How the Towering P-Adic Numbers Work
October 21, 2020 11:42 PM Subscribe
An Infinite Universe of Number Systems - "The p-adics form an infinite collection of number systems based on prime numbers. They're at the heart of modern number theory."
I enjoyed reading the article, but could not give a coherent explanation of what I just read. Is there a word for that?
posted by Harald74 at 2:01 AM on October 22, 2020
posted by Harald74 at 2:01 AM on October 22, 2020
The p-adic numbers are my favorite counterpoint to the often-naive assertion that "numbers are real because mathematics is the language of reality." It's a real pleasure to see the gears catch when you point out that there are an infinity of number systems that not only behave sensibly but are indeed useful, even essential, in making further discoveries.
posted by belarius at 2:15 AM on October 22, 2020 [4 favorites]
posted by belarius at 2:15 AM on October 22, 2020 [4 favorites]
As a student I kept seeing mention of p-adic numbers in passing (talks, papers, OEIS sequences...), but they never showed up in my work or the number theory courses I took. At the time (and by the looks of it, still now) the wikipedia article suffered from wikipedia maths article syndrome and wasn't very enlightening, so it's good to read a clear explanation now. I think they're not far off from how my brain organises information about integers.
So where's my p-adic incremental game? Why hasn't this been made?
posted by polytope subirb enby-of-piano-dice at 2:15 AM on October 22, 2020
So where's my p-adic incremental game? Why hasn't this been made?
posted by polytope subirb enby-of-piano-dice at 2:15 AM on October 22, 2020
So where's my p-adic incremental game?
I literally just finished the SM12 in Ordinal Markup, I don't need this.
posted by Literaryhero at 5:24 AM on October 22, 2020 [3 favorites]
I literally just finished the SM12 in Ordinal Markup, I don't need this.
posted by Literaryhero at 5:24 AM on October 22, 2020 [3 favorites]
Sometimes, I feel stupid because I don't get the difference between p-adic numbers and selecting another base for your numbers. Isn't this just using a more precise word, for instance implying prime bases? Of course there's an infinite number of bases. Somewhere I believe on Metafilter I even saw links to folks using negative or real or imaginary numbers as a base, which I found a lot more interesting than primes.
Maybe someone can help me understand how rational numbers in a different base make a huge difference in relation to real numbers - sure, you can write 1/3 as a rational number in base 3, but then suddenly 1/2 is real in that base? (oh wait I was thinking of decimals)
*crawls back behind a stone*
posted by flamewise at 5:47 AM on October 22, 2020
Maybe someone can help me understand how rational numbers in a different base make a huge difference in relation to real numbers - sure, you can write 1/3 as a rational number in base 3, but then suddenly 1/2 is real in that base? (oh wait I was thinking of decimals)
*crawls back behind a stone*
posted by flamewise at 5:47 AM on October 22, 2020
I enjoyed reading the article, but could not give a coherent explanation of what I just read. Is there a word for that?
Oh, the inumeracy!
posted by y2karl at 5:50 AM on October 22, 2020
Oh, the inumeracy!
posted by y2karl at 5:50 AM on October 22, 2020
What I would like to know, Harald74, is whether the number of persons nationwide who understand this could numbered in more than five figures. I certainly am not one of them.
posted by y2karl at 6:00 AM on October 22, 2020
posted by y2karl at 6:00 AM on October 22, 2020
Bases are different representations of a particular value, these are fundamentally different systems.
posted by Horselover Fat at 6:00 AM on October 22, 2020 [1 favorite]
posted by Horselover Fat at 6:00 AM on October 22, 2020 [1 favorite]
"It's the reals that are the outliers."
This all day, regardless of context.
posted by riverlife at 9:35 AM on October 22, 2020 [2 favorites]
This all day, regardless of context.
posted by riverlife at 9:35 AM on October 22, 2020 [2 favorites]
P-adic numbers are very strange beasts. One way it's been explained is that they are like decimal fractions, but instead of being allowed to go infinitely far to the right of the decimal point (like 1/3 = 0.33333....), in the p-adics they are allowed to go infinitely far to the left: .....3333333. (And the base is some prime p, not base 10).
Another thing that makes these numbers strange is that the distance between two p-adic numbers a and b is how many common factors of p they share, not the difference a-b. They are used (among other things) to study problems related to divisibility.
One can go pretty far in math without ever dealing with the p-adics! (I'm more of a quaternion/octonion fan myself).
posted by crazy_yeti at 9:36 AM on October 22, 2020
Another thing that makes these numbers strange is that the distance between two p-adic numbers a and b is how many common factors of p they share, not the difference a-b. They are used (among other things) to study problems related to divisibility.
One can go pretty far in math without ever dealing with the p-adics! (I'm more of a quaternion/octonion fan myself).
posted by crazy_yeti at 9:36 AM on October 22, 2020
I'm trying to remember when I read about p-adic systems, because I'm pretty sure I have and also don't remember any details. The article rang some familiar bells but not enough to jog my memory. Maybe the Erdös biography?
But I'm a sucker for a fractal representation, so I'm excited to run into them again and wondering if I can find a way to incorporate this into some art. Or maybe for April Fools next year we can render everyone's userid in 19-adic in celebration of the first post.
Looking around for some further approachable reading, I skimmed past a lot of formal mathematical writing that I can't even start to follow to find this nice pop math article that talks through p-adic distance approachably and, pandering to me, uses the Cantor set as a reference for the 2-adics, and which also links to this readable short sketch of some arithmetic operations with p-adics. (Also found a paper that is on the whole unreadably dense for me but which features on page 22 a lovely hint of a Sierpinski triangle as a visualization of the 3-adics. I'll take what I can get.)
There's also a 3Blue1Brown video, What does it feel like to invent math?, that talks about infinite series and distance functions on the way to reaching 2-adic numbers at the end that I found a little helpful. I found that in this Sci Am article which digs in on the idea of p-adic absolute value as a way of talking about distances between numbers as well.
My two trailing thoughts on this are (1) the idea of fractal spaces + unconventional definitions of locality and distance really makes me wonder if this analogizes directly in any way to Hilbert curves and other space-filling curves and (2) the fact that Blaseball has introduced modulo arithmetic to scorekeeping and win/loss records this season feels like some sort of eldritch kismet.
posted by cortex at 9:37 AM on October 22, 2020 [5 favorites]
But I'm a sucker for a fractal representation, so I'm excited to run into them again and wondering if I can find a way to incorporate this into some art. Or maybe for April Fools next year we can render everyone's userid in 19-adic in celebration of the first post.
Looking around for some further approachable reading, I skimmed past a lot of formal mathematical writing that I can't even start to follow to find this nice pop math article that talks through p-adic distance approachably and, pandering to me, uses the Cantor set as a reference for the 2-adics, and which also links to this readable short sketch of some arithmetic operations with p-adics. (Also found a paper that is on the whole unreadably dense for me but which features on page 22 a lovely hint of a Sierpinski triangle as a visualization of the 3-adics. I'll take what I can get.)
There's also a 3Blue1Brown video, What does it feel like to invent math?, that talks about infinite series and distance functions on the way to reaching 2-adic numbers at the end that I found a little helpful. I found that in this Sci Am article which digs in on the idea of p-adic absolute value as a way of talking about distances between numbers as well.
My two trailing thoughts on this are (1) the idea of fractal spaces + unconventional definitions of locality and distance really makes me wonder if this analogizes directly in any way to Hilbert curves and other space-filling curves and (2) the fact that Blaseball has introduced modulo arithmetic to scorekeeping and win/loss records this season feels like some sort of eldritch kismet.
posted by cortex at 9:37 AM on October 22, 2020 [5 favorites]
Flamewise, the p-adics are indeed closely related to base p!
Let me try to weave together the "tower" depiction from the article with the "numbers that continue infinitely to the left" explanation from crazy_yeti through a couple examples. I'll mainly use base 10 for straightforwardness (the p-adics are usually studied for prime p, but you can have 10-adic integers).
Example 1. What integer satisfies x+1=0? Well, –1 of course, but here's another answer that's not crazy: ···99999. This is what you get if you calculate 0–1 by long subtraction, perpetually borrowing as you move from right to left. Conversely, if you check ···99999+1 by long addition, you get ···00000 as the 1 carries off into the sunset. (It's satisfying to imagine an infinitely long odometer ticking up from ···99999!)
This should look familiar if you've worked with, say, 31-bit integers, where 0–1 does indeed yield 11111···11 (a binary integer with 31 bits). In a sense, 31-bit integers are trying to be 2-adic integers, they're just subject to the limitations of the computing architecture. They're the 31st level in the tower.
Example 2. What integer made of 3's and 6's is divisible by every power of 2? That seems like an absurd question... and yet:
6 is divisible by 2
36 is divisible by 4
336 is divisible by 8
6336 is divisible by 16
It turns out you can keep going forever. At each stage, either a 3 or a 6 prepended on the left will buy us another factor of 2. Changing the digits on the right doesn't pay; any multiple of 4 will have to end in 36, for example, and therefore so will the multiples of higher powers of 2, since those are necessarily multiples of 4.
Since the established digits never change, there is some unique infinitely-long-to-the-left number made of 3's and 6's that encapsulates everything there is to know about this problem; its last N digits will always be divisible by 2^N. This number "starts" ···333636366336 and doesn't have a predictable pattern. This is no more disturbing to me than knowing that there's a real number called π that encapsulates everything there is to know about the ratio of a circle's circumference to its diameter, but has infinitely many digits that require effort to compute and don't have a predictable pattern!
Example 3. There's an intriguing phenomenon of integers that appear at the end of their own squares:
52 = 25
252 = 625
6252 = 390625
906252 = 8212890625
8906252 = 793212890625
I bet you can guess what comes next in this sequence! (Just look at the squares on the right side, they're telling you what to do.)
Again we have the phenomenon of adding digits toward the left, while the existing digits remain stable. So we can say there is a nontrivial 10-adic solution ···2890625 to the equation x2 = x. (The "trivial" solutions are x = 0 and x = 1.) There's one other nontrivial solution. Y'all might enjoy finding it and figuring out how it relates to the solution ···2890625.
posted by aws17576 at 10:20 AM on October 22, 2020 [16 favorites]
Let me try to weave together the "tower" depiction from the article with the "numbers that continue infinitely to the left" explanation from crazy_yeti through a couple examples. I'll mainly use base 10 for straightforwardness (the p-adics are usually studied for prime p, but you can have 10-adic integers).
Example 1. What integer satisfies x+1=0? Well, –1 of course, but here's another answer that's not crazy: ···99999. This is what you get if you calculate 0–1 by long subtraction, perpetually borrowing as you move from right to left. Conversely, if you check ···99999+1 by long addition, you get ···00000 as the 1 carries off into the sunset. (It's satisfying to imagine an infinitely long odometer ticking up from ···99999!)
This should look familiar if you've worked with, say, 31-bit integers, where 0–1 does indeed yield 11111···11 (a binary integer with 31 bits). In a sense, 31-bit integers are trying to be 2-adic integers, they're just subject to the limitations of the computing architecture. They're the 31st level in the tower.
Example 2. What integer made of 3's and 6's is divisible by every power of 2? That seems like an absurd question... and yet:
6 is divisible by 2
36 is divisible by 4
336 is divisible by 8
6336 is divisible by 16
It turns out you can keep going forever. At each stage, either a 3 or a 6 prepended on the left will buy us another factor of 2. Changing the digits on the right doesn't pay; any multiple of 4 will have to end in 36, for example, and therefore so will the multiples of higher powers of 2, since those are necessarily multiples of 4.
Since the established digits never change, there is some unique infinitely-long-to-the-left number made of 3's and 6's that encapsulates everything there is to know about this problem; its last N digits will always be divisible by 2^N. This number "starts" ···333636366336 and doesn't have a predictable pattern. This is no more disturbing to me than knowing that there's a real number called π that encapsulates everything there is to know about the ratio of a circle's circumference to its diameter, but has infinitely many digits that require effort to compute and don't have a predictable pattern!
Example 3. There's an intriguing phenomenon of integers that appear at the end of their own squares:
52 = 25
252 = 625
6252 = 390625
906252 = 8212890625
8906252 = 793212890625
I bet you can guess what comes next in this sequence! (Just look at the squares on the right side, they're telling you what to do.)
Again we have the phenomenon of adding digits toward the left, while the existing digits remain stable. So we can say there is a nontrivial 10-adic solution ···2890625 to the equation x2 = x. (The "trivial" solutions are x = 0 and x = 1.) There's one other nontrivial solution. Y'all might enjoy finding it and figuring out how it relates to the solution ···2890625.
posted by aws17576 at 10:20 AM on October 22, 2020 [16 favorites]
wikipedia maths article syndrome
Yes! This is a huge pet peeve of mine. An encyclopedia should offer an introduction to a topic to someone coming in cold. And in math, Wikipedia fails spectacularly. For sure, there are topics that you simply can’t come to cold so it’s a hard problem but Wikipedia doesn’t even try. They might as well just redirect to Wolfram MathWorld.
posted by sjswitzer at 11:58 AM on October 22, 2020 [5 favorites]
Yes! This is a huge pet peeve of mine. An encyclopedia should offer an introduction to a topic to someone coming in cold. And in math, Wikipedia fails spectacularly. For sure, there are topics that you simply can’t come to cold so it’s a hard problem but Wikipedia doesn’t even try. They might as well just redirect to Wolfram MathWorld.
posted by sjswitzer at 11:58 AM on October 22, 2020 [5 favorites]
But FWIW, I actually did find the Wikipedia article pretty helpful in this case. The introductory section is quite clear (to me).
posted by sjswitzer at 3:42 PM on October 22, 2020
posted by sjswitzer at 3:42 PM on October 22, 2020
Another way to think about the p-adics is to focus the different metric that applies to p-adic spaces. Two numbers are p-adically close if they share the same powers of p. The p-adic absolute value is the reciprocal of the highest power of p in the factorization of the number.
40 is 2-adically close to 24 because they factorize 2^3 * 5 and 2^3 * 3 respectively. The higher the power they share, the closer. However 40 and 41 are distant from each other because 41 contains no power of 2 in its prime factorization.
This alternate definition of distance leads to an alternate version of calculus and geometry, which along one path at least, leads to an alternate p-adic version of string theory with p-adics instead of the real number system.
posted by likethemagician at 9:39 AM on October 23, 2020
40 is 2-adically close to 24 because they factorize 2^3 * 5 and 2^3 * 3 respectively. The higher the power they share, the closer. However 40 and 41 are distant from each other because 41 contains no power of 2 in its prime factorization.
This alternate definition of distance leads to an alternate version of calculus and geometry, which along one path at least, leads to an alternate p-adic version of string theory with p-adics instead of the real number system.
posted by likethemagician at 9:39 AM on October 23, 2020
Oh man, I have to backpedal even more. Wolfram MathWorld gives extremely short shrift to p-adics.
posted by sjswitzer at 5:26 PM on October 26, 2020 [1 favorite]
posted by sjswitzer at 5:26 PM on October 26, 2020 [1 favorite]
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Because, man, I took three quarters of number theory and loved them to bits and I'm not sure p-adics were ever explained to me as a tower? This may be because it's a new conceptual tool or it may be because my professor didn't teach it that way or it may be because my mind is going sure to my advanced age but IN ANY CASE this article made me feel a way about my math education and like it'd be kind of fun to get back into number theory. Where are the cool young number theorists discussing things? Is there a subreddit?
posted by potrzebie at 12:33 AM on October 22, 2020 [1 favorite]