Those who know golden ratio $\phi$ (phi) constant, know for sure that it is an interesting constant. It is roughly $\phi=1.618034...$ . It is present almost everywhere in nature and it has many very interesting properties.

One of the properties of $\phi$ is: $$\phi^2=\phi+1$$ Is there a constant like $\phi$ which squared gives the same answer as itself plus one or is phi special? Is it a real number?

Don't downvote for no reason please.

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    $\begingroup$ How about $\frac{1-\sqrt5}{2}$? $\endgroup$ – sqtrat Apr 1 '16 at 0:02
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    $\begingroup$ $1-\phi$ also works. It is a quadratic equation, and quadratics never have more than 2 solutions. $\endgroup$ – Doug M Apr 1 '16 at 0:03
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    $\begingroup$ This is an elementary quadratic equation...just solve it--there should be two solutions (clearly one of them is negative...as has already been provided). $\endgroup$ – Jared Apr 1 '16 at 0:04
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    $\begingroup$ The other day you were interested in continued fractions. $\phi$ has the simplest one, and you can see how it leads to $\phi = 1 + \frac{1}{\phi}.$ $\endgroup$ – Will Jagy Apr 1 '16 at 0:17
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    $\begingroup$ Well, something similar happens in the continued fraction for any $$ x = \frac{a + \sqrt b}{c} $$ with integers $a,b,c,$ also $b > 0$ but not a square, finally $x > 1$ but $$ -1 < \frac{a - \sqrt b}{c} < 0. $$ The continued fraction is "purely periodic" $\endgroup$ – Will Jagy Apr 1 '16 at 0:27

This is weird, but I'm answering my own question.

Actually just because $\phi$ is a constant kind of misled me, and I forgot that this is normal quadratic equation.

You can solve it like this:

$$-x^2+x+1=0$$ $$\frac{-1\,\pm\,\sqrt{1-4(-1)(1)}}{2(-1)}$$ $$\frac{-1\,\pm\,\sqrt{5}}{-2}$$ $$\frac{1\,\pm\sqrt{5}}{2}$$

Which gives $\phi$ and $1-\phi$.

  • $\begingroup$ Congratulations @KKZiomek, you have just earned a self-learner badge! :-) $\endgroup$ – Oscar Lanzi Apr 1 '16 at 0:30
  • $\begingroup$ I didn't even know that's a thing :) thanks. $\endgroup$ – KKZiomek Apr 1 '16 at 0:34
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    $\begingroup$ Look at the Badges section. Sf-learner means you answer your own question and draw +3 votes. $\endgroup$ – Oscar Lanzi Apr 1 '16 at 0:38
  • $\begingroup$ Now I see. I thought it was weird to answer my own question. $\endgroup$ – KKZiomek Apr 1 '16 at 0:40

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