decimal form of 1/97 is a 96-digit string. Find fraction(s) having longer reptend string(s)?
We can find the fraction for an arbitrary repeating string with the following method:
So, for example, if we wanted a 1000-digit repeating string, we could let
give a repeating sequence of
If we let
would generate a a repeating string of the first
We can also use a similar method to find the fraction for any rational value with a repeating string.
Given a general real number with a repeating string
Note that by the above work, we have
Then we can rewrite our number as