Trick or treat?

October 31, 2022 12:12 AM

1 Billion is Tiny in an Alternate Universe: Introduction to p-adic Numbers - "The p-adic numbers are bizarre alternative number systems that are extremely useful in number theory. They arise by changing our notion of what it means for a number to be large. As a real number, 1 billion is huge. But as a 10-adic number, it is tiny!" (previously ;)
posted by kliuless (11 comments total)

Need more than one booklet to write out that answer...
posted by sammyo at 1:52 AM on October 31, 2022

An Exact Formula for the Primes: Willans' Formula".

The comments are phenomenal. A deep dive into genealogy for figuring out who Willans is, and good jokes.

I don't follow most of the math, but it's still an interesting look at a clever though inefficient formula.
posted by Nancy Lebovitz at 4:12 AM on October 31, 2022 [3 favorites]

I once made extensive use of the fact that the endomorphism ring of a Prüfer p-group Z(p^∞)^n is the ring n-by-n matrices over the p-adic integers Z_p.
posted by jeffburdges at 4:15 AM on October 31, 2022 [2 favorites]

Metafilter: I once made extensive use of the fact that the endomorphism ring of a Prüfer p-group Z(p^∞)^n is the ring n-by-n matrices over the p-adic integers Z_p
posted by Reasonably Everything Happens at 5:15 AM on October 31, 2022 [7 favorites]

Numberphile has ruined me on the idea that any number can ever actually be "large" but I'm definitely enamored of the idea that there are novel ways of expressing numbers that grant us new insights and affordances.

I never would have come up with this in a large-by-my-standards number of years, though. I'd have thought that the number line is just the number line, you know? Negative numbers just mean the number line goes in two directions, and square roots of negative numbers just means we've got a second, perpendicular number line, but that's understandable too... but here there's a whole new way to express and order these numbers we already have? I guess there's nothing preventing that! I just... never would have guessed.

(Update: does this remind anyone of Hamming Codes writ large, maybe multiplexed in a sense?)
posted by mhoye at 7:52 AM on October 31, 2022

The 2’s complement arithmetic your computer uses can be understood as the right-truncation of 2-adic numbers.
posted by sjswitzer at 9:06 AM on October 31, 2022 [3 favorites]

Although I have difficulty comprehending this, I had a lovely time falling asleep to it and dreaming vaguely of Tetris.
posted by Countess Elena at 9:36 AM on October 31, 2022 [1 favorite]

I have been so enured by a career in mathematics that it seems weird to me to call the p-adic numbers 'bizarre'. I mean, their construction is pretty straightforwardly formal power series, and it doesn't feel like anything counter-intuitive is going on. I should WTFV, but the previously mentioned career in mathematics makes it hard for me to consume mathematics media intended for a general audience (I am looking at you especially Quanta Magazine).
posted by 3j0hn at 9:51 AM on October 31, 2022 [1 favorite]

Imagine trying to explain them to someone who doesn't have any familiarity with formal power series though.
posted by Pyry at 11:13 AM on October 31, 2022

And I thought learning to count in hex was hard.
posted by Pouteria at 9:53 PM on October 31, 2022

It’s much more costly for a computer to divide than to multiply. Remember long division? Well, it’s like that. But if you need to divide by a constant you can do it with a multiply and (sometimes) a shift, thanks to 2-adic numbers. Great explainer here. This is an optimization that all modern compilers know.
posted by sjswitzer at 9:59 PM on October 31, 2022 [1 favorite]