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The golden and silver ratios are the roots of the equation $x^2-x-1=0$: $$\frac{1\pm\sqrt{5}}{2}.$$ They show up in the formula of Fibonacci numbers: $$F_n=\frac{1}{\sqrt{5}}\left(\frac{1+\sqrt{5}}{2}\right)^n+\frac{1}{\sqrt{5}}\left(\frac{1-\sqrt{5}}{2}\right)^n$$

Are the roots of the equation $x^2+x-1=0$ any significant and do they have some special names: $$\frac{-1\pm\sqrt{5}}{2}?$$

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    $\begingroup$ The silver ratio is $1+\sqrt{2}$. $\endgroup$ – Raskolnikov Jun 25 '18 at 16:42
  • $\begingroup$ Changing the equation from $x^2-x-1=0$ to $x^2+x-1=0$ just changes the signs of the two roots, so they don't need much of a separate naming from the golden ratio. One just needs a bit of care in explaining an application of them. $\endgroup$ – hardmath Jun 25 '18 at 16:46
  • $\begingroup$ @Raskolnikov, in this post it was called that way. $\endgroup$ – bilgamish Jun 25 '18 at 16:47
  • $\begingroup$ The ratio can not be negative though. So, I assume the term "silver" for $\frac{1-\sqrt{5}}{2}$ is not appropriate. $\endgroup$ – bilgamish Jun 25 '18 at 18:57
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EDIT: For $x^2 + x - 1 = 0$, there is nothing really significant or noteworthy about the roots, other than one of them is $-\phi = \dfrac {-1-{\sqrt 5}}{2}$ or $-1.618$, and the other is $\dfrac {1}{\phi} = \dfrac {-1+{\sqrt 5}}{2}$ or $0.618$.

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  • $\begingroup$ Ok, the terminology needs to be fixed. How about the main question: is it significant and does it have any name or is it applied in any known problem? $\endgroup$ – bilgamish Jun 25 '18 at 18:59
  • $\begingroup$ Whoops. I've changed the answer...the short explanation is that there's nothing significant other than sign. $\endgroup$ – bjcolby15 Jun 25 '18 at 20:24

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