[Subtitles contributed by: Zacháry Dorris] You are watching a Mathologer video, and that probably means you're familiar with infinite sums, but did you ever encounter infinite fractions? Not many people have. Now infinite fractions are incredibly powerful tools for uncovering structure and patterns hidden in real numbers. And they are particularly good at picking out things that have to do with the irrationality of numbers. So what I want to use them for today is to chase down the most irrational of all real numbers. To get started, let's have a look at this identity here, and save the right part, in the box. So the box is equal to 1, so wherever I see a 1, I can replace it by the bit in the box. So, for example, here. Replace, I see another 1, replace, and you can see I can do this forever. And what that seems to say is that 1 is equal to 2/(3-(2/(3-(2/(3-...)...), and so on. Now just to remind ourselves, what did we start with? This guy here. Now it turns out that if I replace all the ones here by twos, the identity actually stays an identity, and I can repeat my game, so I replace, I replace again, I replace all the way to infinity, and well, let's have a look. The right sides here...are actually identical. Which means, of course, that 1 is equal to 2. *Scoffs* So, I start exactly the same way as the last video, but unlike last time, I'm not going to tell you what's wrong here - obviously, something is wrong. You are supposed to work this out yourself in the comments. What I'll do instead is now talk about infinite fractions, and by the end of this video, you should be able to figure out where the mistake is. Any number whatsoever has a representation as an infinite fraction, as a 'continued' fraction. So let me just show you how you generate an infinite fraction using the √2. Okay, √2 is equal to this guy here, so what I do is I separate out the integer part from the rest of the number So there we go, and now I rewrite this one here, as...well, that's not quite it, but that's it. 1/(1/something) is something, right? Okay, now, I evaluate this one here, and that gives me 2., and now, √2 gives me something very remarkable here, the digits that are coming up here now are now exactly the same digits as in √2. Not bad, huh? I play the same game again, separate out the integer part, rewrite this guy, evaluate, and I keep on going like this...forever, and that gives me this continued fraction representation of √2. Now this guy is a very special kind of infinite fraction. It's a 'simple' continued fraction. What makes it simple is the fact that all the numerators here are ones, and you don't have any minuses here, so it's all pluses. So let's just try this for a couple of other 'superheroes' among the real numbers, So for example, the golden ratio (Φ). Φ plays a very important role in all this, because it's got the sort of, the simplest infinite, continued fraction. It's got all ones down there. It doesn't get any simpler than this. When you have a close look, it's actually just (1+√5)/2, so more or less, another square root, like √2, you get something periodic here; in fact, any square root, or slightly mucked up square root, like this will give rise to a periodic continued fraction, infinite one, maybe try this with √3, √5, and √7. Now let's take something that doesn't have anything to do with square roots, let's go for e. Another superhero, right? 2.718... If you have a look at the continued fraction of this guy, hmmm... no pattern? Well, there is a pattern. Starts around there, so let's pick out those numbers. And looks are not deceiving, it actually continues like this. If you compare, these continued fractions to the decimal expansions of these numbers, right, decimal expansions are a complete mess, these continued fractions are beautiful. Beautiful and periodic. Infinite too, whatever. Now, we can actually use these continued fractions and produce proofs that these numbers are irrational. How do we do this? Well, first of all, we have to have a look at a rational number. So what's a rational number? It's a number that can be written as a fraction. So take a fraction, and unleash the scheme on this. What's the continued fraction that corresponds to this number? There it is. Now, maybe there's a bit of a surprise, this thing ends. It doesn't go on forever. And that's actually going to be the same for all fractions. So if you start with a fraction, and produce the continued fraction that corresponds to it, that continued fraction will be finite. Really really, quite nice, isn't it? And you can check this out yourself, maybe just do this one here. And run the scheme, just don't turn this thing into a decimal number [29.46], just keep running with fraction forms. And you pretty much see at a glance why this thing has to terminate, and why all fractions have to terminate in terms of the continued fraction expansion. Once we know this, we have proofs, basically, that √2, Φ, and e, and all these other square roots are actually irrational numbers. Why? Because well, their continued fraction expansion continues forever, whereas if they were rational numbers, they would terminate. Neat, isn't it? Okay, Now, at this point in time, I'm now going to ask for the most irrational number. And that question may sound a little bit idiotic at first glance because either a number is rational, or it's irrational. There is nothing in-between, there is no grey zone here. So how can one number be more irrational than another number? To explain this grey zone, let's have a look at this identity. How do we actually check whether we have got an identity here or not? Well, what we do is we roll this thing up here from the bottom, okay? So 1+1/3 is... 4/3, and then 1/that is this guy here, and then we calculate this, and we keep on going, like this and we find, yes, it's true. A friend of mine just took a number and produced a continued fraction expansion, an infinite one, and he gives it to me and asks me, "You figure out what number I started with." So there it is. And now, well, how do I figure out what number he started with? I can't roll this thing up from the bottom because there is no bottom! But now it turns out that these continued fractions have another really amazing property, which actually makes them very useful for all sorts of purposes. If you chop off things at the pluses, you create a sequence of partial fractions, so the first partial fraction is this guy, second partial fraction is this one here, all of these guys you can calculate, and the sequence of partial fractions always converges to the number that my friend started with here. [Ummm...why is he staring at me?] Quick but very important interlude... for us to write the equal sign here is really only justified because the sequence of partial fractions here converges to √2 in this case. Remember, we were always pushing this term down there ahead of us, and eventually, I just kind of threw it away and replaced it with the three dots. Well, that's really only justified if we pin down exactly what we mean for things to still be equal at that point. [More interlude] The first partial fraction is just 3. Second partial fraction is 3 + 1/7, which is 22/7. The third partial fraction is - well, just the value of this guy here, rolling it up from the bottom, That guy here. And you keep on going like this, well, just one more. So this guy here. So these are fractions that are getting closer and closer to whatever number we are after, and you probably guessed it already - 22/7 is a giveaway at what we're approximating here is π. So this is the continued fraction expansion of π, a simple one. To see how good these approximations are, let's just turn them into decimals, there we go, and here, I've highlighted to what digit they actually correspond to the decimal expansion of π. And you can see, this one here - ridiculously good. Now these fractions that you see here on the left side, they are actually incredibly famous within the history of π, in a very strict sense they are the best approximations to π. And just in general, it turns out that the partial fractions that come out of a continued fraction expansion of a number are the best rational approximations to that number. Now in what sense? Obviously, if you take larger and larger denominators, you can get closer and closer with fractions to whatever number you're interested in. But the point is that you're using very small denominators, to really get incredibly close. So you wouldn't expect, with just one digit, to be able to get as close as that. Or with, like, a 5-digit number, to get as close as that, so they're really punching way above their weight, these, these fractions. And in the description, I say a little bit more about the precise mathematical definitions. So now we actually get this grey zone happening that I was talking about before... What we do is we take two numbers, and we generate these partial fractions which are the 'best' possible approxi - rational approximations, and then we compare, well, which of these two numbers is easy to approximate, and which is not so easy to approximate, uh, using these, these partial fractions. Okay, well, let's just compare those two guys for example, right, so there's Φ, there's order all over here, and this one here, a bit of a mess, which one do you think is more irrational? Okay, I'd say most people would say, "well, π is more irrational", but actually, they would be wrong. [Plot twist] This guy is the most irrational number - it's very hard to approximate this guy here with fractions, where, as we've just seen, it's very easy to approximate this one really really well with fractions, and just to really drive home this point here, I've got a table of partial fractions next to each other, right, here on the left side, you see the ones for π, right, really zooming in to π at an incredible speed, On the other hand, these 'best' approximations for Φ, they are really struggling to get close to Φ, and you can replace π by pretty much any other number here, Φ will always do a lot worse than anything else. And the reason for it, doing a lot worse, when you have a really really close look, is actually hidden in plain sight. It's got to do with these numbers here, so when you kind of scroll through these numbers, like 3, 7, 15, and so on, the larger the numbers you have here, the closer you jump towards the real value when you evaluate the partial fraction. So, something like this is an incredible jump towards the real value of π when you evaluate this partial fraction. So when these numbers get small, the jumps get small. And they get as small as possible if you just choose them as small as possible, if it's all 1, it doesn't get any smaller than this. And so this makes Φ the most irrational of all irrational numbers. Who cares, right? *laughs* Well, mathematicians definitely care. [And so do I!] But you may also have heard that Φ, the golden ratio, is present in nature all over the place, and in fact, whenever Φ comes up, the Fibonacci numbers come up. And a lot of the phenomena that go with Φ and the Fibonacci numbers coming up together in nature can actually be explained with these continued fractions. And just to give you a taste, I'm not going to do this today, but I'll do it in another video, just to show you where the Fibonacci numbers are hiding in here, so if you actually produce the partial fractions, There you go, you can see Fibonacci numbers. Straight away, right, there's the Fibonacci numbers, 1+1 is 2, 1+2 is 3, and so on, and so what is an example of a natural occurrence of these things? Well, look at a flower head like this - this guy here, and count spirals that you see here, twirling in one direction That's a Fibonacci number, twirling in the other direction, that's another Fibonacci number, and most people don't know this, but if you actually focus in on the middle part of a flower head like this, you actually see different Fibonacci numbers popping out. And in a flower head like this, this is actually grown with something called the 'divergence angle', and the divergence angle of a flower head like this, and many flower heads coming up in nature, is actually Φ. Again, I'll talk about this in a follow-up video, it's either the next video, or the video after that. Okay, but at this point in time, you should actually be ready for the puzzle. So you figure out, and you tell us in the comments what's wrong here, what's right here, and I'm really looking forward to this. And that's it for today. [Outro music]
