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.
Mathologer's videos are so well-researched and put together and he does such an awesome job providing mathematical truths beyond the catchy headline (cough Numberphile cough). I respect both channels and everything that goes into both, but I think Mathologer has a more positive and insightful impact on the non-math and lay-math community on a per-video basis.
But can't p = 2p not only be fulfilled by p=0 but also by p=infinity, if you include infinity in R? Because this again would make perfect sense since the infinite product of 2 diverges.
That xxx...=8 problem is driving me crazy. I've tried the obvious thing xxx...=x8 and therefore x = 81/8. However, when you calculate the partial powers you get something that seems to converge to 1.462... Does anybody know what is going on here?
12:23 Hah! Thanks but no thanks.