View source for Square root of 5

{{Short description|Positive real number which when multiplied by itself gives 5}}
{{infobox non-integer number
|rationality=Irrational
|algebraic=<math>\sqrt{5}</math>
|decimal={{gaps|2.23606|79774|99789|69...}}
|continued_fraction=<math>2 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \cfrac{1}{4 + \ddots}}}}</math>
}}

The '''square root of 5''' is the positive [[real number]] that, when multiplied by itself, gives the prime number [[5 (number)|5]]. It is more precisely called the '''principal square root of 5''', to distinguish it from the negative number with the same property. This number appears in the fractional expression for the [[golden ratio]]. It can be denoted in [[nth root|surd]] form as <math display=inline>\sqrt{5}</math>.

It is an [[irrational number|irrational]] [[algebraic number]].<ref>Dauben, Joseph W. (June 1983) [[Scientific American]] ''Georg Cantor and the origins of transfinite set theory.'' Volume 248; Page 122.</ref> The first sixty significant digits of its [[decimal expansion]] are:

:{{gaps|2.23606|79774|99789|69640|91736|68731|27623|54406|18359|61152|57242|7089...}} {{OEIS|id=A002163}},

which can be rounded down to 2.236 to within 99.99% accuracy. The approximation {{sfrac|161|72}} (≈ 2.23611) for the square root of five can be used. Despite having a [[denominator]] of only 72, it differs from the correct value by less than {{sfrac|1|10,000}} (approx. {{val|4.3e-5}}). As of January 2022, the numerical value in decimal of the square root of 5 has been computed to at least 2,250,000,000,000 digits.<ref>{{cite web |last1=Yee |first1=Alexander |title=Records Set by y-cruncher |url=http://numberworld.org/y-cruncher/records.html}}</ref>

==Rational approximations==
The square root of 5 can be expressed as the [[simple continued fraction]]

: <math> [2; 4, 4, 4, 4, 4,\ldots] = 2 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + {{} \atop \displaystyle\ddots}}}}}. </math> {{OEIS|id=A040002}}

The successive partial evaluations of the continued fraction, which are called its [[Continued_fraction#Infinite_continued_fractions_and_convergents|''convergents'']], approach <math>\sqrt{5}</math>:
:<math>\frac{2}{1}, \frac{9}{4}, \frac{38}{17}, \frac{161}{72}, \frac{682}{305}, \frac{2889}{1292}, \frac{12238}{5473}, \frac{51841}{23184}, \dots</math>
Their numerators are 2, 9, 38, 161, … {{OEIS|id=A001077}}, and their denominators are 1, 4, 17, 72, … {{OEIS|id=A001076}}.

Each of these is a [[Continued fraction#Best rational approximations|best rational approximation]] of <math>\sqrt{5}</math>; in other words, it is closer to <math>\sqrt{5}</math> than any [[rational number]] with a smaller denominator.

The convergents, expressed as {{math|{{sfrac|''x''|''y''}}}}, satisfy alternately the [[Pell's equation]]s<ref name=conrad>{{cite web |last1=Conrad |first1=Keith |title=Pell's Equation II |url=https://kconrad.math.uconn.edu/blurbs/ugradnumthy/pelleqn2.pdf |website=uconn.edu |access-date=17 March 2022 }}</ref>
:<math>x^2 - 5y^2 = -1\quad \text{and} \quad x^2 - 5y^2 = 1</math>

When <math>\sqrt{5}</math> is approximated with the [[Methods of computing square roots#Heron's method|Babylonian method]], starting with {{math|''x''<sub>0</sub> {{=}} 2}} and using {{math|''x''<sub>''n''+1</sub> {{=}} {{sfrac|1|2}}{{big|{{big|(}}}}''x''<sub>''n''</sub> + {{sfrac|5|''x''<sub>''n''</sub>}}{{big|{{big|)}}}}}}, the {{math|''n''}}th approximant {{math|''x''<sub>''n''</sub>}} is equal to the {{math|2<sup>''n''</sup>}}th convergent of the continued fraction:
:<math>x_0 = 2.0,\quad x_1 = \frac{9}{4} = 2.25,\quad x_2 = \frac{161}{72} = 2.23611\dots,\quad x_3 = \frac{51841}{23184} = 2.2360679779 \ldots,\quad x_4 = \frac{5374978561}{2403763488} = 2.23606797749979 \ldots</math>

The Babylonian method is equivalent to [[Newton's method]] for [[root of a polynomial|root]] finding applied to the [[polynomial]] <math>x^2-5</math>.  The Newton's method update, <math>x_{n+1} = x_n - f(x_n)/f'(x_n)</math>, is equal to <math>(x_n + 5/x_n)/2</math> when <math>f(x) = x^2 - 5</math>.  The method therefore [[Rate_of_convergence#Q-convergence_definitions|converges quadratically]].

==Relation to the golden ratio and Fibonacci numbers==
[[File:Golden Rectangle Construction.svg|thumb|The <math>\frac{\sqrt{5}}{2}</math> diagonal of a half [[square]] forms the basis for the geometrical construction of a [[golden rectangle]].]]
The [[golden ratio]] {{math|φ}} is the [[arithmetic mean]] of 1 and <math>\sqrt{5}</math>.<ref>Browne, Malcolm W. (July 30, 1985) [[New York Times]] ''Puzzling Crystals Plunge Scientists into Uncertainty.'' Section: C; Page 1. (Note: this is a widely cited article).</ref> The [[algebra]]ic relationship between <math>\sqrt{5}</math>, the golden ratio and the [[golden ratio#Golden ratio conjugate and powers|conjugate of the golden ratio]] ({{math|Φ {{=}} −{{sfrac|1|''φ''}} {{=}} 1 − ''φ''}}) is expressed in the following formulae:

: <math>
\begin{align}
\sqrt{5} & = \varphi - \Phi = 2\varphi - 1 = 1 - 2\Phi \\[5pt]
\varphi & = \frac{1 + \sqrt{5}}{2} \\[5pt]
\Phi & = \frac{1 - \sqrt{5}}{2}.
\end{align}
</math>
(See the section below for their geometrical interpretation as decompositions of a <math>\sqrt{5}</math> [[rectangle]].)

<math>\sqrt{5}</math> then naturally figures in the closed form expression for the [[Fibonacci number]]s, a formula which is usually written in terms of the golden ratio:

: <math>F(n) = \frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}.</math>

The quotient of <math>\sqrt{5}</math> and {{math|''φ''}} (or the product of <math>\sqrt{5}</math> and {{math|Φ}}), and its [[multiplicative inverse|reciprocal]], provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the [[Lucas number]]s:<ref>[[Richard K. Guy]]: "The Strong Law of Small Numbers". ''[[American Mathematical Monthly]]'', vol. 95, 1988, pp. 675–712</ref>

: <math>
\begin{align}
\frac{\sqrt{5}}{\varphi} = \Phi \cdot \sqrt{5} = \frac{5 - \sqrt{5}}{2} & = 1.3819660112501051518\dots \\
& = [1; 2, 1, 1, 1, 1, 1, 1, 1, \ldots] \\[5pt]
\frac{\varphi}{\sqrt{5}} = \frac{1}{\Phi \cdot \sqrt{5}} = \frac{5 + \sqrt{5}}{10} & = 0.72360679774997896964\ldots \\
& = [0; 1, 2, 1, 1, 1, 1, 1, 1, \ldots].
\end{align}
</math>

The series of convergents to these values feature the series of Fibonacci numbers and the series of Lucas numbers as numerators and denominators, and vice versa, respectively:

: <math>
\begin{align}
& {1, \frac{3}{2}, \frac{4}{3}, \frac{7}{5}, \frac{11}{8}, \frac{18}{13}, \frac{29}{21}, \frac{47}{34}, \frac{76}{55}, \frac{123}{89}}, \ldots \ldots [1; 2, 1, 1, 1, 1, 1, 1, 1, \ldots] \\[8pt]
& {1, \frac{2}{3}, \frac{3}{4}, \frac{5}{7}, \frac{8}{11}, \frac{13}{18}, \frac{21}{29}, \frac{34}{47}, \frac{55}{76}, \frac{89}{123}}, \dots \dots [0; 1, 2, 1, 1, 1, 1, 1, 1,\dots].
\end{align}
</math>

In fact, the [[limit of a sequence|limit]] of the quotient of the <math>n^{th}</math> Lucas number <math>L_n</math> and the <math>n^{th}</math> Fibonacci number <math>F_n</math> is directly equal to the square root of <math>5</math>:
: <math>\lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}.</math>

==Geometry==
[[File:pinwheel 1.svg|250px|thumb|right|Decomposition of a <math>1\times 2</math> right triangle into five similar triangles, the basis for the aperiodic [[pinwheel tiling]]]]
[[File:Root rectangles Hambidge 1920.png|250px|thumb|right|[[Jay Hambidge|Jay Hambidge's]] construction of "root rectangles"]]
{{distances_between_double_cube_corners.svg}}

[[Geometry|Geometrically]], <math>\sqrt{5}</math> corresponds to the [[diagonal]] of a rectangle whose sides are of length [[1 (number)|1]] and [[2 (number)|2]], as is evident from the [[Pythagorean theorem]]. Such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. This can be used to subdivide a square grid into a tilted square grid with five times as many squares, forming the basis for a [[subdivision surface]].<ref>{{citation
 | last1 = Ivrissimtzis | first1 = Ioannis P.
 | last2 = Dodgson | first2 = Neil A.
 | last3 = Sabin | first3 = Malcolm
 | editor1-last = Dodgson | editor1-first = Neil A.
 | editor2-last = Floater | editor2-first = Michael S.
 | editor3-last = Sabin | editor3-first = Malcolm A.
 | contribution = <math>\sqrt5</math>-subdivision
 | doi = 10.1007/3-540-26808-1_16
 | location = Berlin
 | mr = 2112357
 | pages = 285–299
 | publisher = Springer
 | series = Mathematics and Visualization
 | title = Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003
 | year = 2005| isbn = 3-540-21462-3
 }}</ref> Together with the algebraic relationship between <math>\sqrt{5}</math> and {{math|''φ''}}, this forms the basis for the geometrical construction of a [[golden rectangle]] from a square, and for the construction of a [[regular polygon|regular]] [[pentagon]] given its side (since the side-to-diagonal ratio in a regular pentagon is {{math|''φ''}}).

Since two adjacent faces of a [[cube]] would unfold into a 1:2 rectangle, the ratio between the length of the cube's [[edge (geometry)|edge]] and the shortest distance from one of its [[vertex (geometry)|vertices]] to the opposite one, when traversing the cube ''surface'', is <math>\sqrt{5}</math>. By contrast, the shortest distance when traversing through the ''inside'' of the cube corresponds to the length of the cube diagonal, which is the [[square root of three]] times the edge.<ref name=":0">{{Cite book |last=Sutton |first=David |url=https://books.google.com/books?id=vgo7bTxDmIsC&pg=PA55 |title=Platonic & Archimedean Solids |publisher=Walker & Company |year=2002 |isbn=0802713866 |pages=55 |language=en}}</ref>

A rectangle with side proportions 1:<math>\sqrt{5}</math> is called a ''root-five rectangle'' and is part of the series of root rectangles, a subset of [[dynamic rectangle]]s, which are based on {{nowrap|1=<math>\sqrt{1}</math> (= 1), <math>\sqrt{2}</math>, <math>\sqrt{3}</math>, <math>\sqrt{4}</math> (= 2), <math>\sqrt{5}</math>...}} and successively constructed using the diagonal of the previous root rectangle, starting from a square.<ref>{{Citation | url = https://books.google.com/books?id=1KI0JVuWYGkC&q=intitle:%22Geometry+of+Design%22+%22root+5%22&pg=PA41 | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1-56898-249-6 }}</ref> A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions {{nowrap|{{math|Φ}} × 1}}), or into two golden rectangles of different sizes (of dimensions {{nowrap|{{math|Φ}} × 1}} and {{nowrap|1 × {{math|''φ''}}}}).<ref>{{Citation | title = The Elements of Dynamic Symmetry | author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0-486-21776-0 | url = https://books.google.com/books?id=VYJK2F-dh2oC&q=%22root+five+rectangle%22++section+inauthor:hambidge&pg=PA26 }}</ref> It can also be decomposed as the union of two equal golden rectangles (of dimensions {{nowrap|1 × {{math|φ}}}}) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between <math>\sqrt{5}</math>, {{math|''φ''}} and {{math|Φ}} mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length <math>\sqrt{5}/2</math> to both sides.

==Trigonometry==
Like <math>\sqrt{2}</math> and <math>\sqrt{3}</math>, the square root of 5 appears extensively in the formulae for [[exact trigonometric constants]], including in the [[sine and cosine|sines and cosines]] of every [[angle]] whose measure in [[degree (angle)|degrees]] is [[divisible]] by 3 but not by 15.<ref>[http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Julian D. A. Wiseman, "Sin and cos in surds"]</ref> The simplest of these are
:<math>\begin{align}
\sin\frac{\pi}{10} = \sin 18^\circ &= \tfrac{1}{4}(\sqrt5-1) = \frac{1}{\sqrt5+1}, \\[5pt]
\sin\frac{\pi}{5} = \sin 36^\circ &= \tfrac{1}{4}\sqrt{2(5-\sqrt5)}, \\[5pt]
\sin\frac{3\pi}{10} = \sin 54^\circ &= \tfrac{1}{4}(\sqrt5+1) = \frac{1}{\sqrt5-1}, \\[5pt]
\sin\frac{2\pi}{5} = \sin 72^\circ &= \tfrac{1}{4}\sqrt{2(5+\sqrt5)}\, . \end{align}</math>

As such, the computation of its value is important for generating [[trigonometric tables]]. Since <math>\sqrt{5}</math> is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a [[dodecahedron]].<ref name=":0" />

==Diophantine approximations==
[[Hurwitz's theorem (number theory)|Hurwitz's theorem]] in [[Diophantine approximations]] states that every [[irrational number]] {{math|''x''}} can be approximated by [[infinite set|infinitely many]] [[rational number]]s {{math|{{sfrac|''m''|''n''}}}} in [[lowest terms]] in such a way that
:<math> \left|x - \frac{m}{n}\right| < \frac{1}{\sqrt{5}\,n^2} </math>
and that <math>\sqrt{5}</math> is best possible, in the sense that for any larger constant than <math>\sqrt{5}</math>, there are some irrational numbers {{math|''x''}} for which only finitely many such approximations exist.<ref>{{Citation | last1=LeVeque | first1=William Judson | title=Topics in number theory | publisher=Addison-Wesley Publishing Co., Inc., Reading, Mass. |mr=0080682 | year=1956}}</ref>

Closely related to this is the theorem<ref name=khinchin/> that of any three consecutive [[convergent (continued fraction)|convergent]]s {{math|{{sfrac|''p''<sub>''i''</sub>|''q''<sub>''i''</sub>}}}}, {{math|{{sfrac|''p''<sub>''i''+1</sub>|''q''<sub>''i''+1</sub>}}}}, {{math|{{sfrac|''p''<sub>''i''+2</sub>|''q''<sub>''i''+2</sub>}}}}, of a number {{math|''α''}}, at least one of the three inequalities holds:
:<math>\left|\alpha - {p_i\over q_i}\right| < {1\over \sqrt5 q_i^2}, \quad
\left|\alpha - {p_{i+1}\over q_{i+1}}\right| < {1\over \sqrt5 q_{i+1}^2}, \quad
\left|\alpha - {p_{i+2}\over q_{i+2}}\right| < {1\over \sqrt5 q_{i+2}^2}.</math>

And the <math>\sqrt{5}</math> in the denominator is the best bound possible since the convergents of the golden ratio make the difference on the left-hand side arbitrarily close to the value on the right-hand side.  In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.<ref name=khinchin>{{Citation | last1=Khinchin | author-link=A. Ya. Khinchin | first1=Aleksandr Yakovlevich | title=Continued Fractions | publisher = University of Chicago Press, Chicago and London | year = 1964}}</ref>

==Algebra==
The [[ring (mathematics)|ring]] <math>\mathbb{Z}[\sqrt{-5}]</math> contains numbers of the form <math>a + b\sqrt{-5}</math>, where {{math|''a''}} and {{math|''b''}} are [[integer]]s and <math>\sqrt{-5}</math> is the [[imaginary number]] <math>i\sqrt{5}</math>. This ring is a frequently cited example of an [[integral domain]] that is not a [[unique factorization domain]].<ref>{{citation
 | last1 = Chapman | first1 = Scott T.
 | last2 = Gotti | first2 = Felix
 | last3 = Gotti | first3 = Marly
 | editor1-last = Badawi | editor1-first = Ayman
 | editor2-last = Coykendall | editor2-first = Jim
 | contribution = How do elements really factor in <math>\mathbb{Z}[\sqrt{-5}]</math>?
 | doi = 10.1007/978-981-13-7028-1_9
 | location = Singapore
 | mr = 3991169
 | quote = Most undergraduate level abstract algebra texts use <math>\mathbb{Z}[\sqrt{-5}]</math> as an example of an integral domain which is not a unique factorization domain
 | pages = 171–195
 | publisher = Birkhäuser/Springer
 | series = Trends in Mathematics
 | title = Advances in Commutative Algebra: Dedicated to David F. Anderson
 | year = 2019| arxiv = 1711.10842
 | isbn = 978-981-13-7027-4
 | s2cid = 119142526
 }}</ref> The number 6 has two inequivalent factorizations within this ring:
: <math>6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}). \, </math>
On the other hand, the real [[Quadratic integer#Quadratic integer rings|quadratic integer]] ring <math>\Z[\tfrac{\sqrt{5}+1}2]</math>, adjoining the [[Golden ratio]] <math>\phi = \tfrac{\sqrt{5}+1}2</math>, was shown to be [[Euclidean domain|Euclidean]], and hence a unique factorization domain, by Dedekind.

The [[field (mathematics)|field]] <math>\mathbb{Q}[\sqrt{-5}],</math> like any other [[quadratic field]], is an [[abelian extension]] of the rational numbers. The [[Kronecker–Weber theorem]] therefore guarantees that the square root of five can be written as a rational [[linear combination]] of [[roots of unity]]:
:<math>\sqrt5 = e^{\frac{2\pi}{5}i} - e^{\frac{4\pi}{5}i} - e^{\frac{6\pi}{5}i} + e^{\frac{8\pi}{5}i}. \, </math>
==Identities of Ramanujan==
The square root of 5 appears in various identities discovered by [[Srinivasa Ramanujan]] involving [[continued fraction]]s.<ref>{{Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction |mr=813071 | year=1984 | journal= Proceedings of the Indian Academy of Sciences, Section A| issn=0253-4142 | volume=93 | issue=2 | pages=67–77 | doi=10.1007/BF02840651| s2cid=121808904 }}</ref><ref>{{Citation | url=http://mathworld.wolfram.com/RamanujanContinuedFractions.html | author=Eric W. Weisstein | title=Ramanujan Continued Fractions}} at [[MathWorld]]</ref>

For example, this case of the [[Rogers–Ramanujan continued fraction]]:
:<math>\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1 + \cfrac{e^{-6\pi}}{1 + { {} \atop \displaystyle \ddots}}}}}
= \left( \sqrt{\frac{5 + \sqrt{5}}{2}} - \frac{\sqrt{5} + 1}{2} \right)e^{\frac{2\pi}{5}} = e^{\frac{2\pi}{5}}\left( \sqrt{\varphi\sqrt{5}} - \varphi \right).</math>

<!-- extra blank line between two fairly complicated [[TeX]] displays for ease of legibility -->

:<math>\cfrac{1}{1 + \cfrac{e^{-2\pi\sqrt{5}}}{1 + \cfrac{e^{-4\pi\sqrt{5}}}{1 + \cfrac{e^{-6\pi\sqrt{5}}}{1 + { {} \atop \displaystyle \ddots}}}}}
= \left( {\sqrt{5} \over 1 + \sqrt[5]{5^{\frac34}(\varphi - 1)^{\frac52} - 1}} - \varphi \right)e^{\frac{2\pi}{\sqrt{5}}}.</math>

<!-- extra blank line between two fairly complicated [[TeX]] displays for ease of legibility -->

:<math>4\int_0^\infty\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx
= \cfrac{1}{1 + \cfrac{1^2}{1 + \cfrac{1^2}{1 + \cfrac{2^2}{1 + \cfrac{2^2}{1 + \cfrac{3^2}{1 + \cfrac{3^2}{1 + {{} \atop \displaystyle \ddots} }}}}}}} \, .</math>

==See also==
<!-- We seem to have a bit of an edit war here.  I don't think the fact that these are also linked to in the body of the article means they _must_ be excluded from this section.  It's not unusual, and it's sometimes useful, to have them in both places. [[user:Michael Hardy]] -->

* [[Golden ratio]]
* [[Square root]]
* [[Square root of 2]]
* [[Square root of 3]]
* [[Square root of 6]]
* [[Square root of 7]]

==References==
{{Reflist}}

{{Algebraic numbers}}
{{Irrational number}}

[[Category:Mathematical constants]]
[[Category:Quadratic irrational numbers]]