Approximate pi to 24 digits via keyboard

[Dieter]

(01-30-2015 06:35 PM)Rick314 Wrote:  This works with several HP calculators I've tried it on, but not on other scientific calculators like a TI-83+ for example.

It works on calculators with a sufficient number of internal digits of pi. For instance, HP's classic 10-digit machines (at least since the mid-Seventies) only use 13 digits, so sin 3,141592653 will not return more than 5,90 E–10. 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.

BTW, instead of the sine you can also use the tangent – just with a negative sign. Actually the number of digits roughly doubles with each step. Using the tangent finally is nothing but a Newton iteration for solving sin x = 0: the quotient of the function and its derivative is sin(x)/cos(x) = tan(x). ;-)

FWIW, the WP34s in double precision mode (34 digits) returns 17 additional correct digits (both with sin and tan). So I assume the internal representation of pi has about 50 digits – or the sine/tangent algorithms may have somewhat limited accuracy for border cases like these.

Dieter