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Marvelous Arithmetics of Distance
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Re: Marvelous Arithmetics of Distance
#12For another accessible introduction to p-adic numbers, check out Eric Rowland's overview [1].
Re: Marvelous Arithmetics of Distance
#13For another accessible introduction to p-adic numbers, check out Eric Rowland's overview [1]. [1]: https://www.youtube.com/watch?v=3gyHKCDq1YA
It's also not a full introduction, but for those who want to have a taste of p-adic numbers, see 3b1b's video https://www.youtube.com/watch?v=XFDM1ip5HdU which motivates 2-adic numbers. There's also a neat connection between twos complement and 2-adic numbers.
Re: Marvelous Arithmetics of Distance
#14I must have missed p-adics in school. This was a fun, interesting, and clear read.
They are typically not taught in school, at least in the US but probably elsewhere too. I don’t even know of a commonly taken undergrad-level course that I’d expect to cover them.
They're often -- but not always -- touched on in advanced undergraduate classes, e.g. after real analysis and abstract algebra. Not every math undergraduate takes number theory these days.
Re: Marvelous Arithmetics of Distance
#15Earlier quoted context omitted.
They are typically not taught in school, at least in the US but probably elsewhere too. I don’t even know of a commonly taken undergrad-level course that I’d expect to cover them.
They're often -- but not always -- touched on in advanced undergraduate classes, e.g. after real analysis and abstract algebra. Not every math undergraduate takes number theory these days.
Which classes are you thinking of?
Re: Marvelous Arithmetics of Distance
#16Shameless promotion of a blog post I made a few years ago connecting 2-adic numbers with computationally fast exponentiation: https://kevinventullo.com/2020/12/21/2-adic-logarithms-and-f...
Re: Marvelous Arithmetics of Distance
#17> Likewise, if you start with a positive integer that ends in 6 and repeatedly raise it to the fifth power, you converge digit by digit toward the strange number
> b = ···743740081787109376
No you don't. Trying with 6, 16, 26, any number, you don't converge.
Re: Marvelous Arithmetics of Distance
#18> Likewise, if you start with a positive integer that ends in 6 and repeatedly raise it to the fifth power, you converge digit by digit toward the strange number > b = ···743740081787109376 No you don't. Trying with 6, 16, 26, any number, you don't converge.
6^5 ends in ...76
6^25 ends in ...376
6^125 ends in ...9376
6^625 ends in ...09376
and so on
Re: Marvelous Arithmetics of Distance
#19I was hoping for something on metric spaces, I was completely wrong but not disappointed. Great post.
Edit: Arguably there is a weird kind of metric in that, so not completely wrong I guess...
Re: Marvelous Arithmetics of Distance
#20there must be a rule that bans using “marvelous, magic, mysterious, never seen before, miraculous, genius, shocking …” words used in math posts. There’s nothing marvelous about any of this—it’s just math. Imo it’s better to let people face math head on than try to sugar coat it and click bait them into liking math.
The marvel of Math is in its non-obviousness. Obvious truths are not marvelous but non-obvious ones are. "Magic" is also a word we can use about Math because we know no true magic exists. Therefore when something is said to be "magic" we know it means something "looks like magic". For the Magician (/Mathematician) it of course doesn't look like magic because they clearly understand and see how it works. So, you are p…
> Mathematician/Magician
Or mathemagician. ;-)