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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
x 2x − 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
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)

     
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.

See also


  • {{Fibonacci}} (mathematical function template)