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Marvelous Arithmetics of Distance
- itjustisthough 3y agoI must have missed p-adics in school. This was a fun, interesting, and clear read.
- wging 3y agoThey 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.
- dividendpayee 3y agoThey'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.
- wging 3y agoWhich classes are you thinking of?
- scheme271 3y agoMy 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.
- behnamoh 3y agothere 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.
- pavel_lishin 3y agoI will continue to marvel at it, though.
- Koshkin 3y ago> just math I don't know what this is supposed to convey. Modern mathematics is one of the most amazing, marvelous, beautiful etc. achievements of the mind. I wish more people could appreciate that.
- galaxyLogic 3y agoThe 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 perhaps the Mathematician/Magician for whom it doesn't look like magic but for the rest of many of us it does. We are even willing to pay for the ticket to see magicians perform their "magic" and to marvel at it.
- mbork_pl 3y ago> Mathematician/Magician Or mathemagician. ;-)
- galaxyLogic 3y agoGreat word
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- jamespropp 3y agoYeah, I admit the title is a bit click-baity. But I got the word “marvelous” from the Audre Lourde poem, so it’s not completely gratuitous.
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- hgjjugtyh 3y ago[flagged]
- jxf 3y agoFor another accessible introduction to p-adic numbers, check out Eric Rowland's overview [1]. [1]: https://www.youtube.com/watch?v=3gyHKCDq1YA https://www.youtube.com/watch?v=3gyHKCDq1YA
- krackers 3y agoIt'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 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.
- kevinventullo 3y agoShameless 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-fast-exponentiation/ https://kevinventullo.com/2020/12/21/2-adic-logarithms-and-f...
- j2kun 3y agoOh 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-unordered-multisets/ https://jeremykun.com/2021/10/14/group-actions-and-hashing-u... Thanks for writing it and clueing me into the idea!
- xeyownt 3y ago> 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.
- LodeOfCode 3y ago6^5 ends in ...76 6^25 ends in ...376 6^125 ends in ...9376 6^625 ends in ...09376 and so on
- BlackFly 3y agoI 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...
- layer8 3y agoI 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.
- nulbyte 3y ago> 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?
- nh23423fefe 3y agoThere 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.
- svat 3y agoGiven 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.)
- Bvlynicole 3y ago[dead]