it’s perfectly legitimate to do 10-adic integers, or for any n (see here).
Ah, the wonders of modern science! Whatever insane idea you get, pretty sure someone already published it decades ago. (Related: Contra Hoel On Aristocratic Tutoring at ACX)
That doesn’t happen if you use a prime number instead, so you get all “fractions” as “integers”.
What would be 1⁄2 in base 2, or 1⁄5 in base 5? I think that p-adic numbers also make an exception for fractions that contain the base in the denominator.
Number theorists will sometimes talk as if this extra completion came from a mysterious new prime number, called the “prime at infinity”.
Not sure if this is related, but writing −1 as …999 already feels kinda like “modulo infinity”. And of course ω is a prime number; it’s not like you can divide it by 2 or something. :D
(Don’t mind me, I don’t really understand most of this. What’s in this article is exactly as far as I got.)
What would be 1⁄2 in base 2, or 1⁄5 in base 5? I think that p-adic numbers also make an exception for fractions that contain the base in the denominator.
Ah yeah, what I said was wrong. I was thinking of the completion Qn which is a field (i.e. allows for division by all non-zero members, among other things) iff n is prime. The problem with 10-adics can be seen by checking that …2222 * …5555 = …0000 = 0, so doing the completion has no hope of making it a field.
There’s no 1⁄2 in the 2-adic integers, but there’s a 1⁄2 in the 2-adic numbers where you’re allowed finitely many places after the decimal (binary) point. So although e.g. 1⁄3 = …0101010101 with nothing right of the binary point, 1⁄2 = 0.1 with a single place to the right. So 5⁄6 = …0101010101.1.
Ah, the wonders of modern science! Whatever insane idea you get, pretty sure someone already published it decades ago. (Related: Contra Hoel On Aristocratic Tutoring at ACX)
What would be 1⁄2 in base 2, or 1⁄5 in base 5? I think that p-adic numbers also make an exception for fractions that contain the base in the denominator.
Not sure if this is related, but writing −1 as …999 already feels kinda like “modulo infinity”. And of course ω is a prime number; it’s not like you can divide it by 2 or something. :D
(Don’t mind me, I don’t really understand most of this. What’s in this article is exactly as far as I got.)
Ah yeah, what I said was wrong. I was thinking of the completion Qn which is a field (i.e. allows for division by all non-zero members, among other things) iff n is prime. The problem with 10-adics can be seen by checking that …2222 * …5555 = …0000 = 0, so doing the completion has no hope of making it a field.
There’s no 1⁄2 in the 2-adic integers, but there’s a 1⁄2 in the 2-adic numbers where you’re allowed finitely many places after the decimal (binary) point. So although e.g. 1⁄3 = …0101010101 with nothing right of the binary point, 1⁄2 = 0.1 with a single place to the right. So 5⁄6 = …0101010101.1.