Golden ratio (Phi, 1618...) and its inverse number (phi, 0,618...) have multiple identities and peculiarities

Phi -1 = 1/Phi = phi 

phi +1 = 1/phi = Phi = x

Phi^2 = Phi +1 and thus x^2 -x -1 = 0, which can be generalized: x^n - x^n-1 = x^n-2, n is a real number bigger than 2.

                                    x^2 -x -1 = 0 is the first and only polynomial, with a unique solutions:

using quadratic formula, we get (1 + or - squareroot(5))/2, where

                                    x+ =  Phi

                                    x- = -phi

If n > 2, other solutions ( x(n-2), x(n-3, ... x2, x1) are zero.

as said earlier, x - 1 = 1 / x (which is x^2 - x - 1 = 0 divided by x), one can make a continued fraction representation: x = 1 + 1 / x thus x = 1 + (1 / (1 + 1 / x) thus x = 1 + (1 / (1 + (1 / 1 + ...

                    , x^2 = x +1, one can make nested radical representation: x = squareroot(1 + x), therefore x = squareroot(1 + squareroot(1 + squareroot(1 + ...

thus one can say that x = 1 + (1 / (1 + (1 / 1 + ... = squareroot(1 + squareroot(1 + ...

Because x + (1/x) = 1, it can be used to combine Phi with e in Euler's identity, as Richard Feynman did: e^i 2pi = Phi - phi.