Pi - Numberphile


Uploaded by numberphile on 12.03.2012

Transcript:

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]