Infinite fractions and the most irrational number

by birtanpublished on September 2, 2020

you're watching mythology video and that probably means you 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're particularly good at picking out things that have to do with the rationality of numbers so what I want to use them for today is 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 a bit in the box so for example here replace I see another one replace and

You can see I can do this forever and what it seems to say is that 1 is equal to 2/3 minus 2/3 minus 2/3 and so on now just to remind ourselves what do we start with this guy here now turns out that if I replace all the ones here by

Twos the identity actually stays in 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 I actually identical which means of course that 1

Is equal to 2 hmm so I start exactly the same way as in the last video but unlike last time I'm not going to tell you what's wrong here obviously something is wrong you 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 a 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 square of 2 ok squared of 2 is equal to this guy here so what I'll do is I separate out the integer part from the rest of the number so there we go

And I'll rewrite this one here as well that's not quite it but that's it 1 over 1 over something is something okay now I evaluate this one here and that gives me two point and now squared of two gives me something very

Remarkable here the digits that are coming up here now exactly the same digit as in square of two know better I play the same game again separate integer part rewrite this guy evaluate and I keep on going like this forever

And that gives me this continued fraction representation of square of two now this guy is a very special kind of infinite fraction it's a simple continued fraction what makes it simple as the fact that all the numerators here

Are ones and you don't have any – this year so that's all pluses let us try this for a couple of other superheroes among the real numbers so for example the golden ratio Gordon ratio 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 closed loop it's actually just one plus square of five divided by two so

More or less another square root like square of to 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 square of three square root of five and square of seven 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 mm-hmm no patent well there is a pattern that's around there so let's pick out those numbers and looks are not deceiving extra continuous like this you compare these continued fractions to decimal

Expansions of these numbers I decimal expansion self complete mess these continued fractions are beautiful and beautiful play otic infinite to forever 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 the rational number so what's a rational number is a number that 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 is now maybe it'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 now getting 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 just keep running with fraction forms and you'll actually see pretty much 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 square of two golden ratio and E and all these other square roots are actually a rational numbers why because well their

Continued fraction expansion continues forever whereas if there 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's nothing in between there's no gray zone here so how can one number be more irrational than another number to explain this gray zone

Let's have another look at this identity how do we actually check whether we've got an identity here or not well what we do is we roll this thing up from the bottom okay so 1 plus 1/3 is 4/3 then 1 over 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 two continuous fraction expansion infinite one and she gives it to me and asked 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 was I can't roll this thing up from the bottom because there is no problem 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 or the first partial fractions this guy second partial fraction is this

One here all of these guys you can calculate and the sequence of partial fractions always always converges to the number that my friend started was here quickbuy very important interlude for us to write the equal sign here is really

Only justified because the sequence of partial fractions here converges to square root of 2 in this case remember we were always pushing this term down there ahead of us and eventually I just kind of throw it away and replace it by

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 the first partial fraction is just three second partial fraction is three plus

One sevens which is 22 over 7 the third partial fraction is well just evaluate this guy here rolling it up from the bottom that guy here and you keep on going like this or just one more so this guy here so these are fractions

That are getting closer and closer to whatever number we're after and you probably guessed it already 22 over 7 as a giveaway what we're approximating here is pi so this is the continued fraction expansion of pi 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 pi you can see this one here ridiculously good

Now these fractions that you see here on the left side they actually incredibly famous within the history of Pi in a very strict sense they're the best approximations to Pi and just in general it turns out that the partial fractions

That come out of a continuous 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 five digit

Number to get as close to that so they really punching way above their weight these distractions and in the description I I say a little bit more about the precise mathematical definition so now we actually get this

Gray zone happening that was talking about before what we do is we take two numbers and we generate these partial fractions which are the best possible approximation rational approximations and then we compare well which of these

Two numbers is easy to approximate and which is not so easy to approximate using these these partial fractions okay well let's just you compare those two guys for example all right so there's Phi let's order all over here and this

One here a bit of a mess which one do you think is more rational okay I'd say most people would say pi is more rational but actually would be wrong this guy is the most irrational number it's very hard to approximate

This guy here with fractions whether it's 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 pi right really zooming into pi at an incredible speed on the other hand these best approximations for Phi they're really struggling to get close to fine you can replace pi by

Pretty much any other number here Phi will always do a lot worse than anything else and the reason for it's 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 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 like something like this is

This incredible jump towards the real value of pi 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 ones it doesn't get any smaller than this and so this makes Phi the most irrational of all irrational numbers who cares right we're mathematicians definitely care but you may also have heard that five golden ratio is present

In nature all over the place and in fact whenever Phi comes up the Fibonacci numbers to come up and a lot of the phenomena that go with Phi and if you Bonacci 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'm going to loosen up just show you where the Fibonacci numbers are hiding in in here so if you actually produce the partial fractions

There you go you can see Fibonacci numbers straight away right there's a Fibonacci numbers 1 plus 1 is 2 is 1 plus 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 it's a Fibonacci number twirling in the other direction another Fibonacci number and most people

Don't know this but if you actually focus it 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 a

Divergence angle and the divergence angle of a flower head like this enough many flower heads coming up in nature is actually fine again I'll talk about this in the follow-up video it's either the next video or a video after that ok but

At this point in time you should actually be ready for 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

you

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