silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory Hmmm... Anybody know any resource

silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory Hmmm... Anybody know any resources related to that? I'm interested in the properties of base-pi. I found this: https://math.stackexchange.com/a/2188852 https://math.stackexchange.com/a/2188852 Strange bases are pretty cool. Any integer can be converted into base-sqrt(2) by interleaving zeoes into its binary representation. E.g. 5 in base-sqrt(2) is 010001, 21 (0b10101) is 0b0100010001, etc. Possibly useless, but neat. Also: https://www.quora.com/Can-we-have-a-number-system-with-the-base-pi https://www.quora.com/Can-we-have-a-number-system-with-the-b... Knuth considers transcendental bases (ee is a transcendental number) in The Art of Computer Programming – in the section called Positional Number Systems. One of the consequences of using irrational bases is unfortunately that numbers need not have a unique representation. EDIT: Wow, this is awesome: https://www.quora.com/What-is-10-in-base-pi https://www.quora.com/What-is-10-in-base-pi Specifically the answer from Dave Williamson. It's ridiculously comprehensive. He shows how to calculate it using pure math, using a calculator, by writing a program, and demonstrates a partial proof on some of the properties.