Re: Pi

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Re: Pi

Colin Guillas writes:

> Anyway, using a loop, this is PI:
>
>  PI=4(1-1/3+1/5-1/7+1/9-1/11...)
>
> If you represent the fractions as 1/a and keep adding two to a, you can
> keep a running total.
> There are workarounds on 386 and 486 processors using QuickBasic that
> allow you to calculate to 127 (or so) decimal places.  If you run the
> program and leave the computer for the appropriate numbers of
> minutes/hours, you'll comeback to the right value, if you double check
> it.

This is certainly an expression for pi. It was one of the earliest expressions
based on an infinite process to be discovered.  I believe (without looking
it up) it is generally attributed to Gregory, about 1600.

This expression also has the rather dubious distinction of being about the
_slowest_ known process for calculating pi!  Because it is an alternating
series
with decreasing terms, the error at each step is roughly comparable to the
size of the term being evaluated, which only gets very slightly smaller at
each successive step.  To get six or seven places of accuracy you need to add
up some _billions_ of terms.  No matter what processor you chose to use, in
the lifetime of the universe you could not work out any interesting number of
decimal places.

If you _did_ want to investigate convergence of this series, it might make
sense to calculate every _other_ partial sum, to get either an increasing or
a decreasing progression.  Thus

        pi = 4(1-1/3) + 4(1/5-1/7) + ...  = 8/3 + 8/35 + ...
or
        pi = 4 - 4(1/3-1/5) - 4(1/7-1/9) - ... = 4 - 8/15 - 8/63 - ...

In general, planning and forethought are a far better investment of time than
letting your calculator/computer run overnight when it comes to arriving at
difficult-to-compute values. As I have mentioned a few times here, running the
right sort of programs on even a TI-82 can quickly get you hundreds of places
of pi.  Next step -- what do you do with them?  :-)}

RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623

>>>> The plural of mongoose begins with p. <<<<
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