Golden ratio

Definitions

Symbol: GoldenRatio $\varphi$ The golden ratio (1.618...)

Numerical value

$\varphi \in \left[1.6180339887498948482045868343656381177203091798058 \pm 3.72 \cdot 10^{-50}\right]$
$\varphi = \frac{1 + \sqrt{5}}{2}$
$\varphi \notin \mathbb{Q}$

Algebraic equations

$\frac{1}{\varphi} = \varphi - 1$
${\varphi}^{2} - \varphi - 1 = 0$
$\left(\frac{a + b}{a} = \frac{a}{b}\right) \implies \left(\frac{a}{b} = \varphi\right)$
$\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} \left[{x}^{2} - x - 1\right] = \left\{\varphi, 1 - \varphi\right\}$
$\varphi = 1 + \frac{1}{\varphi}$
$\varphi = 1 + \frac{1}{1 + \frac{1}{\varphi}}$
$\operatorname{spec}\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \left\{\varphi, 1 - \varphi\right\}$

Trigonometric formulas

$\varphi = 2 \cos\!\left(\frac{\pi}{5}\right)$
$\varphi = 2 \sin\!\left(\frac{3 \pi}{10}\right)$
$\varphi = 2 \sin\!\left(\frac{\pi}{10}\right) + 1$

Recurrence relations

${\varphi}^{n + 1} = {\varphi}^{n} + {\varphi}^{n - 1}$
${\varphi}^{n} = F_{n} \varphi + F_{n - 1}$

Limit representations

$\varphi = \lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}}$

Special function representations

$\varphi = \frac{1}{5} {\left(\frac{\eta(i)}{\eta\!\left(5 i\right)}\right)}^{2}$

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2020-04-08 16:14:44.404316 UTC