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Because the decimal expansion of 19 repeats after 18 digits. That means all the fractions 1/19 .. 18/19 are just rotations of each other. And 20/19 = 1 + 1/19
by shrughes 18y ago
Because the decimal expansion of 19 repeats after 18 digits.
That means all the fractions 1/19 .. 18/19 are just rotations of each other. And 20/19 = 1 + 1/19, which means that 2/19 is a single rotation away.
If you've ever noticed that 142857 * [1..6] produces different rotations of the same number and wondered why, you might have thought about this. Dyson had probably already done all the hard work on the problem and knew immediately that the answers were rotations of some repeating fraction, and one where multiplying by 2 gives rotations by 1 digit.
- hhm 18y agoWow thanks!
- nkurz 18y agoThanks for the insight. I wouldn't have noticed any similarity to the multiples of 1/19. Accepting this, though, is there a reason it would have been obvious to him that there was no smaller number? The first such number, by the way, does indeed have 18 digits. I got there by considering every final digit and figuring out how long the string needed to be before the final digit reoccurred with no carry. (edit: oops, just read your explanation below which pretty much answers this)
- shrughes 18y agoWell, 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.