Approximate pi to 24 digits via keyboard

[Rick314]

(01-31-2015 07:06 AM)Dieter Wrote:  It works on calculators with a sufficient number of internal digits of pi... Newer 12-digit calculators work differently – I do not know how many digits of pi are stored internally – and so you get 12 additional digits.

Hi Dieter. I think what you said here is incorrect. The "newer 12-digit calculators" store the value of pi internally to 12 digits. The point of this post is that calculators vary in their sin(x) algorithm for values of x very close to pi. Just speaking pure math, not calculators or rounding errors or related, if x = 3.14159265358 (exactly) then sin(x) = 9.793238462643383...e-12.

Update: By the way, eventually the "9.793238462643383..." digits in the previous sentence will deviate from the digits of pi, because sin(pi - eps) only approximately equals eps. In this case eps is about 1 count in the 11th decimal place, or 1e-11. Since sin(x) for small x equals approximately x - x^3/3! the digits will start deviating around (1e-11)^3 or the 33rd decimal place. Sure enough, 1e11 * sin(3.14159265358) = 0.
979323846264338327950272... and pi = 3.14159265358
979323846264338327950288... with deviation at the 34th decimal place of pi.