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

Question

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.

Answer 1

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 $1-\phi$.

Answer 2

If you consider split-complex numbers or tessarines, then there are four solutions:

$$\frac{1+\sqrt{5}}2, \frac{1-\sqrt{5}}2, \frac{1+j\sqrt{5}}2, \frac{1-j\sqrt{5}}2$$

The last two solutions are not real or (usual) complex numbers.