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Golden ratio

Table of contents: Definitions - Numerical value - Algebraic equations - Trigonometric formulas - Recurrence relations - Limit representations

Definitions

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

Numerical value

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φ[1.6180339887498948482045868343656381177203091798058±3.721050]\varphi \in \left[1.6180339887498948482045868343656381177203091798058 \pm 3.72 \cdot 10^{-50}\right]
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φ=1+52\varphi = \frac{1 + \sqrt{5}}{2}
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φQ\varphi \notin \mathbb{Q}

Algebraic equations

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1φ=φ1\frac{1}{\varphi} = \varphi - 1
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φ2φ1=0{\varphi}^{2} - \varphi - 1 = 0
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(a+ba=ab)    (ab=φ)\left(\frac{a + b}{a} = \frac{a}{b}\right) \implies \left(\frac{a}{b} = \varphi\right)
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zerosxC(x2x1)={φ,1φ}\mathop{\operatorname{zeros}\,}\limits_{x \in \mathbb{C}} \left({x}^{2} - x - 1\right) = \left\{\varphi, 1 - \varphi\right\}
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φ=1+1φ\varphi = 1 + \frac{1}{\varphi}
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φ=1+11+1φ\varphi = 1 + \frac{1}{1 + \frac{1}{\varphi}}
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spec(1110)={φ,1φ}\operatorname{spec}\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \left\{\varphi, 1 - \varphi\right\}

Trigonometric formulas

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φ=2cos ⁣(π5)\varphi = 2 \cos\!\left(\frac{\pi}{5}\right)
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φ=2sin ⁣(3π10)\varphi = 2 \sin\!\left(\frac{3 \pi}{10}\right)
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φ=2sin ⁣(π10)+1\varphi = 2 \sin\!\left(\frac{\pi}{10}\right) + 1

Recurrence relations

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φn+1=φn+φn1{\varphi}^{n + 1} = {\varphi}^{n} + {\varphi}^{n - 1}
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φn=Fnφ+Fn1{\varphi}^{n} = F_{n} \varphi + F_{n - 1}

Limit representations

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φ=limnFn+1Fn\varphi = \lim_{n \to \infty} \frac{F_{n + 1}}{F_{n}}

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

2019-07-15 23:42:41.550119 UTC