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> Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can'

> Such as? 60 has more prime factors than 10, and therefore using it as a numeric base will result in you encountering fewer irrational numbers. You really can't argue against that. Actually, I can argue with that. Irrationality is a property of the number, not the representation. An irrational number cannot be written as a terminating "decimal" in any base. And, of course, a rational number can be written as an integer fraction in any base. https://math.stackexchange.com/a/625481 https://math.stackexchange.com/a/625481 EDIT: Ignoring, of course, doing silly things like saying pi is rational in base-pi... Which I think deserves a spot in some circle of mathematical purgatory. Also, I wonder if every number except pi and its multiples are non-terminating in base-pi...

sillysaurus3 9y ago
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.
- Someone 9y ago
  Neither mentions base π, but you may be interested in https://en.wikipedia.org/wiki/Non-integer_representation https://en.wikipedia.org/wiki/Non-integer_representation, which even claims applications: "There are applications of β-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998)." and https://en.wikipedia.org/wiki/Quater-imaginary_base https://en.wikipedia.org/wiki/Quater-imaginary_base. (Surprisingly, there seems to be less content on this on mathworld)

sillysaurus3 9y ago

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.

Someone 9y ago
Neither mentions base π, but you may be interested in https://en.wikipedia.org/wiki/Non-integer_representation https://en.wikipedia.org/wiki/Non-integer_representation, which even claims applications: "There are applications of β-expansions in coding theory (Kautz 1965) and models of quasicrystals (Burdik et al. 1998)." and https://en.wikipedia.org/wiki/Quater-imaginary_base https://en.wikipedia.org/wiki/Quater-imaginary_base. (Surprisingly, there seems to be less content on this on mathworld)