This site is supported by donations to The OEIS Foundation.

Golden ratio

From OeisWiki

Jump to: navigation, search
The golden ratio (golden section, golden mean) is the positive root
ϕ
of the quadratic equation
x2x − 1 = 0,

which has roots

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

Note that

\phi + \varphi = 1,
\phi \, \varphi = -1.

Contents

Decimal expansion of the golden ratio

The decimal expansion of the golden ratio (A001622) is

\phi = 1.6180339887498948482045868343656381177203091798057628621 \ldots

and the decimal expansion of the conjugate root of the golden ratio is

\varphi = - 0.6180339887498948482045868343656381177203091798057628621 \ldots

Since

x \, (x-1) = 1,
the multiplicative inverse of the root
x
is
x − 1
(same fractional part), and since
x + [ − (x − 1)] = 1,
the root
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
ϕ
is the golden ratio and
Fn
is the
n
th Fibonacci number.

Powers of
ϕ

n
ϕn =
Fn −1 + Fn  ϕ
ϕ −n + ϕn
6
5 + 8 ϕ
18
5
3 + 5 ϕ
4
2 + 3 ϕ
7
3
1 + 2 ϕ
2
1 + 1 ϕ
3
1
0 + 1 ϕ
0
1 + 0 ϕ
2
−1
−1 + 1 ϕ
−2
2 + (−1) ϕ
3
−3
−3 + 2 ϕ
−4
5 + (−3) ϕ
7
−5
−8 + 5 ϕ
−6
13 + (−8) ϕ
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 + \left[ 1 + \left[ 1 + \left[ 1 + \left[ 1 + \left[ 1 + \cdots \right]^{-1} \right]^{-1} \right]^{-1} \right]^{-1} \right]^{-1},

since

\phi - 1 = \frac{1}{\phi},

and also the simplest nested radicals expansion (again, the all one's sequence)

\phi = \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}}} = 
\left[ 1 + \left[ 1 + \left[ 1 + \left[ 1 + \left[ 1 + \cdots \right]^{\frac{1}{2}} \right]^{\frac{1}{2}} \right]^{\frac{1}{2}} \right]^{\frac{1}{2}} \right]^{\frac{1}{2}},

since

φ2 − 1 = φ.

Approximations

e - \frac{11}{10} = 1.61828182845904\ldots (1.000153173364\ldots \times \phi),
where
e
is Euler's number.
\sqrt{\frac{5 \pi}{6}} = 1.6180215937964\ldots (0.999992339\ldots \times \phi).

As an infinite series

\phi = \sum_{k=0}^{\infty} \left( \frac{3 - \sqrt{5}}{2} \right)^{k} = \sum_{k=0}^{\infty} \left( 1 + \varphi \right)^{k} = 
\frac{1}{1 - (1 + \varphi)} = \frac{-1}{\varphi} = \phi.
\varphi = \sum_{k=0}^{\infty} \phi^{-2k} = \sum_{k=0}^{\infty} \left( \frac{1}{\phi^2}\right)^{k} = \sum_{k=0}^{\infty} \left( \frac{1}{\phi + 1} \right)^{k} = \frac{1}{1 - (\phi + 1)} = \frac{-1}{\phi} = \varphi.

See also


  • {{Fibonacci}} (mathematical function template)
Personal tools