Square of Golden Mean equals One plus Golden Mean

Theorem

$\phi^2 = \phi + 1$

where $\phi$ denotes the golden mean.

Decimal Expansion

The decimal expansion of $\phi^2$ is given as:

$\phi^2 \approx 2 \cdotp 61803 \, 39887 \, 49894$

Thus the square of the golden mean is the unique number $n$ such that:

$\sqrt n = n - 1$

Proof

 $\ds \phi$ $=$ $\ds \frac 1 {\phi - 1}$ Definition 3 of Golden Mean $\ds \leadstoandfrom \ \$ $\ds \phi \paren {\phi - 1}$ $=$ $\ds 1$ $\ds \leadstoandfrom \ \$ $\ds \phi^2$ $=$ $\ds \phi + 1$

$\blacksquare$