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Golden ratio

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The golden ratio (golden section, golden mean) is the positive root
ϕ
of the quadratic equation

which has roots

Note that

Decimal expansion of the golden ratio

The decimal expansion of the golden ratio (A001622) is

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

Since

the multiplicative inverse of the root
x
is
x − 1
(same fractional part), and since
the root
x
added with the additive inverse of its multiplicative inverse also gives
1
.

Powers of ϕ and Fibonacci numbers

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)

since

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

since

Approximations

where
e
is Euler's number.

As an infinite series

See also


  • {{Fibonacci}} (mathematical function template)