Uploaded by numberphile on 27.01.2012

Transcript:

ROGER BOWLEY: The number I want to talk about now is the

square root of 2.

JAMES GRIME: My favorite fact about root 2?

Well, if you take the A series of paper.

So this is A4 paper.

It's pretty standard in most of the world.

If you take a piece of paper like this, if you look at the

ratio between the long edge and the short edge, which

means if you measure the long edge, and you divide by the

short edge, it will be the square root of 2.

And they picked this on purpose.

ROGER BOWLEY: The square root of 2 is about 1.41

something or other.

And it's the number you get if you work with Pythagoras'

Theorem, which said that if you have a unit length along

there, and a unit length along there.

So this length, the square of this length plus the square of

that is that.

This length is root 2, it's the square root of 2.

JAMES GRIME: If I fold this piece of paper in half--

let's try that out.

This now is something called A5 paper.

We started with A4.

This is now A5 paper.

And if you do the same thing again, if you take the long

edge of this and divide by the short edge, it will again be

the square root of 2.

The ratio is the same.

They do this so that you can scale things up and scale them

down without it being disproportionate.

ROGER BOWLEY: Square root of 2 is this, which is 1.41.

Well, it runs on forever.

It doesn't stop at a particular point.

And there are a whole range of these.

This is just the first of many.

Another one is pi.

JAMES GRIME: In fact, they start off with A0.

An A0 is defined to be a piece of paper, which has a ratio,

square root of 2, and it has an area of one meter squared.

In fact, root 2 is the only ratio where this works.

And I'm going to show that it's the

only ratio this works.

There you go.

Well, our rectangular piece of paper.

This is your long edge, which I'm going to call a, and this

is your short edge, which I'm going to call b.

Now, if I take the long edge, a, and divide by the short

edge, b, I'll get a ratio.

It's actually going to be root 2, but let's pretend we don't

know that yet.

But this is what I want.

If I cut this in half, I want this now to be the long edge.

So b is now the long edge.

The short edge is actually half of a, this

side, half of a.

Now I want that to be the same ratio as I had before.

In other words, I want these two things to be equal.

So all we have to do now is rearrange this,

play with this a little.

Let's see what we get.

OK.

Let's take the b's over this side, the a's over this side.

If you play with it, you will get a squared on the left, and

2 b squared on the right-hand side.

Or, in other words, a squared divided by b squared equals 2.

Or, square root it, square root both sides.

And you'll get a divided by b on the left.

And on the right, you get the square root of 2.

And this was actually the ratio we wanted to find.

The only ratio where this works is the square root of 2.

JAMES GRIME: This is Pythagoras' Theorem.

Pythagoreans were a cult way, way back.

And there might have been a figure called Pythagoras

associated with this.

But they were very strange bunch.

And they believed in this theorem.

They believed in the harmony of natural numbers.

So if you want to do music, you took a bit of string of

certain tension.

If you took half the length, you'll get a

harmonious note with it.

They thought all of nature was composed of numbers.

They thought that everything could either be expressed as a

pure integer or a ratio of integers.

That was their fundamental core belief.

JAMES GRIME: What did they know?

They had the numbers one, two, three, four, five.

They had worked out the fractions, that you could

divide two things together, 3/5, 1/2, that sort of thing.

ROGER BOWLEY: They had other beliefs, as well.

That you shouldn't marry a woman who wore gold jewelry,

for example.

Or you should be a vegetarian.

And you shouldn't eat fava beans.

And you shouldn't urinate towards the sun.

There was a disciple of this strange sect called Hippasus.

And he worked out that if you believed in this theorem,

Pythagoras Theorem, then you could show mathematically a

proof showing that this number is not a

ratio of two integers.

JAMES GRIME: But a number that could not be written as a

fraction, that was a new thing that went on forever.

That's what an irrational number is.

ROGER BOWLEY: And they disliked this so much.

I can't begin to tell you how much they disliked it.

So they took him out to sea, according to legend, and he

didn't come back.

Somehow, either they drowned him or left him

on a deserted island.

He wasn't seen again.

JAMES GRIME: They did, apparently--

they suppressed this information.

They weren't sure that this was a real thing.

ROGER BOWLEY: We call them now irrational numbers because

they don't fit in with this Pythagorean viewpoint.

JAMES GRIME: It's the same sort of problem that people

have today.

People say to me, oh, I've heard of complex numbers, I've

heard of that.

I don't believe they exist.

I don't understand it.

Same problem, they do exist.

It's just you have to get used to it.

ROGER BOWLEY: So he was one of the first people to be

persecuted for proving people wrong with

their previous ideas.

And this has gone on through history.

I mean, I could tell you about physicists who've suffered,

like Bruno Giordano, who went to--

burnt at the stake for saying the universe was infinite.

And the Catholic church said, well, that doesn't leave any

room for God.

JAMES GRIME: We're going to prove that root 2 is

irrational.

It cannot be written as a fraction.

The way we prove this is a really powerful, useful

mathematical proof called contradiction.

We're going to assume the opposite.

I'm going to assume you can write it as a fraction.

Let's assume we can write it as a divided by b .

And this a and b are special.

They are integers.

They are whole numbers.

They're one, two, three, four, five.

They're something like that.

And they're in their smallest possible terms.

So we're not including things like 2 divided by 4 because

that's a half.

That's not in its smallest possible terms.

All right.

Let's see what we can do.

ROGER BOWLEY: If you come up with an idea which is right

but goes against conventional wisdom, you can be sent to a

desert island, or burned at the stake, or executed.

Because people don't like their ideas about how the

world should be upset.

Even if you can disprove it, it's not a good idea to

urinate towards the sun.

JAMES GRIME: I'm going to square both sides.

So on the left-hand side, I get a 2.

On the right-hand side, I get a squared

divided by b squared.

This time, a and b are whole numbers, remember.

Let's play with it.

We get 2b squared on the left equals a squared on the right.

Now what I've shown here is I've shown that

a squared is even.

Because it's a multiple of 2, it's an even number.

And you can show that because a squared is even,

a is even as well.

Well, if it was two odd numbers squared, that would be

an odd number.

So yeah, yeah, if a squared is even, the original number, a,

that was even.

So a is even.

OK.

So a is even.

Let's call it something else, let's call it a

equals 2 times c.

So what do we have now?

2b squared on the left-hand side is 2 times c because it's

even, squared.

This is equal to 4c squared.

In other words, I'm saying b squared

is equal to 2c squared.

Can you see that?

b squared is even.

If b squared is even, like before, b is even.

What I've shown is a is even and b is even.

That's a problem.

They can't both be even.

Like my example of 2 divided by 4, it's not in its smallest

possible terms if they're both even.

So this is an impossible fraction.

It doesn't exist.

You cannot write root 2 as a fraction like this.