What gives the golden ratio its unusual numerical properties

algebraic-number-theoryalgebraic-numbersgolden ratio

Why does the golden ratio (and by extension the other metallic means) have such unusual numerical properties?

For those who don't know, the golden ratio ($\varphi$) is the positive root of the quadratic $\varphi^2-\varphi-1$. It's equal to $\frac{1+\sqrt{5}}{2}$, and appears in nature. The limiting ratio of the Fibonacci numbers is the golden ratio. The golden ratio is a common object of interest of amateur mathematicians.

And in terms of strange properties, I'm talking about the following identities with the golden ratio:

$\varphi^2=\varphi+1$

$\varphi-1=\frac{1}{\varphi}$

$\varphi = 1+\frac{1}{\varphi}$ (follows from the one above it)

$\varphi = \sqrt{1 + \varphi}$

All of these follow from the golden ratio's quadratic, and the golden ratio conjugate $\Phi = \frac{1-\sqrt{5}}{2}$ displays the same properties. Is there a reason for these properties? Or am I just unfamiliar with how this works?

Edit: do all quadratic irrational numbers behave like this? There's actually an infinite set of numbers like the golden ratio called the metallic means, which are the positive roots of the quadratic $x^2-Nx-1$ for some natural number $N$. Metallic means take the form $\frac{N+\sqrt{N^2+4}}{2}$. They all display similar properties. So why does this class of numbers in general behave this way?

Best Answer

All quadratic integers have similar properties. Say that $\psi$ satisfies the equation $x^2=ax+b$, with $a,b\in \mathbb{Z}$.

Then:

  • $\psi-a = \frac{b}{\psi}$
  • $\psi = a + \frac{b}{\psi}$
  • $\psi = \sqrt{a\psi+b}$.

It's just that the equation for $\varphi$ corresponds to $a=b=1$ so the formulas look prettier.

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