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.
Interesting, but completely irrational.
It's also calibrated so that A0 has an area 1m2, then A4 is just cutting that sheet in half 4 times.
It's also useful for calculating paper weight.
Standard printing paper is 80gsm (or g/m2). A sheet of A0 paper is, by definition, 1m2, and thus weighs 80g. A sheet of A1 is half the size, and thus weighs 40g. A sheet of A2 weighs 20g, A3 10g, A4 5g, A5 2.5g etc.
Common cardstock weighs 100gsm, and so an A4 sheet of cardstock would weigh 6.25g.
Sadly, A4 is not very popular here in 'Murca
Every other paper size oscillates between two different ratios each time you fold it. A4 just is the golden size where the two ratios converge.
So what's the story behind 8 1/2 x 11? Last time I looked, the origins of that paper size are unknown.
This idea goes back to the 18th century. Wikipedia.
But not more than 7 times, even with an hydraulic press.
No. It is so that sheets can be CUT to yield more standard-sized sheets. It is also so that folded pages or signatures will yield standard sized pages.