Is there a number besides $\phi$ that either squared or added one gives the same answer?

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.

• How about $\frac{1-\sqrt5}{2}$? – sqtrat Apr 1 '16 at 0:02
• $1-\phi$ also works. It is a quadratic equation, and quadratics never have more than 2 solutions. – Doug M Apr 1 '16 at 0:03
• 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). – Jared Apr 1 '16 at 0:04
• The other day you were interested in continued fractions. $\phi$ has the simplest one, and you can see how it leads to $\phi = 1 + \frac{1}{\phi}.$ – Will Jagy Apr 1 '16 at 0:17
• 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" – 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 $1-\phi$.

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