Golden ratio

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Numbers

Golden ratio

$\varphi$:

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695365

$\varphi^{-1}$:

0.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695365

$\hat{\varphi}$:

-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 its conjugate $\hat{\varphi} = 1-\varphi$.

(2)

$\varphi^{-1} = \varphi - 1$.

(3)

$\hat{\varphi} = -\varphi^{-1} = 1-\varphi$.

Comments

(4)

$\varphi$ 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]

Links

[1]

Wikipedia: Golden ratio

[2]

OEIS: Decimal expansion of the golden ratio phi (or tau) = (1 + sqrt(5))/2.

Data properties

Entries are of type: real number

How well the digits are known: proven

How they were obtained: Checked here against ball arithmetic at 4000 bits, which covers every stored digit rather than the first hundred: all three entries agree to within one unit in the last place of the 300 digits held. The values are algebraic, so any number of further digits can be had on demand.