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.

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.

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.

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

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_

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.

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

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”.

In this case it works. But yeah...

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.

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

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.

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!

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.

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

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.

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.

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 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.

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.

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

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.

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

Cool formula, but only works when the sequence is a constant offset from a geometric progression with ratio divisible by 2.

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.

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.)

Excellent album.

@@TomRocksMaths just as I expected

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. 😬

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)

-If you ignore the ones that converge to a non-integer value...- Edit: Sure, it works out if you consider each value to represent a equiveillance class of numbers like in modular arithmetic. "Just" interpreting the values as one number, a positive number or a twos-complement negative number, like standard would only cover integer values.

@@JNCressey Not really. in 3-bit binary numbers we have 011=1/3

@@viliml2763, -in what way is 011 equal a third and not just 3?- edit: ah, yes. with overflow, 3 lots is 1. I see, it's *both* 1/3 and 3. So 1/3 = 3 in some sense, like a modular arithmetic equivalency class.

@@JNCressey 011+011+011=001 (note the overflow).

@@gavinjared1135, ah yes. that works

Geometrically, it feels like a rigged line that is spiralling inwards.

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)

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.

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.

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.

This is not true. d(1,2) is the 2-adic valuation of 2-1=1, which is 1.

​​@@gavinjared1135 now i see. There is 1/2^m > 0. And now I see where its is not correctly defined. d(x, x) = 1/inf. To humans its zero, but mathematically it is just makes no sense, so it is not really defined in (x, x). That is why this stuff says that limits of sequences are so strange. As i I said, metrics are equivalent Upd: don't write that 1/inf is a limit of sth, its not, there is already limit in denominator and after that we can't do arithmetically anything with this

@@Serg_144 As has been noted in the comments, properly it should also be decreed as a special case that d(x,x)=0. "1/inf" is just an intuition, I guess. The limits look strange to us, because we are mostly thinking of the number line, and that picture presupposes the usual metric. With the 2-adic metric, numbers are not naturally arranged in a line. In fact, the 2-adic integers form a Cantor set. On the other hand, if you look in two's complement, then this sort of thing starts to look like how -1 is represented as ...111.

Yes

Modulus applies to other areas not just the reals so it's more general.

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.

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)

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.

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)

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...

It doesn't. In my opinion this sequence doesn't converge at all.

It's perfectly well defined for at least rational numbers. You just have to take a negative power of two, so you'd start getting distances that are > 1. For reals I guess you'd need to explicitly define when two numbers are infinitely far apart.

Also remember a point could be circle tan

There is a related formula for giving the 2-adic distance between two rationals. The unscripted question about 8s kind of threw the discussion in that direction, without Tom being prepared to answer it. Had he prepared for this case one would extend the formula to deal with rationals, which can appear when unexpected in these calculations, and then derived the -8/9 result.