Talking more about infinite decimals, the other day I was looking at mathy stuff and came across Midy’s Theorem. I couldn’t find any biographical info on Midy whatsoever. His theorem adresses how if you have an integer over a prime integer, adding one half of the digits in a repeating decimal with the other half will result in all 9s.
For example: 1/7=0.142857 142857 142857… and so on.
If you add the first half of those digits (1,4, and 2) with the second half of digits (8, 5, 7), you get 999. 1+8, 4+5, and 2+7. The corresponding numbers will be compliments, sort of like how genes work.
Another example: 1/17=0.0588235294117647…. 05882352+94117647=99999999
The proof for this looks very rigorous, and I don’t understand it.
Also, for an even period of repetition, if you group the digits into couplets and then add them, you get a multiple of 9. It seems like it’s a corollary for Midy’s Theorem.
Ex: For 1/7=.142857, 14+28+57=11*9. For 1/17=0.0588235294117647, 05+88+23+52+94+11+76+47=396=44*9.