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Golden ratio
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The golden ratio (golden section, golden mean) is the positive root
of the quadratic equation
of the golden ratio is
is
(same fractional part), and since
added with the additive inverse of its multiplicative inverse also gives 1.
is the golden ratio and
is the
th Fibonacci number.
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
is Euler’s number.
ϕ |
-
x 2 − x − 1 = 0,
which has roots
-
ϕ =
, φ =1 + √ 52
.1 − √ 52
Note that
|
Contents
Decimal expansion of the golden ratio
The decimal expansion of the golden ratio (A001622) is
-
ϕ = 1.6180339887498948482045868343656381177203091798057628621...
ℚ [ √ 5 ] |
-
φ = − 0.6180339887498948482045868343656381177203091798057628621...
Since
-
x (x − 1) = 1,
x |
x − 1 |
-
x + [− (x − 1)] = 1,
x |
Powers of ϕ and Fibonacci numbers
-
ϕ n =
n = Fn − 1 + Fn ϕ,1 + √ 52
ϕ |
Fn |
n |
ϕ |
n |
ϕ n = Fn − 1 + Fn ϕ |
ϕ − n + ϕ n |
Continued fraction and nested radicals expansions
The golden ratio has the simplest continued fraction expansion (the all ones sequence A000012)
ϕ = 1 +
|
since
-
ϕ − 1 =
,1 ϕ
and also the simplest nested radicals expansion (again, the all one’s sequence)
ϕ = √ 1 + √ 1 + √ 1 + √ 1 + √ ⋯ = 1 + [1 + [1 + [1 + [1 + [1 + ⋯]
|
since
-
ϕ 2 − 1 = ϕ.
Approximations
-
e −
= 1.61828182845904... (1.000153173364... × ϕ),11 10
e |
- √= 1.6180215937964... (0.999992339... × ϕ).
5 π 6
Infinite series
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|
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See also
- {{Fibonacci}} (mathematical function template)