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Golden ratio
From OeisWiki
The golden ratio (golden section, golden mean) is the positive root
ϕ |
- x^{2} − x − 1 = 0,
which has roots
Note that
Contents |
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
x |
x − 1 |
- x + [ − (x − 1)] = 1,
x |
1 |
Powers of ϕ and Fibonacci numbers
ϕ |
Fn |
n |
ϕ |
n |
ϕ n = Fn −1 + Fn ϕ |
ϕ −n + ϕ n |
5 + 8 ϕ |
3 + 5 ϕ |
2 + 3 ϕ |
1 + 2 ϕ |
1 + 1 ϕ |
0 + 1 ϕ |
1 + 0 ϕ |
−1 + 1 ϕ |
2 + (−1) ϕ |
−3 + 2 ϕ |
5 + (−3) ϕ |
−8 + 5 ϕ |
13 + (−8) ϕ |
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
- φ^{2} − 1 = φ.
Approximations
e |
As an infinite series
See also
- {{Fibonacci}} (mathematical function template)