Is there a number that 1 compares to such that the # compares to itself + 1?

Question

example: 1/x= x/(x+1)

Answer 1

There are two: φ and -1/φ

Proof: suppose 1/x=x/(x+1). Then multiplying by x(x+1) we see that x+1=x², so x²-x-1 = 0 and by the quadratic formula, x=(1±√5)/2. φ=(1+√5)/2 by definition, and (1-√5)/2 = (1-5)/(2(1+√5)) = -2/(1+√5) = -1/((1+√5)/2) = -1/φ.