Well, since you asked for an obvious reason, and not a completely separate derivation, here it is. The process of rotating a repeating decimal expansion x by 1

Well, since you asked for an obvious reason, and not a completely separate derivation, here it is. The process of rotating a repeating decimal expansion x by 1 digit means adding some integer n and dividing by 10. (n + x) / 10 = 2x Simplifying, n = 19x, i.e. x = n/19. So you know the number has to correspond to a decimal expansion of n/19, which is 18 digits in length.