Marvelous Arithmetics of Distance • Hacker News Board

Re: Marvelous Arithmetics of Distance

#21

Post by scheme271 » Mon, Oct 23, 2023, 9:07 AM UTC

wging wrote: post #3

itjustisthough wrote: post #2

I 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.

My honors analysis class covered it. Although it the professor that taught the class worked in p-adic analysis so he snuck p-adics into the first quarter of the analysis sequence.

Re: Marvelous Arithmetics of Distance

#22

Post by layer8 » Mon, Oct 23, 2023, 10:11 AM UTC

I like the subject matter, but I find the writing much too long-winded for my taste, with all the “you” and “we” and “let’s” and so on. That conversational style might work in a video, but as written text I find it exhausting.

Re: Marvelous Arithmetics of Distance

#23

Post by nulbyte » Mon, Oct 23, 2023, 11:23 AM UTC

> It seems a + b = 1.

I've seen this before, and it makes no sense. Is it common to ignore that you are carrying a one?

Re: Marvelous Arithmetics of Distance

#24

Post by nh23423fefe » Mon, Oct 23, 2023, 3:32 PM UTC

nulbyte wrote: post #23

> It seems a + b = 1. I've seen this before, and it makes no sense. Is it common to ignore that you are carrying a one?

There isn't some state where you are "carrying the one" and the number is different. Carrying is just an algorithm to canonicalize a number w.r.t. to a radix.

consider the natural number 9. it is always equal to (1+1+1+1+1+1+1+1+1+0)

but the radix is what determines allowable digits in the representation.

in any radix >9 you simply write 9, now consider radix=7

you will compute the fact 9 = 7^1*1 + 1^1*2, which implies the numeral is 12. because 9 = 9 + 0 = 7 + (9 - 7) = 7 + 2 = 10_7 + 2_7 = 12_7

computing coefficients doesn't change a number or a sum.

Re: Marvelous Arithmetics of Distance

#25

Post by svat » Mon, Oct 23, 2023, 5:52 PM UTC

nulbyte wrote: post #23

> It seems a + b = 1. I've seen this before, and it makes no sense. Is it common to ignore that you are carrying a one?

Given that

    a = ···256259918212890625

and

    b = ···743740081787109376

already they "make no sense" as natural numbers. (And they aren't.) We can define p-adic addition more formally, and "following the usual rule for adding numbers" (giving c = ···000000000000000001) is just serving as useful motivation in the meantime (that's why the author says "it seems"). You can look up the formal addition rule for p-adic integers, but meanwhile if you accept that the sum c=a+b is also going to be a p-adic integer, then you can think about: if c is not ···000000000000000001, what else could it be? (If you pick a certain position, say the 100 trillionth digit from the right, what is that digit of c going to be? Same for any other position other than the rightmost one.)

(Another analogy: If you consider that 1.0000 - 0.3333 = 0.6667, and 1.00000 - 0.33333 = 0.66667, etc, when you say 1.00000… - 0.33333… = 0.66666…, do you wonder about “where did the 7” go? Footnote 5 in the post discusses this.)

Re: Marvelous Arithmetics of Distance

#26

Post by Bvlynicole » Mon, Oct 23, 2023, 6:44 PM UTC

nulbyte wrote: post #23

> It seems a + b = 1. I've seen this before, and it makes no sense. Is it common to ignore that you are carrying a one?

[dead]

Re: Marvelous Arithmetics of Distance

#27

Post by galaxyLogic » Tue, Oct 24, 2023, 3:32 AM UTC

mbork_pl wrote: post #20

galaxyLogic wrote: post #8

Earlier quoted context omitted.

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. ;-)

Great word

Re: Marvelous Arithmetics of Distance

#28

Post by j2kun » Tue, Oct 24, 2023, 5:27 AM UTC

kevinventullo wrote: post #16

Shameless 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...

Oh hey, I enjoyed your hashing multisets article and wrote my own take on the topic here: https://jeremykun.com/2021/10/14/group-actions-and-hashing-u...

Thanks for writing it and clueing me into the idea!