Numbers that represent piDate: 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 |
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