Correct way for calculating Phi (golden ratio)
I've seen in some of the resources, that Phi is calculated in a way that the whole (=1) is divided into two parts:
---------------------x----------------------|---------1-x--------------
$\frac{1}{x}=\frac{x}{1-x}\\1-x=x^2 \space\space\space\space\space\space\space\space\space \rightarrow x^2+x-1=0 \space\space\space\space\space\space\space\space\space\rightarrow x_{1,2}=\frac{-1\pm \sqrt{5}}{2}$
This will get the correct values for phi eg.:0.618... and -1.618...
But it doesnt work, if I switch the values like this:
---------------------1-x----------------------|---------x--------------
$\frac{1}{1-x}=\frac{1-x}{x}\\x=x^2-2x+1 \space\space\space\space\space\space\space\space\space \rightarrow x^2-3x+1=0 \space\space\space\space\space\space\space\space\space\rightarrow x_{1,2}=\frac{3\pm \sqrt{5}}{2}$
Now, the wikipedia page gives another way and it works, but I wanted to ask what is wrong with the way I did it here because as I said, I've seen it quite frequently. Thx
$\endgroup$ 41 Answer
$\begingroup$You're getting the right result, but the $x$ in the first computation isn't the same number as the $x$ in the second one. Rather $x_{\rm second}$ should be $1-x_{\rm first}$, and indeed we have $$ 1 - \frac{-1+\sqrt5}{2} = \frac{3-\sqrt5}2 $$ and $$ 1 - \frac{-1-\sqrt5}{2} = \frac{3+\sqrt5}2 $$
$\endgroup$
