yeah, the remainder of pi after p^2 for p>2 will always be the same as the remainder after division by p (p^2 if p=2), so it truncates to the left of the radix, and p-adic rationals can’t go off infinitely to the right of the radix.
what’s the remainder of pi after division by 3 and 3^2? notice how the remainder is the same – i.e. there’s no 3s digit needed. and the same will be true for all higher powers of 3. this is because pi is <3^n for all n > 1. but we failed to express the fractional part of pi. if we extend to the p-adic rationals, we can express it as an infinite expansion but that’s not allowed for p-adics – they can only have a finite number of digits in the part to the right of the radix.
flyos@jlai.lu 1 year ago
Out of curiosity, I guess pi cannot be rational in any p-adic set, right?
silent_water@hexbear.net 1 year ago
yeah, the remainder of pi after p^2 for p>2 will always be the same as the remainder after division by p (p^2 if p=2), so it truncates to the left of the radix, and p-adic rationals can’t go off infinitely to the right of the radix.
flyos@jlai.lu 1 year ago
Not sure I understand all of it, but the last part about pi not being algebraic made sense to me, at least! Thanks!
silent_water@hexbear.net 1 year ago
what’s the remainder of pi after division by 3 and 3^2? notice how the remainder is the same – i.e. there’s no 3s digit needed. and the same will be true for all higher powers of 3. this is because pi is <3^n for all n > 1. but we failed to express the fractional part of pi. if we extend to the p-adic rationals, we can express it as an infinite expansion but that’s not allowed for p-adics – they can only have a finite number of digits in the part to the right of the radix.