> From: David Hobby <[EMAIL PROTECTED]> > > The Fool wrote: > > > > > 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. > > This is the meaning of the word "root".
But we are treating it as a base, a 'known' number that does not need to be solved for. You are very mistakenly treating it as a variable here. > > > > 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 > > No coincidence. The golden mean, phi, is a root of > x^2 - x -1 = 0, which is equivalent to x - 1 = 1/x. Other > algebraic (i.e. not transcendental) numbers will have similar > relationships... And I knew that being that I gave the first two non zero integers in that series. x = 1-1/x (sqrt(1.25) +- sqrt((1.25 - 1))) x = 2-2/x (sqrt(2) +- sqrt((2 - 1)))
