Golden ratio

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The Golden ratio (Golden section, Golden mean) is the positive root $\scriptstyle \phi\,$ of the quadratic equation

$x^2-x-1 = 0, \,$

which has roots

$\phi = \frac{1+\sqrt{5}}{2},\ \varphi = \frac{1-\sqrt{5}}{2}. \,$

Decimal expansion of the Golden ratio

The decimal expansion of the Golden ratio is

$\phi = 1.6180339887498948482045868343656381177203091798057628621 \ldots \,$

A001622 Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.

{1, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, 9, 8, 0, 5, 7, 6, 2, 8, 6, 2, 1, 3, 5, 4, 4, 8, 6, 2, 2, ...}

and the decimal expansion of the conjugate root of the Golden ratio is (it has the same fractional part)

$\varphi = - 0.6180339887498948482045868343656381177203091798057628621 \ldots \,$

Since

$x \, (x-1) = 1, \,$

it means that the multiplicative inverse of the root $\scriptstyle x \,$ is $\scriptstyle x-1 \,$ (same fractional part), and since

$x + [-(x-1)] = 1, \,$

it means that the root $\scriptstyle x \,$ added with the additive inverse of its multiplicative inverse also gives 1.

Powers of φ and Fibonacci numbers

$\phi^n = \bigg(\frac{1+\sqrt{5}}{2}\bigg)^n = F(n-1) + F(n) \, \phi, \,$

where $\scriptstyle \phi\,$ is the Golden ratio and $\scriptstyle F(n) \,$ is the $\scriptstyle n \,$ ^th Fibonacci number.

**Powers of phi**
$n \,$	$\phi^n = \,$ $F(n-1) + F(n) \phi \,$	$\phi^{-n} + \phi^{n} \,$
6	5 + 8 $\scriptstyle \phi \,$	18
5	3 + 5 $\scriptstyle \phi \,$
4	2 + 3 $\scriptstyle \phi \,$	7
3	1 + 2 $\scriptstyle \phi \,$
2	1 + 1 $\scriptstyle \phi \,$	3
1	0 + 1 $\scriptstyle \phi \,$
0	1 + 0 $\scriptstyle \phi \,$	2
-1	-1 + 1 $\scriptstyle \phi \,$
-2	2 + -1 $\scriptstyle \phi \,$	3
-3	-3 + 2 $\scriptstyle \phi \,$
-4	5 + -3 $\scriptstyle \phi \,$	7
-5	-8 + 5 $\scriptstyle \phi \,$
-6	13 + -8 $\scriptstyle \phi \,$	18

Continued fraction and nested radicals expansions

The Golden ratio has the simplest continued fraction expansion (the all ones sequence A000012)

$\phi = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{\ddots}}}}} = 1 + \sqrt[-1]{1 + \sqrt[-1]{1 + \sqrt[-1]{1 + \sqrt[-1]{1 + \sqrt[-1]{1 + \cdots}}}}}, \,$