|Simple: Pi to 24 digits on later HP calc's |
Message #10 Posted by Karl Schneider on 9 Dec 2005, 2:19 a.m.,
in response to message #9 by Crawl
Utilizing the identity
sin(A - B) = sin(A)*cos(B) - cos(A)*sin(B),
sin(pi - x) = sin(pi)*cos(x) - cos(pi)*sin(x)
is easily obtained.
Let "x" represent the entire string of missing digits when the value of pi is truncated (not rounded) to a certain number of decimal places.
sin(x) ~= x for small x, and the approximation improves toward perfection as x approaches zero.
Therefore, entering the first "n" digits of pi and taking its sine in radians mode should accurately yield a string of immediately-subsequent digits.
Best results for this simple procedure are obtained using an HP calculator with a Saturn microprocessor. These offer 12 significant digits and a faster processing speed that enables the functions to be computed more accurately.
In radians mode on a HP-20S,
sin (3.14159265358) = 0.00000000000979323846264
and pi = 3.14159265358979323846264, correct to 24 digits.
The difference between x and sin(x) for this very small value of x is guranteed to be less than (1/3)*x3 (the next term of the Taylor series for sine), which is much smaller than x.
(I assume that one could probably use "EXACT" mode on an HP-49 to obtain more digits of pi.)
On a 10-digit pre-Saturn HP calc, the 11th through 20th digits of pi cannot be obtained by this procedure. The algorithms apparently don't carry out the series for sine to as many terms, probably owing to lack of processor speed. Alas, tradeoffs had to be made...
In radians mode on a HP-15C or 41C,
sin (3.141592653) = 0.00000000059
and pi = 3.14159265359, rounded to 12 digits.
Edited: 9 Dec 2005, 2:37 a.m.