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# Golden ratio

The golden ratio (golden section, golden mean) is the positive root
 ϕ
 x 2 − x − 1  =  0,

which has roots

ϕ  =
 1 + 2√  5 2
, φ =
 1 − 2√  5 2
.

Note that

 ϕ + φ =  1, ϕ  φ =  −1.

## Decimal expansion of the golden ratio

The decimal expansion of the golden ratio (A001622) is

 ϕ  =  1.6180339887498948482045868343656381177203091798057628621...
and the decimal expansion of the conjugate root in
 ℚ [2√  5 ]
of the golden ratio is
 φ  =  − 0.6180339887498948482045868343656381177203091798057628621...

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

ϕn  =
 1 + 2√  5 2
n  =  Fn  − 1 + Fn ϕ,
where
 ϕ
is the golden ratio and
 Fn
is the
 n
th Fibonacci number.

Powers of
 ϕ

 n
 ϕ n = Fn  − 1 + Fn  ϕ
 ϕ  − n + ϕ n
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)

ϕ  =  1 +
1
1 +
1
1 +
1
1 +
1
1 +
 1 ⋱
=  1 + [1 + [1 + [1 + [1 + [1 + ]  − 1  ]  − 1  ]  − 1  ]  − 1  ]  − 1,

since

ϕ − 1  =
 1 ϕ
,

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

ϕ  =    2  1 +   2  1 +   2  1 +   2  1 +
2
=  1 + [1 + [1 + [1 + [1 + [1 + ]
 1 2

]
 1 2

]
 1 2

]
 1 2

]
 1 2
,

since

 ϕ 2 − 1  =  ϕ.

## Approximations

e
 11 10
=  1.61828182845904... (1.000153173364... × ϕ),
where
 e
is Euler’s number.
2
 5 π 6
=  1.6180215937964... (0.999992339... × ϕ).

## Infinite series

 ∞ ∑ k  = 0

 3 − 2√  5 2
k  =
 ∞ ∑ k  = 0

(1 + φ)k  =
 1 1 − (1 + φ)
=
 −1 φ
=  ϕ.

 ∞ ∑ k  = 0

ϕ  − 2k  =
 ∞ ∑ k  = 0

 1 ϕ 2
k =
 ∞ ∑ k  = 0

 1 ϕ + 1
k  =
1
1 −
 1 ϕ + 1
=  1 +
 1 ϕ
=  ϕ.