ALEX BELLOS: Well, I wrote a
whole chapter in my book,
Alex's Adventures in Numberland,
about pi, because
it's kind of the most famous
number in math.
It's the sort of celebrity
number.
Pi is the simplest possible
ratio of the
simplest possible shape.
So, the simplest possible shape
being a circle, really.
And pi is the ratio of all the
way around it to all the way
across it--
its circumference
to its diameter.
ROGER BOWLEY: And the
Babylonians first thought it
was about three.
In fact, they wrote
it down as three.
ALEX BELLOS: What is interesting
is that it's the
simplest possible ratio of the
simplest possible shape, but
it's the most complicated,
kind of ugly, number.
It doesn't slice neatly.
It's what's called an irrational
number, which means
that if you were express it as
a decimal, the decimals just
go on forever without
ever repeating.
And people have been fascinated
by this idea that
something so simple, something
so basic, it sort of lets you
into this mess and madness
and chaos.
ROGER BOWLEY: So I would like,
with your permission, to talk
about the work that was done by
Archimedes which has been
discovered.
And he did it in the
following way.
ALEX BELLOS: And so this was
this brilliant kind of
narrative of human expedition.
Like kind of trying to get to
the moon and then to get to
the next planet, of trying to
find more and more digits.
ROGER BOWLEY: He took
a unit circle--
so I'm going to make a unit
circle on here, and draw it
out in green, so that you can
see, that's a simple circle.
MALE SPEAKER 1: Lovely
circle, Roger.
ROGER BOWLEY: I know,
I couldn't have
drawn it better freehand.
ALEX BELLOS: Also, it was one
of the first numbers to gain
its own symbol.
Because you can't keep writing
3.1415 all the time.
And once it gained the pi,
probably short for periphery,
the symbol, it just became
quite iconic.
ROGER BOWLEY: Now I'm going to
draw inside, a triangle.
And I'll try and make this
have equal sides.
So we call it an equilateral
triangle.
ALEX BELLOS: So just say you
were an ancient Egyptian wheel
maker or something.
Pi is going to come up, because
you're going to need
to be calculating the size
of rims and spokes,
and things like that.
ROGER BOWLEY: Now if I were to
walk around this path, all the
way, all the way around, that
is a longer path for me to
walk than the path that goes
in a straight line between
these points, between
these points, and
between these points.
So you can see this perimeter
is bigger than that.
So then he went to the next
stage and put another little
triangle in here, another one
there, and another one there.
And you get a hexagon, that just
means six equal sides.
And this is getting closer in
length to this perimeter.
And then he went to the
next stage, to make
something with 12 sides.
So you end up with a
12 sided figure.
Then he did 24, then he did
48, then he did 96.
And this length is still less
than the radius of the circle,
because when you go in a
straight line, it's quicker
than going on a curved line.
And he got out pi, which is--
the circumference is 2
pi times the radius--
this pi he gets out is
3 plus 10 over 71.
That's a bound.
Pi has got to be greater than
this because this curved
surface is greater
than this number.
When you have a 96-sided
regular figure.
So, that wasn't the end of the
story because that doesn't
give you the answer.
He then considered a triangle
that goes outside with three
equal sides.
Now clearly, this
is much longer.
You go much further than
going along this path.
So this is going to give a
greater bound than pi.
And then he filled it in with
lines and with lines.
And now you've got a hexagon
that goes around this circle.
And then he filled it in with
more, and he carried on like
this 'til the cows come home.
He went all way up to 96-sided
figures again, and that gave
him the other bound, which
is, pi is less than 3
plus 10 over 70.
ALEX BELLOS: And using that
method, Archimedes sort of
upped the ante, and
got much closer.
And that was for, I think, 1,000
years, was basically how
people estimated pi.
ROGER BOWLEY: And so he got
two limits for this, and
worked out from that, that
pi is about 3.1412.
ALEX BELLOS: The next great jump
in understanding how to
calculate pi required the era
in the Enlightenment and the
invention of calculus.
So even slightly pre-calculus,
the idea of
the infinite series.
The simplest infinite series for
pi is pi over 4 is equal
to 1 minus 1/3 plus 1/5 minus
1/7, plus 1/9, et cetera, et
cetera, et cetera.
So, what you need to do is you
just need to add as many of
the terms on as possible--
which is kind of zigzagging
and still honing in on pi.
And the more you do, the
closer you'll get pi.
ROGER BOWLEY: People do it
on computers and get into
hundreds of places of
decimals or more.
I have no idea how accurately.
This is the realm of
the super geek.
ALEX BELLOS: The better
the computers got, the
more digits they want.
And now, I think 3 or 7 trillion
digits in pi has been
the last ones decided.
Remembering the digits in pi
is just as difficult--
just as easy--
just exactly the same-- as the
digits in the square root of
2, the square root of 3, phi,
E, but no one wants to
memorize those ones.
They want to memorize pi.
MALE SPEAKER 2: I wouldn't be
doing my job properly if I
didn't ask you how many digits
you can remember pi to.
ALEX BELLOS: Five.
3.1459.
Maybe there's a two
after that?
Yeah, my memory is not--
I don't like to spend my spare
time memorizing pi.
ROGER BOWLEY: I got 3.14159.
And I've got a mnemonic.
ALEX BELLOS: The beauty of pi
is fascinating, but just
memorizing things
is not my bag.
ROGER BOWLEY: "How I Like To
Drink, Alcoholic Of Course,
After Two Heavy Lectures
Involving Quantum Mechanics."
ALEX BELLOS: At first, people
were sort of disconcerted by
the fact that pi--
the numbers in pi--
the digits never repeat
ROGER BOWLEY: "How" has three
letters, "I" has one--
3.1.
"Like," four.
"To"--
3.142.
"Drink," five.
"Alcoholic," nine.
ALEX BELLOS: But actually, this
fact they don't repeat is
actually what's fascinating.
ROGER BOWLEY: "Of," two.
And then we're getting
where I don't know.
2-6, "Course." "After," five.
The Heavy, Lectures.
ALEX BELLOS: The pattern.
They're so devoid of any
pattern, that actually, the
digits in pi, if taken as random
numbers, are the most
random numbers that we
know of, really.
They pass sort of
every test for
randomness with flying colors.
ROGER BOWLEY: "After," five.
"The," three.
"Heavy," five.
"Lectures," I can't
count-- eight--
Involving Quantum Mechanics.
And you can get even longer
ones than these.
These are ways of remembering
if you remember the phrase.
But of course, I can't
remember the phrase.
I prefer to remember
the number.
ALEX BELLOS: What you had during
the 1970s and '80s was
a sort of arms race between
America and Japan where the
two great tech nations
developing their
supercomputers.
And really--
no one really cares what digit
the two billionth digit in pi
is, but you want to do it
because it shows how strong
your computer is.
They're not interested in the
digits in pi because it's
going to be any use in
terms of doing any
calculations with circles.
Because just say, your
high-precision wheel design or
something--
or even if you do something
for like a spaceship--
10, pi to 10 decimal
places, that's
probably more than enough.
You're just never going
to need that much.
But now we have it to several
trillion decimal places.
ROGER BOWLEY: To a physicist,
there's
an engineering approach.
That if you've got it on your
calculator to enough
significant figures, you
really don't care.
Because most of the time when
we're working in physics, you
work to two or three significant
figures.
ALEX BELLOS: It's good for
testing computers.
And it's also fantastic
have this set of
beautifully random numbers.
It's kind of perfect chaos.
[BEEPING SOUNDS]
