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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

derive
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    There is nothing wrong with your way. You just have to change the sign. – ctst Oct 05 '16 at 13:35
  • ok I give up. What sign are you talking about? (where?) – derive Oct 05 '16 at 13:47
  • You want to calculate the length of the left side and you got $x_{1,2}=\frac{3\pm \sqrt{5}}{2}$. Now the length of the left side is $1-x=1-\frac{3\pm \sqrt{5}}{2}=\frac{2-3\mp \sqrt{5}}{2}=\frac{-1\mp\sqrt{5}}{2}$. This is just the same as before (mod swapping the ordering/sign at $\sqrt{5}$) – ctst Oct 05 '16 at 14:09
  • yup thats it, I think that I got confused because the first version gives the phi and 1/phi directly as solution of the equation (because it involves 1/x at the left side). THX man – derive Oct 05 '16 at 14:21

1 Answers1

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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 $$ 1 - \frac{-1-\sqrt5}{2} = \frac{3+\sqrt5}2 $$

hmakholm left over Monica
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