You already know that the decimal expansion of a
rational number eventually repeats or terminates
(which can be viewed as a repeating 0).
But I tell you something that perhaps you did not know:
if the denominator of that rational number is not
divisible by 3, then the repeating part of its decimal
expansion is an integer divisible by nine!
- 1/7 = .142857142857...
has repeating part 142857. This is divisible by 9.
- 41/55 = .7454545...
has repeating part 45. This is divisible by 9.
The Math Behind the Fact:
This rather curious fact can be shown easily.
If the rational X is purely repeating of period P and
repeating part R, then
R = 10P X - X = (10P-1) X
= (10P-1) (m/n).
Thus R*n = (10P-1)*m is an integer. Since
(10P-1) is divisible by 9, if n is not divisible
by 3, then R must be. If you like these fun deductions,
you may enjoy a course in number theory!
How to Cite this Page:
Su, Francis E., et al. "Repeating Digits."
Math Fun Facts.