Editing Golden ratio (section)

===Continued fraction and square root===
{{see also|Lucas number#Continued fractions for powers of the golden ratio}}
[[File:Golden mean.png|right|thumb|Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers]]

The formula {{tmath|1= \varphi = 1 + 1/\varphi}} can be expanded recursively to obtain a [[simple continued fraction]] for the golden ratio:<ref name="Concrete Abstractions" />
<math display=block>
\varphi = [1; 1, 1, 1, \dots]
= 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}
</math>

It is in fact the simplest form of a continued fraction, alongside its reciprocal form:
<math display=block>
\varphi^{-1} = [0; 1, 1, 1, \dots]
= 0 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + { \vphantom{1} \atop \ddots}}}}
</math>

The [[Convergent (continued fraction)|convergent]]s of these continued fractions, {{tmath|\tfrac11}}, {{tmath|\tfrac21}}, {{tmath|\tfrac32}}, {{tmath|\tfrac53}}, {{tmath|\tfrac85}}, {{nowrap|{{tmath|\tfrac{13}8}}, ...}} or {{tmath|\tfrac11}}, {{tmath|\tfrac12}}, {{tmath|\tfrac23}}, {{tmath|\tfrac35}}, {{tmath|\tfrac58}}, {{nowrap|{{tmath|\tfrac8{13} }}, ...,}} are ratios of successive [[Fibonacci numbers]]. The consistently small terms in its continued fraction explain why the approximants converge so slowly. This makes the golden ratio an extreme case of the [[Hurwitz's theorem (number theory)|Hurwitz inequality]] for [[Diophantine approximation]]s, which states that for every irrational {{tmath|\xi}}, there are infinitely many distinct fractions {{tmath|p/q}} such that,
<math display=block>\left|\xi-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^2}.</math>

This means that the constant {{tmath|\sqrt5}} cannot be improved without excluding the golden ratio. It is, in fact, the smallest number that must be excluded to generate closer approximations of such [[Lagrange spectrum|Lagrange numbers]].<ref name=hardy />

A [[Nested radical|continued square root]] form for {{tmath|\varphi}} can be obtained from {{tmath|1=\textstyle \varphi^2 = 1 + \varphi}}, yielding:<ref name=sizer/>
<math display=block>
\varphi = \sqrt{1 + \sqrt{\textstyle 1 + \sqrt{ 1 + \cdots \vphantom)}}}.
</math>