Editing Golden ratio (section)

===Other properties===

The golden ratio's ''decimal expansion'' can be calculated via root-finding methods, such as [[Newton's method]] or [[Halley's method]], on the equation {{tmath|1=\textstyle x^2-x-1=0}} or on {{tmath|1=\textstyle x^2-5=0}} (to compute {{tmath|\sqrt5}} first). The time needed to compute {{tmath|n}} digits of the golden ratio using Newton's method is essentially {{tmath|O(M(n))}}, where {{tmath|M(n)}} is [[Multiplication algorithm#Computational complexity|the time complexity of multiplying]] two {{tmath|n}}-digit numbers.<ref name=muller /> This is considerably faster than known algorithms for [[pi|{{mvar|π}}]] and [[e (mathematical constant)|{{mvar|e}}]]. An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers {{tmath|F_{25001} }} and {{tmath|F_{25000} }}, each over {{tmath|5000}} digits, yields over {{tmath|10{,}000}} significant digits of the golden ratio. The decimal expansion of the golden ratio {{tmath|\varphi}}<ref name=a001622 /> has been calculated to an accuracy of twenty trillion ({{tmath|1=\textstyle 2 \times 10^{13} = 20{,}000{,}000{,}000{,}000}}) digits.<ref name=ycruncher />

In the [[complex plane]], the fifth [[Root of unity|roots of unity]] {{tmath|1=\textstyle z = e^{2\pi k i/5} }} (for an integer {{tmath|k}}) satisfying {{tmath|1=\textstyle z^5 = 1}} are the vertices of a pentagon. They do not form a [[ring (mathematics)|ring]] of [[quadratic integer]]s, however the sum of any fifth root of unity and its [[complex conjugate]], {{tmath|z + \bar z}}, ''is'' a quadratic integer, an element of {{tmath|\Z[\varphi]}}. Specifically,

<math display=block>\begin{align}
e^{0} + e^{-0} &= 2, \\[5mu]
e^{2\pi i / 5} + e^{-2\pi i / 5} &= \varphi^{-1} = -1 + \varphi, \\[5mu]
e^{4\pi i / 5} + e^{-4\pi i / 5} &= -\varphi.
\end{align}</math>

This also holds for the remaining tenth roots of unity satisfying {{tmath|1=\textstyle z^{10} = 1}},

<math display=block>\begin{align}
e^{\pi i} + e^{-\pi i} &= -2, \\[5mu]
e^{\pi i / 5} + e^{-\pi i / 5} &= \varphi, \\[5mu]
e^{3\pi i / 5} + e^{-3\pi i / 5} &= -\varphi^{-1} = 1 - \varphi.
\end{align}</math>

For the [[gamma function]] {{tmath|\Gamma}}, the only solutions to the equation {{tmath|1= \Gamma(z-1) = \Gamma(z+1)}} are {{tmath|1= z = \varphi}} and {{tmath|1=\textstyle z = -\varphi^{-1} }}.

When the golden ratio is used as the base of a [[numeral system]] (see [[golden ratio base]], sometimes dubbed ''phinary'' or {{tmath|\varphi}}''-nary''), [[quadratic integer]]s in the ring {{tmath|\Z[\varphi]}} – that is, numbers of the form {{tmath|a + b\varphi}} for {{tmath|a}} and {{tmath|b}} in {{tmath|\Z}} – have [[repeating decimal|terminating]] representations, but rational fractions have non-terminating representations.

The golden ratio also appears in [[hyperbolic geometry]], as the maximum distance from a point on one side of an [[ideal triangle]] to the closer of the other two sides: this distance, the side length of the [[equilateral triangle]] formed by the points of tangency of a circle inscribed within the ideal triangle, is {{tmath|4\log(\varphi)}}.<ref name=horocycle />

The golden ratio appears in the theory of [[Modular form|modular functions]] as well. For <math>|q|<1,</math> let
<math display=block>
R(q) = \cfrac{q^{1/5}}{1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+{ \vphantom{1} \atop \ddots}}}}}.
</math>
Then
<math display=block>
R(e^{-2\pi})
= \sqrt{\varphi\sqrt5}-\varphi ,\quad R(-e^{-\pi})
= \varphi^{-1}-\sqrt{2-\varphi^{-1}}
</math>
and
<math display=block>
R(e^{-2\pi i/\tau})=\frac{1-\varphi R(e^{2\pi i\tau})}{\varphi+R(e^{2\pi i\tau})}
</math>
where {{tmath|\operatorname{Im}\tau>0}} and {{tmath|\textstyle (e^z)^{1/5} }} in the continued fraction should be evaluated as {{tmath|\textstyle e^{z/5} }}. The function {{tmath|\textstyle \tau\mapsto R(e^{2\pi i\tau})}} is invariant under {{tmath|\Gamma(5)}}, a [[Modular group#Congruence subgroups|congruence subgroup of the modular group]]. Also for [[positive real numbers]] {{tmath|a}} and {{tmath|b}} such that {{tmath|1=\textstyle ab = \pi^2,}}<ref name=rrcf />

<math display=block>\begin{align}
\Bigl(\varphi+R{\bigl(e^{-2a}\bigr)}\Bigr)\Bigl(\varphi+R{\bigl(e^{-2b}\bigr)}\Bigr)&=\varphi\sqrt5, \\[5mu]
\Bigl(\varphi^{-1}-R{\bigl({-e^{-a}}\bigr)}\Bigr)\Bigl(\varphi^{-1}-R{\bigl({-e^{-b}}\bigr)}\Bigr)&=\varphi^{-1}\sqrt5.
\end{align}</math>

{{tmath|\varphi}} is a [[Pisot–Vijayaraghavan number]].<ref name=duffin />