Golden ratio

Description

The golden ratio is the real number $$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887.$$ It is the positive root of the polynomial $x^2 - x - 1$, so $\varphi^2 = \varphi + 1$. The other root is $\psi = \frac{1 - \sqrt{5}}{2} = 1 - \varphi = -\frac{1}{\varphi}$.

Remarks

Two positive numbers $a > b$ are in the golden ratio when $\frac{a + b}{a} = \frac{a}{b}$. Then $\frac{a}{b} = \varphi$. The number $\varphi$ is irrational, because $\sqrt{5}$ is irrational. The ratios of consecutive Fibonacci numbers tend to $\varphi$. Binet's formula uses $\varphi$ and $\psi$.

Explanations

L1 human

Cut a stick into a long part and a short part. Do it so that the whole stick relates to the long part as the long part relates to the short part. The ratio of long to short is then $\varphi$, about $1.618$.