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2-Adic Metric: 9, 99, 999 Converges to -1 · Acedra
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2-Adic Metric: 9, 99, 999 Converges to -1
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0:00
If you were to define what it meant to be a banana, you had to tick every box on that list and then you could be nothing but a banana. We're going to look at a sequence and show that it converges to a limit you weren't expecting. So the sequence is as follows: the first term is 9, second term is 99, third term 999. Fourth term - you guessed it - four 9s; and we're going to continue with this pattern. So obviously this sequence goes to infinity.
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We're adding more and more 9s, so in the usual sense this is a divergent sequence; it goes to infinity, it's getting bigger. But I'm going to show you that in fact this tends to -1. This sequence of adding more and more 9s, making the number larger and larger, in a certain situation can be shown to tend to -1. And here's how it works. First of all we need to establish what we mean by saying a sequence tends to a particular number. This might seem like
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something you know, or you've seen before, but this is going to be really important in justifying why this is -1 so bear with me. So if we start with a different sequence; so suppose I have 3/2, 4/3, 5/4, 6/5 and it's going to continue. So we're adding 1 to each of the numerators - numbers on top - and we're adding 1 to each number on the bottom. So the nth term, if we would sort of go along and say pick a certain point in the sequence, what does the nth term look like at that
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position? It's going to be n plus 2 over n plus 1. So first term, plug it in, we get 3 on the top over 2. Third term here this would be n equals 3, we get 5 on top 4 on the bottom. Whatever position down you might say to me, find the 1000th term in the sequence, it's a 1002 over 1001. Now this hopefully you can see is going to tend to 1. As we increase n, we go further and further along the sequence, the top and the bottom get closer and closer together. So we can
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get as close to 1 as we want by going further and further down the sequence. So the way we would show that is you take your nth term - which here is n plus 2 over n plus 1 - and we subtract off what we think the limit is, so here we think it's going to be 1, and then we work out the distance between these two. So I'm calling this our distance function. Now if we're working in the real numbers - which these are all real numbers - then the distance
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function is just the modulus of one number subtract the other. So we take the modulus because it's a distance so it should be positive - you can't really have a negative distance. So here the distance in the real numbers is n plus 2 over n plus 1 minus 1. What is n plus 2 over n plus 1? Well we can simplify that and say, well that's actually the same as n plus 1 plus another 1 all over n plus 1 minus 1. But then we've got an n plus 1 over an n plus 1
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so that's a 1 so we can say that's actually 1 plus 1 over n plus 1 minus 1 so we get 1 over n plus 1.
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So again, with this distance function the usual way of measuring distance, we can show that the nth term of the sequence and what we think the limit is; we can say the distance between them looks like 1 over n plus 1. So as we now go further and further down the sequence what we're doing is we're letting n tend to infinity. We're making n bigger and bigger, what happens to our distance as n gets bigger, the distance tends to zero. So as n tends to infinity, the
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distance between the two things is 1 over n plus 1 which very clearly tends to zero, so we would say this sequence converges to 1. And we've done that using the distance function, which for the real numbers is just modulus of one of them minus the other. I'm going to need more paper. As always. The reason I've introduced the distance function, as some of you may have guessed, is if we're going to show our sequence of 9s is tending to -1 we obviously can't use the
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normal definition of distance because the distance between a larger larger number and -1 is getting bigger so it's it's just not going to work, it's not going to tend to zero. So we need to think about what other definition of distance can we use to show that this sequence is indeed going to -1. And the one we use is called the 2-adic metric - and we will check this is a distance. You say the
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distance between two numbers x and y. So x and y are going to be integers because here in our sequence we're only working with integers so we'll keep things simple. So the distance between two integers in this metric is defined to be 1 divided by 2 to the m, where m is the largest power of 2 that divides x minus y. So this feels a little complicated, so let's look at an example, let's think about what does this mean. Let's suppose x is 100 and y is 4. So in the usual measure of
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distance we would say the modulus of x minus y is the modulus of 100 minus 4, they're 96 apart. Feels hopefully kind of obvious. Now in the 2-adic metric we've changed our definition of distance, so we need to think about what power of 2 divides their difference. So we already know the difference, the difference between them is 96. So if we think of powers of 2 we've got 2, 4, 8, 16, 32, 64. 128 is
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obviously too big so I'm going to stop there. 64, well that's 2 to the 6, but 64 doesn't divide 96. It's smaller than it but 64 times 2 - which would be the next smallest thing you could have - wouldn't give you 96. So we actually need to go down to 2 to the 5, which is 32, which perfectly divides 96. So 2 to the 5 is 32, 32 times 3 gives us 96. 2 to the 5 divides the difference between our
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numbers so, in our definition, we can say in the 2-adic metric the distance between 100 and 4 is 1 over 2 to the biggest power, so here 2 to the 5, so that's going to be 1 over 32. So usual definition of distance that we're all used to, these numbers are 96 apart, 100 and 4. However in this metric they are 1 over 32 apart; which doesn't feel at all natural and- - (Brady: Or useful) - Or useful. I
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mean well that's a whole other conversation. We're entering the realm of pure maths. This- now that you've set me off on this - it is useful because we learn properties about the numbers. So this kind of Investigation, okay this isn't a distance in our world, but we can learn things about distributions of numbers through this. Ao this 2-adic metric can be generalised and is used to actually in fields medal level research to learn about rational numbers. So beyond beyond
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my pay grade, beyond my ability to do the maths, but there is some usefulness in there.
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So we've got our 2-adic metric. So I've just told you this is a way of figuring out how far apart two numbers are, but you need to check from a mathematical perspective that it does everything you would expect a distance measure to do. So we have a set of axioms. In order to be a distance function: it maps from two numbers in our starting space into the real numbers, we get an answer that's a real number. Where d satisfies three things - this one's called positivity. And
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these should hopefully fit with your intuition of what we might expect from a distance. If we have two numbers x and y, the distance between them is greater than or equal to zero. We're talking about distances, okay they might not fit our usual way we think about them, but they still need to be positive. You can't have a non-positive distance, negative distance makes no sense. And in fact you can say and it equals zero if and only if x is equal to y. So
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you only have a zero distance if it's the same point. Again, hopefully fits with our usual notion. Second one is called symmetry, so this says the distance between x and y- so if I start with x and move to a point y, I should obviously get the same thing if I start at y and move back to x. d of x,y is d of y,x - quite straightforward. Doesn't matter where we start. The third one is called the triangle law. Now this one is best seen by picture; if I start at point x and I want to
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go to point z then what this says is moving directly from x to z is always going to be faster than going via some third point y. Again, fits our usual way of thinking about distances. If you're thinking about the usual measure of distance this is saying the shortest way to move between two points in the plane is a straight line. Of course it changes if you're on spheres and changing your geometry. In terms of our function that says the distance between x and z
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has to be less than or equal to - so it's shorter than - going by a detour. So if we did d(x,y) plus d(y,z) written out algebraically. So these three axioms, three rules, need to be satisfied in order for a function d to be called a distance in terms of the mathematical definition of a distance. So we just need to check that our 2-adic metric satisfies these three things; because at this point you're just taking my word that this is a reasonable thing to do, even though we get a
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rather ridiculous answer. So we do want to verify this. Let's write down our definition, 1 over 2 to the m where the m was the largest power of 2 that divides their difference. So is this positive? It's one over a power of 2, that's certainly a positive number. So we can say that works. We're all good. Symmetry: if I do the distance between x and y, is that the same as doing y and x? Well here
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we were finding the largest power of 2 that divided x minus y. So what that means is we're saying 2 to the m divides x minus y. So if we write that out algebraically, what that tells you is that x minus y equals something - let's call it a - times 2 to the m. Because if this is going to be a factor of that there has to be something else. So in our previous example we had this was 96, we had 32 here,
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so a would be 3. It's just that other integer part of our division. So if 2 to the m divides that it, means that. So if I now say, well what's y minus x? Well that's just minus a times 2 to the m. So the same power of 2 still divides the number the other way around, we've just got a minus sign in front. We don't care about that, we're in the integers so we're allowed negative numbers. So- - (So positivity)
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(could go out the window there) - Positivity does still hold, because the function itself was the biggest power of 2 that divides the number. So even though there's a minus here, this- so this might be -96, the biggest power of 2 is 32. So it's still the biggest power of 2 multiplied by some integer, now it's a negative one, but the power of 2 is still positive so we're okay. So that gives us symmetry;
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now triangle law, a little bit of work required. Let's start by thinking about (x,y). Suppose this is 1 over 2 to the m, and let's suppose the same power also divides this one, 1 over 2 to the m as well. So we're sort of thinking here, what is the largest power of 2 that divides both of these two okay? And pick that one to be our distance for these two. Because what we can then say is this
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means x minus y is equal to uh some number - let's call it a - times 2 to the m. This one means y minus z is some number b times 2 to the m. These of course could have powers of 2 in them, there could be a bigger power of 2 that divides this one, but we want a power of 2 that works on both of them. And then if we try and work out this one- so for here we have to consider x minus z. So what's
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x minus z? We've got equations for things here so we can say x minus z is- x is going to be y plus a(2 to the m). And then we want to subtract off z, we've got z - if I take it over there and do that - z is y minus b(2 to the m). So x minus z we're going to lose the ys and we've got minus minus becomes a plus, so what we've got is a plus b times 2 to the m. So what this is saying is if I have a power of 2
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that divides x minus y, and a power- the same power of 2 that also divides y and z, then it definitely divides x minus z. And it can't get smaller; if you sort of look at what's happening here, because it divides these two, by this algebra it's definitely a factor of this one. So it definitely divides that one. But it can get bigger, the power of 2 here could get bigger. Because a plus b, as I
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mentioned here, these could have factors of 2 in. So when you add them together this could contain additional powers of 2, but that would just mean this one had a larger power of 2 - what's our definition of distance? It's 1 over that power. So you could potentially have a bigger power of 2 dividing x minus z, but that means the distance between x and z is smaller which is what we needed. So that one's a little bit harder to get your head around; and I recommend working through
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this for yourselves actually because this is the kind of crux of showing that this metric works. But it will satisfy the triangle law and therefore, as ridiculous as it may sound, the 2-adic metric is a distance function. - (Just because it matches those things- I mean if I- if I said to you all) (bananas are yellow and then I pulled out a yellow Volkswagen, and I said look that's yellow so that)
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(must be a banana as well - just because it happens to meet the criteria it doesn't feel like to me) (it has to still be a distance; you're telling me it does?) - It's- I I see your point but I think if you were to define what it meant to be a banana it would be: it is yellow, it is a fruit, it has the curve- whatever it was there would be several things. And so you would create almost like an exhaustive list such that absolutely everything- you had to tick every box on that
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list and then you could be nothing but a banana. - (You haven't convinced me though that this three) (is exhaustive) - Yeah well that's a fair point. So yeah, I mean this is why they're called axioms, and this is why this is the mathematical definition of a distance. - (For this video I'll take your word for) (it that that list is exhaustive.) - Yes that this- anything satisfying these three things from the
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mathematical perspective we call a distance. Now our real world definition of distance fits these. Okay, when we talk about the numbers and taking the modulus it's obviously positive, symmetry and it does satisfy the triangle law with the picture. So the one we're used to in the real numbers is a distance. Of course that doesn't in any way prove that all distances have to look like this, but the one we used to fits these rules, and it just so happens that we can find other distances.
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Some of them are sensible; like um great circles, great arcs, on a sphere - they will also satisfy the same kind of thing. But it just so happens that here it it fits the axioms; we as mathematicians have said if it fits the axioms it's a distance, so we can now use it to measure distance for this particular sequence. - (All right, I'll let you do it.) - We showed at the beginning that if this is going
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to converge to some limit - which I'm telling you is -1 - then what we need to show is that the distance between the nth term, which we'll work out in just a moment, of the sequence and what we think the limit is which here is -1 we need the distance of those two to tend to zero as n goes to infinity. So we did the example with n plus 2 over n plus 1 and we showed that that does indeed tend to 1 because we got 1 over n plus 1 here. So as we make n bigger and bigger and
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bigger, 1 over a bigger number we get zero. So the nth term: well if this is the first term, it's 10 minus 1. This is the second term- so let's write that, 10 minus 1. This is 10 squared minus 1. 10 cubed minus 1 - so it's actually a nice and easy 10 to the n minus 1 is our nth term. So we need to show this and that tells us that this sequence does indeed go to -1. What's our definition of distance we're using? The
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2-adic metric; because if we use the real numbers, it obviously is infinity, that's no fun. We wanted to get -1. So we need to find the largest power of 2 which divides that number minus that number. So of course -1 plus 1 10 to the n. So what's the largest power of 2 which divides 10 to the n? Well 10 to the n is 2 to the n times 5 to the n; because 10 is 2 times 5, 2 to the n is a factor so our distance in the 2-adic metric between the nth term of the sequence to the limit
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-1 is 1 divided by 2 to the n. What happens to this is n tends to infinity? 1 over a bigger and bigger and bigger number, this tends to zero as n tends to infinity. So what this means is n tends to infinity is saying as the number in the sequence - 10 to the n minus 1 - as this number gets bigger and bigger and bigger, more and more 9s in a row, the distance between this really big number and -1 gets smaller and smaller. Because in this measure of distance, the 2-adic space, in
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this space the way that we measure the distance is 1 over the biggest power of 2 that divides it. So this number is getting huge and it's getting closer and closer to -1 in this space so we can say the sequence of 9s converges to -1. - (Does this happen with a lot of other sequences? Like is this-) (is this some unique property of numbers with lots of 9s in it and the -1? Or could you pull-)
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(could you manufacture other sequences where this happens?) - You can do this with any sequence you want. So the 9s I just picked because it works quite nicely with the algebra. The other thing you can change is here we've done on a 2-adic metric, the more general case is p-adic where p is a prime.
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So you need a prime number for lots of reasons but that again is, I'm going to say, a little bit beyond what I'm trying to get across here in this particular case. But as long as you pick any prime number you can measure distance in the same way. So you- for the 3-adic you would say it's the biggest power of 3 that divides the difference between these two numbers and then your distance is 1 over 3 to that power, similar kind of idea. - (So if you'd done 8 - 88, 888 and used)
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(8s instead of 9s would this necessarily still work? Does it depend on the numbers?) - No that would work. I think that- off the top of my head I want to say that would be zero because 8's perfectly divisible by 2. If you change the numbers in your sequence the limit will change. So -1 here isn't special, that is just the limit for the sequence with all the 9s. If you had a sequence with
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all the 8s like you said you're going to get a different limit - and it might not even converge. It doesn't have to converge to something. Like because I could write down an arbitrary sequence that's not going to necessarily converge. So you do have to be careful, it needs to be something which almost has been like artificially constructed to make sure it's going to converge, and then you need
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to figure out what that limit is. And finding that limit can be quite tricky. So I told you it was -1 but to actually work out why it's -1, a little bit of work involved in that. - (So possibly right) (now we could put on the screen a few examples of other sequences and what they converge to.) - We could.
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Every time we have a pattern puzzle I'm going to write down the list of numbers 1 through 16 in the order that they appear in the tray so for example...
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