[AntonioLao]

The analysis of the inverse function f(x)=1/x and the rational function g(x)=x/(1+x) suggests their natural golden connection to the natural base of logarithm, e. This base is used practically everywhere in the formulations of scientific discoveries for both field and particle theory even in the growth and decay of living things. In quadratic forms, the golden mean can be written as x+x-1=0 if x is less than unity or as x-x-1=0 if x is greater than unity. Equivalently, the first is given by equating (1/x)=x+1 and the second is given by equating (1/x)=x/(1+x). However, equating (x+1)= x/(1+x) would not make any sense unless x is complex. Nonetheless, (1/x)=x/(1+x) is simply the same as equating f(x)= g(x).

Further analyses give the following results: (1) the derivative of f(x) is -1/x (2) the derivative of g(x) is 1/(1+x) (3) the limit of f(x) as x®0 is undefined (4) the limit of g(x) as x®0 is zero (5) the limit of f(x) as x®infinity is zero (6) the limit of g(x) as x®infinity is unity (7) the integral of f(x) is ln|x| ( the integral of g(x) is x-ln|1+x|. Coincidently or maybe not, the clincher is that the natural base of logarithm is defined as the limit of the x-power of the reciprocal of g(x) as x®infinity.