More on PI
Gary Kremen
1) There was a recent posting on the exactness of pi/4 = 4*atan(1/5) -
atan(1/239). This is a true relationship. You can easily prove this by
taking the tangents of both sides and using the tan(a+b) relationship.
Quick proof:
tan x = 1 / 5
tan 2x = 2 tan x / (1 - tan x * tan x) = 5 / 12
tan 4x = 2 tan 2x / (1- tan 2x * tan 2x) = 120 / 119
> tan (4x - PI/4) = (tan 4x - 1) / (1 + tan 4x) = 1 / 239
>>> atan(1 / 239) = 4x - PI/4
= 4 atan(1/5) - PI/4
2) Sheldon Meth talked about the simplicity of using the expansion of
atan(1) which is equal to pi/4. Yes, it is simple but it converges
extremely slow. Machin's 1706 series is much better.
3) There is a new method of calculating PI discovered in 1976. It is
asymetrically faster than using arctan methods. It is pretty complex. If
there is any interest I will post info to the net.
4) In 1983 I computed PI to 225,000 digits using IBM 4341. No problem,
but it took 1.1 days of machine time. It was lucky I was the machine's
only operator. I wrote the program in very optimized FORTRAN/assembly
code. Recently I rewrote it in C, for use on microcomputers.
5) There is a book "A History of Pi" by Peter Beckman. It is well worth
reading even if you don't have a mathematics background. He explains how
history of progress in Pi and mathematics mirrors man's history. For
example - during the dark ages the most enlighted countries have the
most mathematics progress.
6) Current Pi record is around 16 million places.
--
Name: Gary Kremen
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