# The Mathematical Ninja and the Ninety-Sevenths

“A ninety-seventh.” The student scratched her head. “I’d call that 0.01.” A moment more’s thought. “0.0103? Probably good enough.”

For the Mathematical Ninja, this was about as good as could be expected. They sighed all the same and wrote down: $0. \dot 01\, 03\, 09\, 27\, 83\, 50\, 51 \\ 54\, 63\, 91\, 75\, 25\, 77\, 31 \\ 95\, 87\, 62\, 88\, 65\, 97\, 93 \\ 81\, 44\, 32\, 98\, 96\, 90\, 72 \\ 16\, 49\, 48\, 45\, 36\, 08\, 24 \\ 74\, 22\, 68\, 04\, 12\, 37\, 11 \\ 34\, 02\, 06\, 18\, 55\, 6\dot7$

“I’ll take your word for it,” said the student.

“No need!” said the Mathematical Ninja. “You simply treble each pair of digits and anticipate the carry.”

“*Simply*.” said the student, pointedly.

“For example, the fourth pair of digits is 27. Trebling that is 81, but because trebling 81 will give you at least 200, you add 2 to make it 83. Trebling 83 gives 249, but a) we’ve already dealt with the 2, and b) trebling 49 will give us something between 100 and 199, so we add 1 to make it 50.”

“I think I see.”

“Also, when you get midway through the number – in this case, the 32, which is the 24th pair of digits – the number starts antirepeating: the first and the 25th pair add up to 99, the second and 26th also add to 99… and so on.”

“Is that just with 97ths?”

“I believe it’s for any prime reciprocal, and it’s related to Fermat’s Little Theorem. I could tell you more.” The Mathematical Ninja reached for some sunglasses. “But then I’d have to kill you.”