Numbers that represent pi

Date: 08/08/97 
From: Edith N. Ferrer
Subject: Numbers that represent pi

If pi is an inexact and transcendental number, how in the world 
did we obtain a certain series of numbers to represent pi?
 
In other words, what value circumference did we divide by 
what value diameter to arrive at the accepted pi value of 
3.141592653589 etc....? 

Please help. Thank you very much.

Date: 08/08/97 at 12:39:39
From: John Conway
Subject: Re: Numbers that represent pi

This problem has had a long history.  Archimedes proved that
pi was between  3 + 1/7   and   3 + 10/71  by working out bounds
for the area of a 96-sided regular polygon, and various other 
authors used the same method with even larger polygons to obtain
closer bounds, culminating with Ludolph van Ceulen, who worked on 
the problem all his life,  eventually got a value to 35 places, and
had it engraved on his tombstone.

But over the last 300 years far better methods have been found,
so we now work out pi analytically rather than using geometry.  
Here for instance is Machin's formula for pi (from which you can 
quite quickly get its value to (say) 10 places by hand, or a few
hundred thousand by computer:

              1        1         1         1         1
let    a  =  ---  -  -----  +  -----  -  -----  +  -----  -  ...
              5      3x5^3     5x5^5     7x5^7     9x5^9

              1        1         1         1         1
and    b  =  ---  - ------- + ------- - ------- + ------- - ...  .
             239    3x239^3   5x239^5   7x239^7   9x239^9        

    Then  pi = 16a - 4b.                 

John Conway

Date: 08/09/97 at 12:28:10 (PDT)
From: Susan Addington
Subject: Re: Numbers that represent pi

Here is an addition to John Conway's reply to Edith Ferrer.

Pi is _not_ an inexact number. It is not a rational number (fraction),
but just because you can't write a number as a fraction (equivalently,
a terminating or repeating decimal) doesn't mean that it can't be
exactly described by other means.

Another way of exactly describing a number is by giving a polynomial
that it satisfies. For example, the square root of two satisfies
x^2 - 2 = 0 (by definition). However, pi can't be described
this way, either - there are no polynomials with rational number
coefficients that pi satisfies. That's what transcendental means - 
it has no polynomial definition. (A number is called algebraic if
it _can_ be defined by a polynomial.)

Susan Addington