> 3*pi*(pi^3 - pi^2) = 2*pi*(pi^3 - pi^2) + 3(pi^3 - pi^2) + pi > > and pi is a root of a nontrivial polynomial in the rationals. > > Since pi is transcendental, this is impossible. The > argument generalizes...
Aren't you mistaking pi for a variable here? In this case it is the base or just 3.14159265358979323846264338327950... and not what is being solved for. Or we could move away from trancendental numbers if you prefer. How about the golden mean instead? (sqroot(1.25) + sqroot(.25)) ~= 1.6180339887498948482045868343656 Coincidentally the inverse (1/x) of the golden mean (gm) is .6180339887498948482045868343656 Or did you prefer something greater than two? How about (sqroot(2) + 1) ~= 2.4142135623730950488016887242097 Coincidentally the inverse (1/x) is .4142135623730950488016887242097
