Diversion on Pi: Suppose
you where someplace where there were no computers , hand calculators or
handbooks and you wanted to find the value of Pi to one hunred places.
How would you do this? The choice available to you would be to use the
old Archimedes method of inscribing and circumscribing polygons about a
circle or to use the far superior method of a series expansion for this
number. The first approach was taken to its extreme by Ludolph van Ceulen
of Holland in 1596 allowing him to obtain (after a lifetime of calculations)
the value of Pi good to 35 places. In his honor Pi is still referred to
in parts of Europe as the Ludolph number. With the advent of calculus these
calculations could be much speeded up using infinite series. The first
of such series was that of Gregory who gave Pi=4(1-1/3+1/5-1/7+...). Unfortunately
this series is very slowly convergent and so requires a large number of
terms in the series to yield a reasonable approximation. The next improvement
came with Machin who was the first to employ a multiple arctan expansion.
His formula is Pi=16*arctan(1/5)-4*arctan(1/239) and allowed him to get
Pi to better than a hundred places, still taking a considerable amount
of time because of the relatively slow convergence of the arctan(1/5) series.
Later years brought considerable improvement in the convergence rate of
arctan formulas including those given by Euler, Gauss, Stormer etc. The
secret to quick convergence is that the number x in the arctan expansion
arctan(x)=x-x^3/3+x^5/5-... be small. In playing around with this requirement
about twenty years ago, I came up with the interesting four term arctan
formula-
Pi=48*arctan(1/38)+80*arctan(1/57)+28*arctan(1/239)+96*arctan(1/268)
This series gives about 3 extra places of accuracy for
each additional term taken in the arctan series . Thus, the first thirty
three terms in the expansion for the arctan in this formula will be enough
to yield Pi to about one hundred place accuracy. Such a calculation can
be carried out by hand in several weeks (I did not want to waste the time
doing so) or in about a second with computer programs such as Mathematica
or Maple( which I have done below). Can any of you prove the above result?
A book by Petr Beckmann on the History of Pi might help. It should be pointed
out that Pi has now been computed to some 50 billion places (as of 1997)
using recurrence formulas, series for 1/Pi, and supercomputers. Note that
if one where to print out the latest results for Pi at about 5000 digets
per page, it would take about ten million pages to do so, which corresponds
to a stack of papers about a mile high! The string of digits are non repetitive
as is to be expected for irrational numbers such as Pi , e and sqrt(2).
Knowledge of these digits could be of interest for cryptographic encoding.
On the other hand, for all practical engineering purposes , the easily
remembered six place accurate Otto ratio of 355/113 =3.1415929..for Pi
is good enough.
The value of Pi obtained by using a 31 term arctan
expansion(n=0,1,...30) in the above formula yields the 100 term accurate
result-
Pi=3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068