golden ratio

the golden ratio is represented by the greek letter phi, $\varphi$

the golden ratio is derived from the concept that we divide a line into two parts so that the whole length divided by the long length is equal to the long length divided by the short length
we can see that the formula for the golden ratio is $\frac{a+b}{a}=\frac{a}{b}$ for a is the short length and b is the longer length
given we substitute the golden ratio as $\frac{a}{b}$, the formula becomes
$\varphi =1+\frac{1}{\varphi }$
it looks like the golden ratio can only be defined in terms of itself :/
luckily, we can find the exact value of the golden ratio using some algebra
${\varphi }^2=\varphi +1$
${\varphi }^2-\varphi -1=0$
let us use the quadratic formula , discriminant

$\varphi =\frac{1\pm \sqrt{5}}{2}$
since it is a ratio, it must be at least 1
$\varphi =\frac{1+\sqrt{5}}{2}$

$\varphi \approx 1.618$