You're watching a Mathologer video, so that probably means that you're familiar with
Srinivasa Ramanujan, one of the most ingenious mathematicians who ever lived.
You are probably also familiar with his strange counterintuitive identity 1 + 2 + 3
and so on is equal to -1/12. If you don't know about any of these things
check out my video about this. Ramanujan is also famous, at least among
mathematicians, for this strange infinite 1, 2, 3, 4, ... expression. It's an infinite nested
radical and he says it's equal to 3 and I actually can give you his argument
which is very, very pretty.
So he says 3 is equal to square root of 9.
Well, yes, and 9 is equal to 1 plus 8, 8 is equal to 2 times 4, 4 is equal to
square root of 16, 16 is equal to 1 plus 15 and 15 is equal to 3 times 5
and 5 of course is equal to square root of 25. And you can probably see how
this continues. But let's just do one more, 25 is equal to 1 plus 24 and 24 is
4 times 6 and 6 is equal to square root of 36. All right now all these
expressions that we've seen here they're all equal to 3, obviously, and so
since this continues on forever he says this shows that the infinite expression
itself is equal to 3. And that sounds pretty convincing doesn't it, until you
figure out that you can do exactly the same thing for 4 and let me show you that
one too.
So, here we go. So 3 is equal to square root of 9, well 4 is equal to square
root of 16 right and now Ramanujan rewrites like this.
Well square root of 16 I can rewrite like that
okay, so 1 plus 2. Now here he rewrites like that, I rewrite like this and I can
actually go on forever just like Ramanujan goes on forever, so we're
pushing numbers ahead of us, here it's square numbers, here it's weird blue numbers.
But they're always going to be there and if you accept that this argument here shows that the
infinite expression is 3
then you should also accept that the infinite expression is 4.
Well then of course we are in trouble right, 3=4, something is definitely fishy
here, so it seems that even famous mathematicians like Ramanujan sometimes
get it a little bit wrong.
Well okay we'll have a look. Ramanujan was really interested in infinite
expressions and he was a real master of figuring them out. So there's infinite
sums like this there's infinite products, these guys are called power towers, that
one up here is very funny. If you haven't seen it yet there is a
video by heart in which she talks about lots and lots of different infinite
expressions and this one is supposed to be equal to the mysterious number wau.
If you haven't watched that video you absolutely have to watch it and then of
course there's this fraction here that a lot of people have been waiting for.
Just recently I did a video where I start with 1 and then from 1 I grow
this infinite fraction, just like Ramanujan grows this infinite nested
radical starting with 3 but then I also show that you can do exactly the same
thing starting with 2 just like I just showed you that you can grow the
radical from 4. Ok and then I asked does this imply that 1 is equal
to 2. And we got a lot of really good discussions going on and a lot of ground
covered, really good comments but i thought what I do today is try to make
sense of all these infinite expressions kind of give you the tools, next time you
come across one of those infinite expressions to figure them out yourself,
The very first thing you have to realize about all of these things is that to
start with their completely meaningless. If I took away all the dots you would
know exactly what to do with any of them but since the dots are there any single
one of those expression asks you to do something infinitely often and to start
with it's not clear what that actually means. So we have to actually make up our
minds what that's supposed to mean,
executing an operation infinitely often. But maybe before we want to do this, we
want to figure out whether it's actually worth doing all this work and there's
actually a nice trick that can help you with this making up your mind. I just
want to show you this trick.
It doesn't apply to all of these infinite expression but it applies to many of
them, all these periodic ones basically. Let's have a look so we are interested in this
sort of thing. Looks pretty, but should Ireally waste some time on it.
Well, let's see so this guy here I don't know what it is but let's call it
something. Let's call it "r" for root, this guy here "f" for fraction that guy down there
"p" for product. Right, now have a close look here, this yellow bit is actually the
whole thing itself, so we've got the "r" sitting inside itself. So we can actually
rewrite this as "r". That gives me an equation for "r". So I can manipulate that, gives
me a quadratic equation. I solve it and get two solutions, of course, 1 plus
minus square root of 5 divided by 2. A closer look shows that one of these numbers positive, one
is negative.
Nothing negative in sight here, so if it's anything it should be the positive guy. And
actually this one here is of course a super-famous number it's the golden
ratio now here I've got my motivation, I've got something super pretty on this side
I've got the golden ratio on the other side and what I want to figure out really is
are these somehow equal in some sense. So I'm now prepared to actually put the work
in and well let's just do it again for this guy here, for the puzzle one. So here you
also see that you can rewrite like this we get an equation. So whatever this is
supposed to be, that infinite expression it should be equal to one of the solutions
of this equation. Again it's basically a quadratic equation and you actually get
1 and 2 which was a bit sneaky of me but I knew that some people know this trick
they would be applying it and so at that point in time you actually figure out
well it doesn't give me anything new because I already started constructing
these things from 1 and 2 and then finally this guy here looks a bit weird
because 2 times 2 times 2 times 2 well obviously that explodes to infinity,
shouldn't have any value, but let's say we're not very careful, you know, and
we're just going to follow our nose here but then we see that there's a "p" here
and now we ask what's a number that satisfies "p" equal to two "p"
there's only one of them, 0, so you're not really careful you kind of get a
prediction 0 here. Doesn't really matter in terms of motivation, like all of these
right sides here tell me I really want to figure out what's going on but it tells
you you've got to be careful with
these sorts of things. Ok so now how do we actually make sense of these infinite
expressions. Let's look at a fairly simple one, infinite series. We're
supposed to add infinitely many bits here, that sounds hard
don't know what to do, but at least I can get started, right. So I'll just start
adding so we've got 1
ok then i add 1/2 gets me this guy and I just keep on going like this and these guys
are called partial sums of this infinite series and with this particular one is
actually it's easy to see there's a nice pattern here know what comes next and so
on.
Also, these are increasing and they are converging to 2, actually 2 is greater
than all of them and 2 is actually the smallest number greater than all of
these guys here. Now have a look at this guy here. So we don't really know
what it is but if it corresponds to a number, some number, that number should
also be greater than all of these. Now I step back to step and have a look at all
this and you see, well, really, the only number that really qualifies here is 2 so now
we actually define the sum of this guy to be 2.
That's us doing it so we are actually somehow gods in this respect, we actually
giving this a meaning and we're doing this in general for these infinite series,
so get an infinite series, translate it into the sequence of partial sums. Does
the sequence converted to a number? If yes then this number is declared to be
the sum of the infinite series. If not well as series is divergent and you have
to maybe try something else something non-standard. This is the standard
approach, there is also non-standard for that check out to 1+2+3... video. Puzzle fraction
fraction what do we do with this one?
Well, again, we'll just start calculating. So let's start calculating, ok first
result here is 2/3. Do another one that turns out to be 6/7, and we keep on going
like this and again we see a nice pattern here, numbers getting bigger and bigger
sequence converging to 1 and so, obviously, this should be equal to 1. it's a
reasonable way of assigning a value to this probably the most reasonable one, except,
when you really think about it there's actually at least one more totally
different
and reasonable way of associating a sequence of numbers to this one and well
let's have a look.
Well if we stop calculating here, then actually first numbers 2 and then if
we stop there
well 2 divided by 3 minus 2 is 2 again and if you do it again and again and again
actually all the numbers that fall out of here are 2s and so obviously this
guy here
converges to 2 and so another way of associating a number to this infinite fraction
would be 2 and actually if you're the first person to look at this guy here
then, you know, there's no real preference for one or the other or both are pretty
reasonable
ok but in this case you're not the first person to look at it. Actually people
have been looking at it for hundreds of years and there is a whole theory of these
internet fractions here and within that theory it actually turns out that the
first way of chopping up of generating the sequence is the way to go because
it's applicable, it's useful, it's just it. Whereas the second way
of associating a number is not it.
So, by default, if somebody shows you an Infinite fraction like this today, in the
context of the larger theory, the answer is, this guy here is equal to 1. But
just in general, if you are facing an infinite expression there could well be
a couple of different ways of associating a number to it that are perfectly
reasonable. What about Ramanujan's infinite nested radical? Well let's just
calculate. Chop of here, okay,
chop of there and keep on going like this now I just display some of the numbers you
come across here. Well these numbers are creeping up again.
The pattern is not as apparent as it with the other examples I had but it actually
seems that we're creeping up to 3 and you can actually prove that this
sequence of numbers here converges to 3.
Ok so Ramanujan was right about this thing being equal to 3 in some sense.
Is there another way in which you can associate a sequence to this. Yes there is. So
you can chop off here you, can chop of there but you can actually see that when
we do this we actually eventually end up with the same numbers. It's also easy to see
why.
With this particular infinite nested radical there's really only one way of
associating a number to it,
one good way and it's 3. Ok so Ramanujan was right about the 3 but the
argument he gave was not quite complete. The context in which all this stuff comes
up is actually a puzzle you know just like I give these puzzles in my YouTube
videos, mathematicians sometimes challenge other mathematicians in math
journals. So this was a challenge like this and actually Ramanujan challenged
other mathematicians to figure out what it is and after while nobody gave a
response so then he had to give his own answer and that's his answer
which you can check out, linked it in in the the description.
Actually he gave a second challenge, it's this guy here and so if you feel
particularly brave today you can try and figure out what this one is and depending on
your mathematical background you can actually you know prove whatever you
come up with here. And well if that's a bit too hard, then maybe try and figure
out whether this guy here is really equal to the golden ratio or what's all this business
with Wow, so is this equal to Wow or what is Wow. So try and figure that one out.
Maybe also have a look at Vi Hart's video and check out some of these other
infinite expressions and make sense of them. And then maybe finally one of my
favorite equations, solve for x, have fun.
You're probably calculus specialists and you've heard about infinite series, you
know about infinite products and you know about infinite fractions like this,
continued fractions. But what about all this other stuff, do they actually show
up anywhere? Apart from these particular types of infinite expressions?
Yes, heaps. So let's have a look. So this guy for example can be defined totally
in terms of this infinite expression here. So the "c" stands for a complex number
now first of all we need to know how we chop this thing up to get our infinite
sequence of numbers. So the way we chop this up is like this.
The Mandelbrot set is a subset of the complex plane, so a point is either
inside the Mandelbrot set or is outside the Mandelbrot set and to decide whether
inside or outside you can use or you do use this infinite expression here. This
is a complex number, that's "c" okay now we evaluate this guy here which corresponds
to making up this infinite sequence of numbers. If this infinite sequence of
numbers is contained in a finite region of the complex plane, doesn't explode to
infinity, then the point is inside the Mandelbrot set. If, on the other hand, that
sequence of numbers explodes to infinity then it's outside and, in fact, depending
on how fast they explodes to infinity you give the point of different color
which then gives you this strange halo effect around the Mandelbrot set that you
see often in pictures. If you actually check out my Mandelbrot set video or if you
know anything about Mandelbrot sets you probably haven't
seen this infinite expression here. What you have seen is this here.
So what everybody who knows anything about the Mandelbrot set knows is that
we figure out whether a point is inside or outside using this function here. What we
do with this functions is we iterate it and doing so is the same as evaluating this
infinite expression. I just show you how this works. So you initialize with x equal to
0. So 0 squared plus c is equal to c. What we get out we just
feed into this function again so we have to square and then plus c and then this guy here
gets fed in again so we square and plus c again and you can see it's basically the
same thing. Iterating the function is the same thing as you know making one of
those infinite expressions here. Our trick has something to do with these iterated
functions. The answer it actually gives, the trick, is to a question that we're
not really posing. So this guy for example here that corresponds to one of
those iterated functions, the function is this guy here. And you've seen this one
before. That comes up when we set up our trick.
So there's the f so there's the function. When we solve for f the
values we get here
that's 1 and 2 those are the fixed values of the function. What it means is
that if instead of 0 I initialize with one of those fixed values the infinite sequence
of numbers that gets spat out here is constant 1 or constant 2. So these are
very special and actually they are super duper special. So basically when we
are talking about the infinite expression we're talking about one
particular sort of initialization for this iteration process or maybe two or
maybe three specific ones depending on our interpretation of what that infinite
expression might mean but really what the answer here is is about all possible
internet sequences and their behavior given different initializations. The
answer 1 and 2 what it says is that, no matter how i initialize here, if the
sequence of numbers actually converges to anything it's going to be 1 or 2 and then
actually you know 1 and 2 can happen because if you feed in 1 and 2
you know you really get 1 as constant sequence 2 constant sequence. So the
answer that we're getting to our question here you know, you may sometimes get
lots of answers or a slightly misleading answer, it actually does make sense in
this context, it is a complete answer there but it's too much of an answer
really for what we're interested in looking at infinite expressions. To
analyze these sorts of things there is a huge literature there and lots of and lots of really
beautiful stuff. So, basically, what you have to investigate are these fixed
values of functions and there's nice theorems there, fixed-point theorems and
we're going to talk about these in future Mathologer videos. So something to
look forward to and that's really it for today.