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Tom discusses a series that, by one definition, has an unexpected property.
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The one thing that I think would make this more compelling is if there was some explanation of why we would ever want to use something like the 2 adic for distance. There was some hunting toward it being relevant due to p adic being generalizable and better fitted, but starting with that I think would have brought some more context to why we inventing this in the first place. It kind of feels like we are creating this method of finding distances in order to show this strange result rather than this strange result being a product of something that has more obvious uses.
Grant Sanderson has an old video on the 2-adics ("what it feels like to invent math" I believe), and there's one excellent SoME2 submission on the topic
@@blobberberry came here to say that glad I was beaten to it lol
@@blobberberry - I agree. this video didn't cover the idea of representing rationals, or negatives which i find far more compelling. I also remember a video that cover the topic but called them something like "reversemals"
There! What Alan said! Well said. Because you've stated this in the context of "youtube" viewers and youtube presenters. Gold star.
The easiest place where I know that something similar shows up is the way computers interpret negative numbers. -1 is, in some systems and standards, the binary string 11111….111, with as many ones as you can store in one integer in that system. Since addition can‘t (in this system) handle carries to the place a space to the left of that, adding 1 to 111…111 gives you 000…000 (with binary addition). So the number 111…111 has a reason to be called -1. The reason can be strengthened by talking 2-adic, since the binary sequence 1, 11, 111, …, (2^n -1) converges 2-adically to -1.
I think the biggest problem with this video is that Tom didn't fully explain what the p-adic numbers actually are, which makes this distance function seem arbitrary without context. It's not just that this function is technically a distance function because it follows rules x, y, and z; this function literally defines what distance means for the p-adic numbers. Eric Rowland posted a great video on the p-adics a while back that I think gives much needed context here.
You don't need to introduce the full p-adic numbers to introduce p-adic distances. It's just like how you don't need all the real numbers before you introduce the usual absolute value. BTW, it turns out the the ONLY absolute values on the integers (equivalently rational numbers) are the usual absolute value, and the p-adic absolute values for each prime p, so p-adic absolute values are "natural" in some sense. Perhaps some applications of the p-adic distance could be motivated, so here's why might care. In number theory, to show that equations don't have integer solutions, a common technique is to look at their remainders. Consider x^2 + y^2 = 3z^2 and suppose (x,y,z) is a solution with smallest absolute value (in the usual sense). Squares can only have a remainder of 0 or 1 when divided by 4 (if x is even, say x = 2y, then x^2 = 4y^2 = 0 mod 4, if x is odd, say x = 2y+1, then x^2 = 4(y^2+y)+1 = 1 mod 4). So the LHS can be 0,1,2 mod 4, and the RHS can be 0,3 mod 4. These are only equal if both sides are zero mod 4, but then x,y,z are even, so divide to get a smaller solution, contradicting minimality. A natural question now arises. Is it enough to consider only congruences (remainders) to show that polynomial equations have no integer solutions. The answer is no (3x^3 + 4y^3 + 5z^3 = 0 has no integer solutions, but does have mod n solutions for all n), but using congruences is still an extremely useful technique. The p-adic numbers allow you to see if an equation has these congruence solutions for all powers of a prime p. Doing this for all primes allows you to see if such an equation has mod n solutions for any n. Here's a formal statement: If F(x_1,...,x_n) is a polynomial in any number of variables, then F(x_1,...,x_n) = 0 mod p^m has an integer solution for all m if and only if F admits a p-adic solution. Note that the congruence only has to fail once for there to be no integer solutions, so this is a very powerful technique indeed! Moreover, the p-adic numbers have a "geometry" and "topology" much like the real and complex numbers. As such, just like there is a study of differentiable manifolds, complex manifolds/varieties, there is an analogous subject of p-adic manifolds (usually called rigid analytic spaces) and p-adic varieties. The geometry of these varieties give immense insight into number theoretical problems (and pure geometry problems as well, even for real and complex geometry). Peter Scholze, arguably the leading mathematician in the world today has done essentially all his research in the world of p-adic numbers, p-adic geometry, p-adic Hodge theory, etc. This is one of the most active fields in mathematics today. p-adic numbers were omnipresent in Wiles' proof of Fermat's last theorem. They are everywhere.
@@theflaggeddragon9472 I'm not the biggest fan of reading but You have my respect for writing this text to explain something to someone.
@@satyam2922 LOL I hate reading too, dw
numberphile has officially exceeded my education level
I also wish Tom solved for -1 as the limit instead of plugging it into the distance function and checking.
P-adic numbers are cool because they don’t just define a new distance, but an entirely new _calculus_ where you can still take derivatives and infinite series, but limits which didn’t exist in the real numbers suddenly exist in p-adics. There’s even a sense in which e and pi can be found in certain extensions of p-adic numbers.
What is e and pi in extensions of the p-adics? Closest I've seen to e is log[eps] in Bcris
@@theflaggeddragon9472 The Taylor series represention of the function f(x)=e^x yields a series that happens to converge in pZ_p (the p-adic integers with valuation at least 1). This means e^p (or at least something similar to it) is a p-adic number, meaning e is algebraic over Q_p (again, _in a sense_ ). You can do something similar with pi/p using sin and arcsin, but it has a few more steps.
My favourite thing about p-adic numbers (and ultrametric spaces in general) is that every triangle is isosceles. Fun stuff.
1d is point and line 3rd point get plane 2d
Also, the center of a circle isn't unique: every point inside a circle is a center.
@@ExplosiveBrohoof I think you mean inside a _ball_
The missing part is that there isn’t just one way to organize numbers on a number line. If you reorganize them to adhere to a p-adic system, they will now be in a point cloud where the distances you measure between them is now aligned with the p-adic formula.
Thank you. That makes so much more sense.
Fun fact: it's not just any point cloud, but a quite familiar one. The p-adic integers are topologically exactly a Cantor set. :) (This is easiest to see for the 2-adics; rewrite everything as binary, then reverse the digits, replace 1's with 2's, and reinterpret as ternary. The digit reversal makes it follow the normal metric again.)
@@gavinjared1135 mind is pleasantly blown, of course it's Cantor
For all you programmers out there, this is very related to “two’s complement”!
In what way?
I feel like I remember reading about exactly this in Hacker’s Delight. Also finding the largest power of two that divides an integer in binary is very simple, it’s just the ‘count trailing zeros’ function, which is actually a single instruction on many ISAs.
N-bit signed integers in two's complement is just modulo 2ⁿ, but thinking about it as infinitely long 2-adic integers truncated to the last N bits opens so many possibilities. Now fractions, square roots etc. can be represented by integer types as long as they are 2-adic integers.
@@wearwolf2500 computers deal with n-bit integers, which form the ring of integers mod 2ⁿ. That's just chopping off the 2-adics after the first n bits. They are an approximation of the 2-adics. Same way you approximate numbers with n decimals.
@@wearwolf2500 Simplifying down *a lot*, we can think of computers as if they’re doing math in base 2, mod 2^n for however many bits “n” are reserved for a number. If you ask this computer to remember the number 2^n, it will overflow back to zero, because it can only hold onto n of the digits, starting from the least significant. Thinking in terms of the 2-adic distance, this computer can’t distinguish numbers as “close” as 2^(-n) ! Two’s complement is a programming trick to turn subtraction (hard) into addition (easy). The idea is to take advantage of the fact that this computer thinks in mod 2^n, like a really really big clock. Instead of using -x, you find what 2^n -x is, which there’s a simple algorithm for. Then you perform that addition, and the extra 2^n goes away because overflow. -1 is the same as 1111…1111 mod 2^n, which is the same as saying “we’re only paying attention to the least significant digits in base 2”, which is the same as saying “really close in 2-adic distance”.
Additional fun fact about the p-adic metric: In some ways is better than the usual distance of d(x,y) = |x-y| because for any p-adic distance we have the strong triangle inequality d(x,z)
But actual, physical distance doesn't follow this law. If A, B, C are on a straight line then d(A,C) = d(A,B) + d(B,C).
That’s correct. Our usual Euclidean metric does not satisfy the ultrametric condition. This does not mean the p-adic numbers are not a metric space though. They just have the stronger property of being an ultrametric space. To see how much weirder this is than our usual notion of distance think about what an open ball of radius 1 around 0 would be in the 2-adic numbers vs the open ball around 0 in the real numbers.
@@miloweising9781 Yup. Because of the strong triangle inequality the unit ball in the 2-adics actually forms a ring because it's closed under addition! The unit ball in the reals is definitely not a ring.
@@normanstevens4924 Yes, the "usual" metric doesn't obey the strong triangle inequality. Just the weak one.
And because of (thanks to) that, things go 'really fast' in the p-adic world, which allows for cool stuff to happen analytically.
This has -1/12 vibes EDIT: Alright everybody chill, I know it's p-adic distances, I just said it has the vibes of the "-1/12" video because of the original silly statement. Of course you can also say that 5+5=12 but in the octal number system, relax please
Is the -1/12 the only controversy in the You-Two-ber community?
@@asheep7797 no iDubbz is a simp, that was also a pretty big controversy
But he stresses that it tends and is not equal. So, def a responsible take on the problem.
@@zerosiii lol
It seems like he uses book cover to wright math
1:00 Numbers getting bigger converging to a negative number? Oh no, here we go again!
In this case it works. But yeah...
I feel like one thing that's missing in the explanation of "why are these the rules for what a distance function is", is that these are exactly the rules we need in order to be able to speak about convergence, and have nice properties like for example uniqueness of limit points.
I'd like to address the question of "why are these our axioms for distance functions?". The answer is pragmatic: because it works in a lot of contexts. You can't illustrate this very easily, but in a lot of contexts that involve "distance", these properties are enough to prove things and develop tools like convergence. Mathematical definitions are often like that: something is studied in lots of different contexts until someone brings it all into a single theory. In that sense, the definitions are more important than theorems, and are harder to fully appreciate.
Sneakily glossed over proving that d(x,y) = 0 iff x = y. I guess you’d need to separately define this as true for the 2-adic metric, since 1/2^m can’t ever be zero.
That definitely deserved mention. There's no proof per se since it's actually supposed to be part of the definition. But since every power of 2 divides 0 the only reasonable value for d(x,x) is lim[k -> oo] 1/2^k = 0
@@martinepstein9826 yes, it bothered me too, until I did the calculation. But the answer was either going to be annoyingly hand-wavy or too long-winded for this video.
I was looking for this comment😊
@@martinepstein9826 yep, that could have been the argument but it was left out for whatever reason.
@@necropola exercise for the viewer
Agreeing with a lot of the comments here. Some understanding of what p-adic numbers are used for, either in the real world or some basic understanding of what they're used for in math, would have gone a long way to dispelling "this is a cool limit off a technicality"
This video is not about p-adic numbers, though. It is about p-adic metrics. How are you people failing to understand this? The video was pretty explicitly about what it is about.
Somehow you stay on the most topical mathematical ideas present in even the furthest removed places of mathematic academia. Thanks, for that or at least what I think of it.
For what it's worth, the p-adics were first described explicitly in 1897. That's not quite ancient history, but in terms of math the frontiers have expanded quite a bit since then. That being said I'm very glad this topic is being shown to more eyes. If it's new to you, if it makes you excited and interested, that's all that matters!
Eric Rowland has a beautiful video on explaining p-adic numbers using 3b1b animations
Finally, a video on the _p_-adic numbers!
I notice that Tom uses the word "modulus" for what I have always heard referred to as "absolute value" (unless I misunderstand what he's doing). Is this a difference between American and British mathematical terminology? (I'm in the US.)
Modulus is the same as absolute value for real numbers, but it is more general, for example it applies to complex numbers. The modulus of 3+4i is 5.
@@aaronhorak710 Interesting. In the programming word 'modulus' tends to be used for the modulo operator - eg: "5 modulo 3 = 2". The 'remainder' part that's returned is the 'modulus'. Never heard it used for an 'absolute value' equivalent. Learn something new every day!
@@michaelcondon9806 I'm in the same boat as you. I thought modulus was related to modulo (or maybe synonymous with modulo), meaning that it would have something to do with remainders. I appreciate Aaron's explanation. If I understand correctly, modulus then would be the distance of a point from the origin, in the complex plane (with the distance expressed as a positive real number).
@@Paul71H They are very much related, mathematically and etymologically. The number to which one counts in modular arithmetic is called the modulus. _Modulus_ is a diminutive Latin noun derived from the Latin noun _modus_ "measure". One might use modulus if translating something like "4 beats to the measure" or "one measure of lemon juice". Modulus is a common word in physics and engineering meaning a thing measured. There were a lot of quantities being discovered that it is interesting to measure but because the need to measure them only arose in the modern era they had no historical name. That's why you have at least four types of elastic moduli (bulk, shear, flexural, Young). It's not all that different to the prevalence of "parameter" in so many disciplines. The second half of "parameter" itself comes from an Ancient Greek word, also a noun, for "measure".
@@michaelcondon9806 Compsci is forever doing things to make mathematicians cry. In the expression 15 mod 7 = 1, the modulus is 7. Would probably have to look it up in Latin (since that's what Gauss wrote his book in) but in modern English 15 is the dividend and 1 is the remainder.
I worried that this was going to go approximately like the -1/12 video but this was very well-explained. Nice!
What is the distance function that is needed for getting -1/12 as the the sum of the whole numbers?
I also heard the start and immediately thought "they're not doing -1/12 again, are they?" lol
@@mesplin3 There is no distance involved in the sum of whole numbers. Sum of whole number is just equal to -1/12.
@@bunderbah I disagree. When adding a whole number, the resulting sum is larger and a series of sums of whole numbers starts at 1. So any chosen value will be surpassed at some point. 1 = 1 1+2 = 3 1+2+3 = 6 ... So I would say that this series diverges, rather than converges. In addition, -1/12 is already smaller than the first term of a series that increasing.
"We're going to look at a sequence and show that it converts to a limit you weren't expecting." He forgot the "...in this space that you had no reason to think about."
Paper looks 2d
@@thej3799 Indeed. So?
This is the maths equivalent of clickbait
@@Nick-LabI still say that's total bunk. The sum of two natural numbers > 0 is another natural number strictly greater than the two being added. P-adics are total nonsense that lead to being able to prove things like 1+1=1.
This seems related to -1 in 2's complement binary representation being represented as all ones the same as the largest positive binary number for a given number of bits. To me it makes a lot of sense thinking about it this way. 2-adic distance probably reveals something about the similarity of the binary representation of the numbers
That's exactly what it is. Two's complement is like truncated p-adic numbers. In two's complement we have 2^n=0, in 2-adic numbers we have 2^infinity=0
Worth mentioning: The reason these "distances" matter isn't just "pure math." There are different number systems (not everything is Base-10, such as Binary, making this math valuable for computer science) and when things enter into "real world" numbers (like, say physics equations), it can be very difficult to see how things relate in "normal" number-space because they seem to have no similarities, but if you transform them into "arbitrary" number-spaces, you can find relationships and trends and such that can help you work BACKWARD to find something that WASN'T related, but all of a sudden can be shown to matter in whatever thing you're doing. I'd hazard you might find a ton of this math used in "real world" applications via things like String Theory or Quantum Mechanics, or, as I'm sure Tom is familiar, Navier-Stokes work.
It's the -1/12th thing all over again!
This is pure math. The pursuit of abstract beauty
Does this video demonstrate that the p-adic metric satisfies d(x, x) = 0? I feel like it was brushed over
It was entirely skipped. fun times.
It's not so bad to 'show', though. If two numbers are equal, their difference is 0. All powers of 2 divide 0, so, strictly speaking there is no largest power of 2 dividing the difference. That said, you can probably somewhat convince yourself that since all powers of 2 divide the difference, even really really big ones, then you have 1/really big as your distance, which approaches 0. The reverse implication i.e. that having a distance of 0 implies the numbers are equal is even tougher to give a wishy-washy explanation for, so I'm just gonna not do it.
And now I'm curious about the 2-adic space convergence values for various entries in the OEIS.
You've always been able to explain these things very well. I feel even if I were a few tabs deep, I'd understand clearly.
I think what would make this interesting and understandable for the layman is representing the number in base p (even base 10 would be fine in this case) and see the following: Normally things converge when the left digits are constant and the rightmost digits approach the goal. Here a sequence converges if its rightmost digits are constant and the *leftmost* digits approach the goal!
Brady’s Banana-Volkswagen example is a perfect example of “the fallacy of the missing middle.” He nailed it … but for the “check ALL boxes” requirement. So the math holds.
Brady’s skepticism here is all of us
Slight revision of what Tom has written: the 2-adic metric on x,y is defined as 1/2^m where 2^m, not m as he’s written, is the largest power of 2 that divides x-y.
Not being an expert in p-adic number, I just couldn't get my head around how ⋯999 could have the limit -1. I was thinking about 2-adic numbers (like exemplified in the video). I took me way to long to understand that ⋯999→ -1 only for 10-adic numbers. I feel that the video could have explained this important distinction a bit more carefully as this would have save me some hours of doodeling. I vaguely remember having similar problems with other video where Tom was just a bit to imprecise to allow for own further investigation.
10:10 What about d(x,y)=0 x=y ? m is infinity because the reminder of 0 divided by 2^m is always 0 ?
If m is the largest power of two which divided x - y, then we should have gotten 1/2^2^5 in the example. Also in the video wad not shown, why this distance function is zero iff x = y. If x = y, then x - y = 0 and then m is undefined as every power of two divides 0. But d(x, y) = 0 will never hold, as 1/2^m will never be equal to zero assuming there is **one** m. You could take the limit with m to infinity, but i don't see why this would be reasonable here ad any power of two matches the condition.
M is the largest power of two that divides zero so m would be the limit of infinity. But he should have addressed this not obvious point. Also he miswrote the definition as you noted.
No power of two matches the condition: You are supposed to take the largest power of two that matches. Because of this, it's normally definite explicitly that the distance functions returns 0 if both numbers are equal (or the slightly more formally correct path via the full p-adic numbers and their subtraction and absolute value for which |0| = 0 is a pretty comfortable decision)
@@megaing1322 Yes, you could define that d(x, x) = 0 for the 2-adic metric, but it should hsve been explained. And in the form they presented here, this was not given.
@@kippy1997 The greatest number means to me, to find the maximum, not the supremum. Therefore we cannot find m as the maximum doesn't exists for {2^k}.
I am glad someone else noticed the 0 distance problem. I just spent few minutes searching for an explanation online and then proceeded to search through comments.
I think Tom missed the chance to glimpse that 'distance' could mean the 'proximitiness' or 'likeliness' between two numbers in any sense whatsoever, one of which the shared presence of them in a given interval of the real line (which is the 'distance' we are used to call by this name, and which has a physical correspondance to our world). This also explains why the distance between a number and itself must be zero (a number shares all of its properties with itself), as well as the commutative property remains valid (they don't need to be ordered to be compared). For the triangular property, it seems they want to restrain the comparison to non-cyclic variations.
One of the best motivations for why you would want to introduce the 2-adic or any other p-adic distance is Ostrowski's theorem which says that any "absolute value" on the rational numbers is equivalent either to the normal absolute value or a p-adic distance. Where "absolute value" has the special meaning of any function |x| that satisfies: (1) |x|>= 0 with |x| = 0 only if x = 0, (2) the triangle inequality |x+y|
I couldn't quite follow this but I'm intrigued and will revisit the video. Thank you!
I find this with all Toms videos. Miss James Grimes he was able to explain it right down to everyones level and then build on that.
This was the first Numberphile video that made me feel uncomfortable. If it went completely over my head I'd feel better.
I'm so glad a video on this topic has been added to the channel! Super interesting stuff!
I think what makes the 2-adic metric not feel like what I mean when I say distance is that d(x,y+n) decreases as n increases. That "as n increses" is what I mean by distance, so if the distance function disagrees, it's hard to take it seriously as a distance at all. Not that it still isn't an interesting concept, cuz it is and I love this video
Eyyyy, p-adic numbers! Been waiting so long for that
What's the algorithm for counting Tom's new tats? 🤔
I didn't see why the distance from a point to itself is zero in your p-atic distance formula.
Strictly speaking, it isn't. But it kinda is if you can pretend infinity is allowed as a power of 2 - which is admittedly complete nonsense.
@@fahrenheit2101 ah, if we can pretend, then we can pretend anything
So much effort to avoid another -1/2 standoff situation ... i like it
Thank you for not picking the obvious clickbait title.
I feel like saying that these three properties for a metric space are exhaustive at defining the notion of distance is not the best way to go about saying why we use those conditions. In reality, the reason that we use those three properties is because we found that most of what we want to do with distance relies solely on those properties, so we can generalize our work on traditional distance to other functions that happen to satisfy those same conditions.
Indeed, especially since there are weaker (topology) as well as stronger (norms / scalar products) concepts that will also be used depending on the requirements of the problem at hand.
Okay, I think I figured out how this measures distance. It turns the numbers into binary, subtracts them and then counts how many 0s the result has on the right. For example 33 and 5 → 100001 - 101 = 10100 and that has two 0s on the right, so the distance is 2 and actually 28 is divided by 2² as the highest power of 2. How is that at all useful? Somebody knows, I guess.
That's bound to be useful in deep level programming, the design of processor chips and the like. An efficient way to measure the size of binary numbers is bound to come up a lot.
This feels like one of the best examples of how professional mathematicians make up absurd rules, faff about with those rules to find the absurd edge cases, and then use those absurd results to show something fundamental, amazing, or even just curious about mathematics. It takes a certain amount of: "ok, it's superficially ridiculous, but just follow me for a moment..." and then magic happens. :)
If you want to learn more about p-adic numbers, Eric Rowland has an excellent video explaining their properties.
You don't really need to go through so many steps to know the distance will get smaller and smaller because you can notice straight away that the modulus remains unchanged while the divisor keeps getting larger and larger, and the Difference between the dividend and the divisor is always 1
"At least, it still fits our concept of distance ... you can't have a non-positive distance, negative distance makes no sense", said while he absolutely thrashes with the concept of distance
Dude looks like he listened to BMTH's Count Your Blessings and both loved it and took it literally.
Excellent album.
@@TomRocksMaths just as I expected
Learnt this in 3B1B's Video
10:06 it is possitive but it wouldnt give you 0 if x = y right?
The most reasonable extension of this definition gives 0. The definition as written does not give you anything for x=y, since you can't find a largest power of two that divides 0.
@@megaing1322 Assume d(x, x) = b > 0. This means an m exists so that b > 1/(2^m) > 0. As 2^m divides x-x it must follow that d(x, x)
What I now wonder is whether the distance axioms are enough to ensure that every sequence under every distance function are guaranteed to converge to at most 1 limit. Or whether there is some Sequence S and distance function D to where as S_n(to denote the nth element of the sequence) continues as n tends towards infinity that D(S_n,L_1) tends towards 0 and D(S_n,L_2) tends towards 0 for numbers L_1 and L_2 where L_1 =/= L_2.
In any case, here is a simple proof of uniqueness: Let (X,d) be a metric space, where X is a set and d is a metric. Let (x_n) be a sequence that converges to x in X, with respect to d. Suppose there exists y in X such that (x_n) also converges to y with respect to d. Then, d(x,y) 0 and d(x_n,y)->0 in the usual Euclidean sense. This also means that d(x,x_n)+d(x_n,y)->0 as n goes to infinity (this is just limit algebra for real numbers). Given that the aforementioned sequence converges to 0, for any natural number m, we can find indices n_m such that d(x,x_(n_m))+d(x_(n_m),y) < 1/m. In that regard, d(x,y)
It’s not the normal distance, it’s a particular type so distance, that will be useful in the next video?
FTW I have always seen the triangle law called the 'triangle inequality'.
I left this video with more questions than answers. Not saying it as a bad thing. In fact, it makes it quite interesting.
It is worth watching other videos on p-adic numbers. It is a counter-intuitive topic so hearing other explanations might help wrap your head around the topic
Excellent video, very much enjoyed it. I do feel like explaining the Manhattan norm at around 15:00 (and why it's named the Manhattan norm) could have bridged the gap a bit on justifying the definition of distance. Otherwise fantastic.
So from my understanding 2 adic measures sort of the average distance a function deviates over a unit, so it is a distance to an extent but I am unsure how it would hold up for different functions.
If I understand it right, by redefining the way you calculate the distance you kind of "bend" the number line/field such that you redifine the concept of infinity along the negative x-axis of the distance values sorted by how fast the sequence grows?
It is more subtle than that. For example usually sequences that diverge to infinity still don't converge and when they do you can also get a positive limit. Convergence against 0 in this distance means that the terms of a sequence eventually become divisible by arbitrary large powers of 2 so they usually do have to grow as a consequence (or become 0). Convergence against any other limit just means that the sequence that is offset by that limit has that property.
With the 2-adic method don't your distances wobble up and down? If the difference between the two values is odd then m always has to be 0 because there are no powers of 2 in odd numbers. So as you increase one of the numbers it's going to keep jumping between 1 and 1/(2 to a larger and larger number). Seems like a strange thing for a distance function to do but I guess that's based on the general idea of distance.
The distance doesn't have to increase/decrease/stay the same. You can define a very "basic" distance like this: If (a=b) then dist=0, if not, then dist=1 This is called Discrete metric (metric means distance, basically) Depending on your sequence, the distance can wobble all you want even with the standard euclidean distance. How far is the sin function from the X axis at any given point? It's gonna wobble between 1 and -1
You need both infinities to have a number then to count them
The distance "wobbles" from the perspective of our intuitive(*) conception of distance that we've picked up from our everyday experiences, which we like to think of as a 'number line'. However, under the perspective of a 2-adic metric, there is no wobble or number line - the numbers that are close together under this metric just are close together, and the numbers that are far apart just are far apart. (*)We should also remember that some of this intuition is from our modern perspective, given that the number line itself is only a few hundred years old and even negative numbers took a while before they were accepted. Obviously there's still something that's more widely intuitive about (approximately) Euclidean space, but we should keep in mind that at least some of our intuitions aren't wholly "natural" if you know what I mean.
The thing is that the distance function induces a kind of shape or geometry on a given set. Our usual distance induces the shape of the straight line on the numbers - the number line. But the p-adic method induces a totally different shape. I don't know if we can even imagine that shape. Certainly it's not a line. If it was a line again then the p-adic distance probably wouldn't be very interesting. What I want to say is, the wobble is not a bug, it's a feature. :)
Wobbling it has to. Like if you know the factorization of a number n, you know almost nothing about the factors of n+1 (apart form being different from the ones from n.)
Regarding HOW CAN THERE BE DIFFERENT DISTANCES (I'm not sure why Dr Crawford struggled with explaining this) - this should "click" for you: if two people live on 2 sides of a city, the shortest distance between them by LENGTH will indeed be close to a straight line. But if the city center has a speed limit and/or lots of traffic lights, then driving through it is probably not the "best" route. It's better to get out to a highway that encircle the city, which looks like a detour (LENGTH-wise) but it's actually shorter by TIME. There's might even be a third shortest distance by FUEL ECONOMY (or in math terms, the same graph can have different "shortest distance" between nodes, depending on the set of weight per edge).
An issue with using that comparison is that all those metrics are still equivalent in the sense that if a sequence converges to a limit in one metric it will also converge to the same limit in the others. So those examples are somewhat helpful but they do not really do justice as to how fundamentally different the 2-adic metric is to the usual distance metric. It seems actually quite hard to come up with an easy to understand example that captures how different these distance measures behave.
A lot of the confusion arises from the fact that most people are simply not familiar with axiomatic definitions.
Oh thank goodness you guys use the 2-adic system
19:10 I strongly think the sequence 8, 88, 888,... doesn't converge at all. It certainly doesn't converge to -8/9 (in 2-adic metric). You'd need some increasing power of 2 which divides the members of this sequence after adding some constant. I don't see how any (even negative) power of 2 should divide e.g. 888 - (-8/9) = 888.88888...
Yh that bugged me, too.
This question about the 8s was an unscripted question which requires a slight extension of the 2-adic theory described here to rational numbers. Had this been scripted then it could be answered by discussing this extension. There is a related formula for 2-adic distances between rationals. Now the example you give is the rational 8000/9 = 2^6 * (125/9) so m =6 here. From the sequence argument conclude the distance limit is 0, so the sequence limit is indeed -8/9 in the 2-adic rationals.
@@roys4244 Thanks for pointing that out. I was wrong and realize now that the sequence 10*n * 8/9 converges to 0. 😬
Did Tom do some Parker Square stuff at 19:23 ??
8:22 "negative distance makes no sense" Of course, because everything else in this is making sense
As for 2 adic metric distance between 0 and 8 is less than distance between 0 and 4 in a number line
This is true. In fact, for any two 2-adic integers, their 2-adic distance will be less than between 0 and 2 with the regular absolute value. As long as there is no power of 2 in the denominator of a-b, d(a,b)
For the complete verification of the definition of distance, the property whereby d(x,y)=0 if and only if x=y is missing. But if x=y then d(x,y) is the greatest power of 2 that divides 0, and what is this power? Is it considered to be "2^infinity" (in some sense) because 0 divided by something always equals 0? And therefore d(x,x)=0 because "1 divided by infinity" equals 0 (within the usual limits with the metric of real numbers)?
Yes, technically this would require a more precise phrasing of the definition but I can see why they skipped that point because that would lead to more confusion in the general audience.
I don't know if I am mistaken but the second rule of the first Axiom doesn't seem to hold. the 2adic distance between 1 and 1 seems to be 1/2^0 and not 0
We're looking for the largest power of 2 that divides 0. 2^1 divides 0, so d(1,1) =< 1/2^1 2^2 divides 0, so d(1,1) =< 1/2^2 2^3 divides 0, so d(1,1) =< 1/2^3 etc...
I've a question, how many dimension would it require to represent n points in a graph(coordinate not network) in this 2-adic metric? if Anyone have the Answer please tell me [I worked it out for 1,2,3,4,5 seems like we need 5 dimensions so maybe the Answer itself is n but I don't know distance between 2 and 4 was 1/2 this was interesting otherwise I only would've needed 4 dimensions]
This has really piqued my interest, I'd love to see some follow up on p-adic numbers!
One should underline a property of distance : if a sequence converges (in the sense of any given distance) to a given limit, that limit is unique. Not hard to work out with the three properties of distances, though.
Things like this is why I quit math.
This topic is crazy interesting
I'm not great at math but I usually follow these pretty well but to me this just feels like inventing some random thing that gets you the answer you want rather than finding what it really converges to. So like an answer looking for a problem rather than finding the answer to the problem. I'm sure I just don't get it.
I feel inventing "arbitrary" ideas in Mathematics that seem counter-intuituve , but still follow "rules", we are familiar with, is the most powerful thing that mathematicians do. Complex numbers, distance functions and analytic continuation as examples. It's like intuition is a hindrance to unwrapping the mysteries of the universe.
19:10 curious about the workings for the 8 sequence
It doesn't. In my opinion this sequence doesn't converge at all.
For me, I would define something as a distance, if i could add that distance to one of the inputs to obtain the other. For example, if we have 4 and 100 as our inputs, and we wish to find a distance, that distance must be able to transform 4 into 100, AND (in the other direction) 100 into 4. for 1/32 to be our distance, i would check whether 4 can be altered by 1/32 to make 100, and whether 100 can be altered by 1/32 (but in the opposite direction) to make 4. What is the function that makes 1/32 work for this requirement?
In the p-adics that requirement that a point and given distance does not determine another unique point. For example d(4, 36)= 1/32 as well. In general so is d(4, 4+32q) has this distance for q coprime witth 2. Eg q=3, 1, 5 etc.
17:28 When using the 2-adic distance function (extended to rational numbers), wouldn't the sequence of 1/2^n tends to infinity instead of to 0? The largest power of 2 that divides (1/2^n - 0) is 2^(-n), so: d(1/2^n, 0) = 1/2^(-n) = 2^n So the distances both becomes smaller, and tends to infinity?
You're right that the sequence 1/2^n gets further from 0 in the 2-adic metric, but the slightly confusing point is that a metric takes a pair of points to a real number. So when you measure the distances to be 1/2^n for the original sequence you are thinking of those distances as real numbers with the usual metric, where they do converge to 0. (which then implies that the original sequence converges in the 2-adic metric)
Take three numbers in such a progression, a, b, c. Calculate (b^2-ac)/(2b-a-c). This works for the nines and all but the factorial example at the end. It also predicts 8,88,888 going to -8/9.
Cool formula, but only works when the sequence is a constant offset from a geometric progression with ratio divisible by 2.
thank you for the amazing video ! it's always great pleasure and very instructive to watch. in my point of view, the only thing missing is demonstrating that d(x,x) = 0 even though i just realized that it might be a natural consequence of d(x,z)
What about d(x,y) =0 when x =y? Does we assume that any 2^m divide 0 so 1/2^m tends to 0 cause m could be bigger as we want?
Ok, let me parse this, this distance, which doesn't need geometric interpretation per se, is a measure of relatedness between two quantities based on the closeness of one quantity to the position (in the real line) of a power of a "common factor" with the other. And it is a distance from 0 to 1. With 1 being completely unrelated, hence the maximum distance, to 0 being the closest possible, when the two quantities are equal. It doesn't matter how far from each other (it matters but let's wait a moment), in the common distance metric |x - y|, they are, what is important is how close to the closest power of the "common factor" they are. But it takes into account another feature: if they are really apart in the "real line" but they are close to a power of the common factor, in this n-adic metric that coincidence weights the closedness further, they are more related. I hope I am not that far from understanding the definition and then I can see the usefulness of it when studying theory of numbers.
Geometrically, it feels like a rigged line that is spiralling inwards.
One thing is missing: there's no mention of Hausdorff-ness! We don't know that a sequence will converge to a unique limit in this metric space (though it's true!
An interesting fact is that actually all metric spaces are Hausdorff!
@@floyo Yea, as I said at the very end, though it’s true!
@@sleepymalc Oh oke, I thought you meant that we don't know it for a general metric space but that it's true for the 2-adic one.
@@floyo Ah right, I use “this metric space.” Anyway, I think mentioning this fact would be great for such a video since it’s for general audience.
Is there a distance metric that would verify the convergence of the Ramanujan's series { Sum(n = 1..inf) = -1/12 }
When he says "modulus" it's what I would call "absolute value". Is that right?
Yes
Modulus applies to other areas not just the reals so it's more general.
Next video on why we ever want to use 2-adic metric (or p-adic numbers) pls
It definitely could have helped to mention the Manhattan metric, the chess metric, and the SNCF metrics to motivate that specific axiomatic definition of distance. It also could have helped to clearly define the set of numbers we were working with.
This can b used to prove converse of collatz conjecture?
Big fan of you Sir. Inspired by you I have also opened my youtube channel. I don't know if it will work or not but you inspire me to do the hardwork and just don't think about the result ❤
it makes sense if you think about binary numbers 1, 11, 111, ... since every appendment of a one is exactly increasing by 2^1. Hence adding nines in the decimal case tends to the binary case for large enough numbers.
Didn't this sequence overflown the sign bit at the last, the infinite, step?
Sign bit? We're talking pure math here, not programming. There is no sign bit, though apparently there is definitely an analogue with 2's complement binary, but I can't say I'm too familiar to comment on that.
Question on the triangle law (at 8:50). Can it not be that y lies in the path to z such that x,y is a shorter distance than x,z?
Sure, that is why you also have to add the distance from y to z on the right hand side.
Okay so im just trying to get the issue here.. its that its like an inverse distance? right, like as the "distance" increases p-adic metric gets smaller, then when you apply the limit = 0 diference calc from the start that assumed that convergent series had 0 difference?
It's kind of like that, but not quite. Not every number that is large under the usual metric is large 2-adically. For instance, the 2-adic distance between any odd number and 0 is 1, as no odd number is divisible by any power of 2 greater than 2^0=1. So some numbers that are very large are very small, while some that are very large are as big as a 2-adic integer can get.
At about a minute into this video, you had me thinking of _p_-adic numbers.
My brain is cracked and I'm only 50% thru the video. I definitely need a 101 level video to understand this.
I knew it will be the p-adics the moment I saw the title
Is there a version of p-adic numbers with no decimal? Just an infinite sequence of figures with no beginning or end?
My head hurts. Can't wait for the next video.
All those questions and didn't ask if it's possible to converge to 2 different values in the same p-adic distance metric...