Can you calculate Pi using a Solver?

[Valentin Albillo]

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Self-quoting a little:

(12-13-2019 11:30 PM)Valentin Albillo Wrote:  Unless the Solver can work with complex numbers I don't think there's such an equation using just real numbers and no trigs. Equations whose roots are arbitrarily accurate approximations to Pi (say 34 digits) can be produced but ones returning exactly Pi (in theory, limited to 10-12 digits in practice) is a no-go IMHO.

Well, to make myself clearer, the previous quotation refers to polynomials or rational functions, which indeed can't produce Pi as an exact root because Pi is transcendental.

But if we involve other functions but polynomials, rational functions and trigs, there are many ways to concoct an equation whose root is exactly Pi. For instance, if your Solver admits nesting and can use the Gamma function, then this will do:

Code for the HP-71B, where FNROOT (Find Root) is the "official" HP-71B's "Solver":

     1   DEF FNF(X) = FNROOT(1, 1, GAMMA(FVAR) - X)
     2   DISP FNROOT(3, 4, FNF(SQR(FVAR)) - 1/2)

>RUN

          3.14159265359

This produces as a root the theoretically exact (not an approximation) value of Pi.

V.