Golden ratio
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Numbers
Golden ratio
$\varphi$:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695365
$\varphi^{-1}$:
-0.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695365
Definition
$\varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio.
Formulas
(1)
$\varphi^2-\varphi-1=0$, which has two roots, $\varphi$ and $\varphi^{-1}$.
(2)
$\varphi^{-1} = 1 - \varphi$.
(3)
$\varphi$ equals the ratio $a/b$ of those positive real numbers $a$ and $b < a$, such that $\frac{a+b}{a} = \frac{a}{b}$.
Programs
(P1)
Sage
numbers = [golden_ratio]