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DR. JAMES GRIME: So Zeno's
paradox is a problem that's
not just of interest to
mathematicians, like myself.
But it's also of interest to
physicists and philosophers.
And it has been for thousands
of years.
So Zeno was a Greek
philosopher.
He lived 2,500 years ago.
And he came up with a
list of paradoxes.
There's about nine of
these paradoxes.
The ones I want to talk about,
though, are kind of similar.
The first one, though, I love,
because there's a nice little
story with it.
So the first one is called
Achilles and the Tortoise.
You've got Achilles and the
tortoise in a race.
Now, the tortoise is
slower, of course.
So he's given a head start.
He's got a 100 meter
head start.
And then they start the race.
Now, Achilles sprints 100
meters, and he catches up to
where the tortoise was.
But in that time, the tortoise
has moved on.
He's moved on 10 meters.
So Achilles now has to sprint
to catch up with
the tortoise, again.
So he sprints another 10 meters,
and he catches up to
where the tortoise was.
Now, since that time,
the tortoise has
moved on another meter.
So Achilles has to sprint
to catch up to where
the tortoise was.
And this keeps happening
over and over again.
So the paradox is does Achilles
ever catch up with
the tortoise?
And that's insane.
And obviously, he must do.
We all live in the real world.
He must catch up with
the tortoise.
Another of these paradoxes is
basically the same, but it's a
bit simpler.
I'm holding my hands
apart like that.
Now, I'm going to keep
my left hand still.
I'm going to move my right hand
towards my left hand so
that they clap.
Now, when I do that--
actually, you can think of it
like I'm halving the distance.
I then halve the distance
again and then halve the
distance again and again
and again and again.
And I halve the distance an
infinite number of times,
infinitely many times.
So does that mean my
hands never clap?
Is there some sort of
force field here
stopping my hands meeting?
Well, obviously, they do.
This is the paradox.
So what's going on here?
How can an infinite
process end?
What's going on?
And this is part of
Zeno's paradoxes.
I want to give you the
mathematician's point of view
for this, because, well, some
say that the mathematicians
have sorted this out.
Let's say I start with my
hands 2 meters apart.
And then I'm going to
join them together.
So they start 2 meters apart,
and then I halve the distance.
So I've traveled 1 meter.
I then halve the distance again,
so we've now traveled
another half a meter.
Then you do that again, you
halve the distance again.
So now, your hands have traveled
1/4 of a meter.
And then it's an 1/8
and 1/16 and 1/32.
And then you keep doing
this forever, halving
the distance forever.
Let's say this does
have a value.
Let's give it a value.
Let's call it S--
S for sum.
Here's a trick that
you can do.
First of all, I'm going
to multiply by 1/2.
I'm going to multiply the
whole thing by 1/2.
So if I multiply the left hand
side by 1/2, you get 1/2 of S.
On the right hand side,
I'm going to multiply
term by term by 1/2.
So 1 multiplied by 1/2 is 1/2.
I'm just going to write that
there-- going to leave a
little gap.
Now, I'm going to multiply
the 1/2 by 1/2,
which gives me 1/4.
And I'm doing it by term by
term, so 1/4 times 1/2 is 1/8,
and 1/8 by 1/2 is 1/16.
Now, I'm going to subtract
those two things.
If I subtract, from the left,
I get S minus 1/2 S, which
gives you 1/2 S. On the right
hand side, I'm going to
subtract these two lines.
So I've got 1.
And I've got plus 1/2 minus 1/2
plus 1/4 minus 1/4 plus
1/8 minus 1/8.
And all the terms cancel out.
So all you're left with is 1,
which means, if everything's
canceled out then, you can
work out what S is.
S is equal to 2.
So this would be 2 meters.
So your hands would
travel 2 meters.
Even though it's an infinite
process, your
hands do travel 2 meters.
Now, time is important,
as well.
If each step was taking a
second, then that's an
infinite number of seconds,
or if it was
longer than a second.
So you would complete your
2 meters, but it takes an
infinite amount of time to
complete your 2 meters.
So time is important as well.
So let's say we'll do it a
little bit faster than that.
Let's say I move 1
meter per second.
So I'm going to move
1 meter per second.
It's the same, working out.
If I halve the distance, that
would take me 1 second.
And then I halved the distance
again-- it would take me 1/2
second, then 1/4 second.
So I could travel 2 meters
in 2 seconds.
And my hands would clap.
So something like this-- an
infinite sum-- behaves well
when, if you take the sum and
then you keep adding one term
at a time, so you've got lots
of different sums getting
closer and closer
to your answer.
If that's the case, if
your partial sums--
that's what they're called-- are
getting closer and closer
to a value, then we say that's
a well-behaved sum, and at
infinity, it is equal
to it exactly.
And it's not just getting closer
and closer but not
quite reaching.
It is actually the whole
thing properly.
So it sounds like I'm saying
that you can complete an
infinite process.
Now, an infinite process doesn't
have a last step.
So how can something without
a last step be completed?
And that's the paradox.
BRADY HARAN: What is the--
and is there a solution?
DR. JAMES GRIME: I have no easy
answers for you here.
This is a paradox that has
baffled philosophers and
mathematicians for
2,500 years.
I wrestle with it as well.
There are greater minds than you
and I that have wrestled
with this problem in the
last 2,500 years.
If this melts your brain,
don't worry.
You are in very good company.
BRADY HARAN: Some of these
numbers where the digits
apparently go on forever
and ever--
pi, the square root of 2--
how do they fit into all this?
DR. JAMES GRIME: Yeah.
So if something goes on forever,
it's almost like
you're asking, can I even draw
something with these decimal
representations that
never finish?
And it's the same sort of
problem that we've got here.
But again, even though
that's an infinite
process, it can be completed.
And I know that's bizarre.
But look.
If I draw a triangle, that has
length 1, that has length 1.
It's a right angle.
And this has length the
square root of 2.
And there's no reason I can't
do that, even though the
square root of 2 is irrational,
which means the
decimals go on forever.
Even though it's like that,
there's nothing stopping me
from drawing it.
In the same way, there's nothing
stopping me from
clapping my hands.
You can think of this as
an infinite process.
I'm halving the distance an
infinite number of times.
But I can still clap my hands.
So these 19th century
mathematicians had to work out
when these infinite sums behaved
well and when they
behaved badly.
And so there's a lot of
technical detail behind that.
And it just turned out to be
rigorous and consistent, which
is very important.
But some of the tests that
you can use to prove that
something is a well-behaved
sum--
well, they're quite easy.
I'll show you one.
What you can do is if you take
one of these terms--
I'm actually going
to call it a.
You take one of them,
the nth term--
an.
You divide that by the one that
came before it, and if
it's negative, well, you'll
make it positive, then.
If you take those numbers--
if those numbers go
towards a value--
let's call it r, as these
n's get bigger.
If these numbers tend towards a
value, and if that number is
less than 1, then it's
well-behaved.
So this was well-behaved.
Why was this well-behaved?
Because if you take one of those
terms and divide by the
one that came before it,
you always get 1/2.
Look.
1/2 divided by 1 is 1/2.
1/4 divided by 1/2
is also 1/2.
These are all halves,
actually.
You get a common
factor of 1/2--
common ratio of 1/2
for each one.
That passes the test.
And that's what this is.
This is a test.
So it passes the test.
This is well-behaved.
If this was bigger than 1, then
it's not well-behaved.
And that could mean--
well, it could mean it
goes on to infinity.
Or it could mean it's periodic
or does something else weird.
So it's not well-behaved.
And the other case is if this
number was equal to 1, we
don't know.
It could be anything.
It could be either.
It could be well-behaved
or not.
It depends.
So r equals to 1,
we're not sure.
BRADY HARAN: What's a series
where r is greater than 1?
Can you show me, what's a
badly behaved series?
DR. JAMES GRIME: So if it was
bigger than 1, than the terms
would be getting bigger and
bigger and bigger each time,
maybe not in a common
ratio way.
Well, we were talking about the
Fibonacci sequence earlier.
So if we had something
like that, like 1
and 1 and 2 and 3.
If you divide those, they don't
have a common ratio.
But actually, the ratio
is tending to
1.618, the golden ratio.
So 1.618 would be
bigger than 1.
And this would be
badly behaved.
If you added them together, you
would get a number that
was going off to infinity
in that case.
And that would be called
divergent.
This is well-behaved,
our example.
And it's called convergent.
My question to physicists is if
we were literal about this
and considered the real world
so that we are actually
dividing a distance repeatedly
in half over and over again
infinitely many times, or
dividing the time it takes to
complete each step infinitely
many times so it's getting
quicker, can you do that?
Can you divide space and time
infinitely many times?
I don't know the
answer to that.
So I want a physicist
to tell me.
We're in slightly the area of
calculus, as well, which some
of you may meet when you get
higher up at school.
Calculus helps you work out the
area by adding up really
thin strips together.
And it's in the same area.
So Newton and Leibniz were
trying to add up areas by
taking strips that were
really, really thin--
not quite 0, but close to 0.
And this idea was later replaced
with what we did
here, the 19th century
idea called limits.
So this has taken a long
time to work out.