Square root of 5

The square root of 5 can be expressed as the simple continued fraction

: $[2;\; 4,\; 4,\; 4,\; 4,\; 4,\backslash ldots]\; =\; 2\; +\; \backslash cfrac\; 1\; \{4\; +\; \backslash cfrac\; 1\; \{4\; +\; \backslash cfrac\; 1\; \{4\; +\; \backslash cfrac\; 1\; \{4\; +\; \{\{\}\; \backslash atop\; \backslash displaystyle\backslash ddots\}\}\}\}\}.$ {{OEIS|id=A040002}}

The successive partial evaluations of the continued fraction, which are called its convergents, approach $\backslash sqrt\{5\}$:

:$\backslash frac\{2\}\{1\},\; \backslash frac\{9\}\{4\},\; \backslash frac\{38\}\{17\},\; \backslash frac\{161\}\{72\},\; \backslash frac\{682\}\{305\},\; \backslash frac\{2889\}\{1292\},\; \backslash frac\{12238\}\{5473\},\; \backslash frac\{51841\}\{23184\},\; \backslash dots$

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 best rational approximation of $\backslash sqrt\{5\}$; in other words, it is closer to $\backslash sqrt\{5\}$ than any rational number with a smaller denominator.

The convergents, expressed as {{math|{{sfrac|x|y}}}}, satisfy alternately the Pell's equations{{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 }}

:$x^2\; -\; 5y^2\; =\; -1\backslash quad\; \backslash text\{and\}\; \backslash quad\; x^2\; -\; 5y^2\; =\; 1$

When $\backslash sqrt\{5\}$ is approximated with the Babylonian method, starting with {{math|x₀ {{=}} 2}} and using {{math|x_n+1 {{=}} {{sfrac|1|2}}{{big|{{big|(}}}}x_n + {{sfrac|5|x_n}}{{big|{{big|)}}}}}}, the {{math|n}}th approximant {{math|x_n}} is equal to the {{math|2ⁿ}}th convergent of the continued fraction:

:$x\_0\; =\; 2.0,\backslash quad\; x\_1\; =\; \backslash frac\{9\}\{4\}\; =\; 2.25,\backslash quad\; x\_2\; =\; \backslash frac\{161\}\{72\}\; =\; 2.23611\backslash dots,\backslash quad\; x\_3\; =\; \backslash frac\{51841\}\{23184\}\; =\; 2.2360679779\; \backslash ldots,\backslash quad\; x\_4\; =\; \backslash frac\{5374978561\}\{2403763488\}\; =\; 2.23606797749979\; \backslash ldots$

The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial $x^2-5$. The Newton's method update, $x\_\{n+1\}\; =\; x\_n\; -\; f(x\_n)/f\text{'}(x\_n)$, is equal to $(x\_n\; +\; 5/x\_n)/2$ when $f(x)\; =\; x^2\; -\; 5$. The method therefore converges quadratically.

File:Golden Rectangle Construction.svg forms the basis for the geometrical construction of a golden rectangle.]]

The golden ratio {{math|φ}} is the arithmetic mean of 1 and $\backslash sqrt\{5\}$.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). The algebraic relationship between $\backslash sqrt\{5\}$, the golden ratio and the conjugate of the golden ratio ({{math|Φ {{=}} −{{sfrac|1|φ}} {{=}} 1 − φ}}) is expressed in the following formulae:

: $$

\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}

(See the section below for their geometrical interpretation as decompositions of a $\backslash sqrt\{5\}$ rectangle.)

$\backslash sqrt\{5\}$ then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio:

: $F(n)\; =\; \backslash frac\{\backslash varphi^n-(1-\backslash varphi)^n\}\{\backslash sqrt\; 5\}.$

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

: $$

\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}

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:

: $$

\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}

In fact, the limit of the quotient of the $n^\{th\}$ Lucas number $L\_n$ and the $n^\{th\}$ Fibonacci number $F\_n$ is directly equal to the square root of $5$:

: $\backslash lim\_\{n\backslash to\backslash infty\}\; \backslash frac\{L\_n\}\{F\_n\}=\backslash sqrt\{5\}.$

File:pinwheel 1.svg.]]

File:Root rectangles Hambidge 1920.png construction of "root rectangles"]]

{{distances_between_double_cube_corners.svg}}

Geometrically, $\backslash sqrt\{5\}$ corresponds to the diagonal of a rectangle whose sides are of length 1 and 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.{{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 = $\backslash sqrt5$-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

}} Together with the algebraic relationship between $\backslash sqrt\{5\}$ 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 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 and the shortest distance from one of its vertices to the opposite one, when traversing the cube surface, is $\backslash sqrt\{5\}$. 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.{{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}}

A rectangle with side proportions 1:$\backslash sqrt\{5\}$ is called a root-five rectangle and is part of the series of root rectangles, a subset of dynamic rectangles, which are based on {{nowrap|1=$\backslash sqrt\{1\}$ (= 1), $\backslash sqrt\{2\}$, $\backslash sqrt\{3\}$, $\backslash sqrt\{4\}$ (= 2), $\backslash sqrt\{5\}$...}} and successively constructed using the diagonal of the previous root rectangle, starting from a square.{{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 }} 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}}}}).{{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 }} 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 $\backslash sqrt\{5\}$, {{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 $\backslash sqrt\{5\}/2$ to both sides.

Like $\backslash sqrt\{2\}$ and $\backslash sqrt\{3\}$, the square root of 5 appears extensively in the formulae for exact trigonometric constants, including in the sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.[http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html Julian D. A. Wiseman, "Sin and cos in surds"] The simplest of these are

:$\backslash begin\{align\}$

\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}

As such, the computation of its value is important for generating trigonometric tables. Since $\backslash sqrt\{5\}$ 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.

Hurwitz's theorem in Diophantine approximations states that every irrational number {{math|x}} can be approximated by infinitely many rational numbers {{math|{{sfrac|m|n}}}} in lowest terms in such a way that

:

and that $\backslash sqrt\{5\}$ is best possible, in the sense that for any larger constant than $\backslash sqrt\{5\}$, there are some irrational numbers {{math|x}} for which only finitely many such approximations exist.{{Citation | last1=LeVeque | first1=William Judson | title=Topics in number theory | publisher=Addison-Wesley Publishing Co., Inc., Reading, Mass. |mr=0080682 | year=1956}}

Closely related to this is the theorem that of any three consecutive convergents {{math|{{sfrac|p_i|q_i}}}}, {{math|{{sfrac|p_i+1|q_i+1}}}}, {{math|{{sfrac|p_i+2|q_i+2}}}}, of a number {{math|α}}, at least one of the three inequalities holds:

:

\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}.

And the $\backslash sqrt\{5\}$ 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.{{Citation | last1=Khinchin | author-link=A. Ya. Khinchin | first1=Aleksandr Yakovlevich | title=Continued Fractions | publisher = University of Chicago Press, Chicago and London | year = 1964}}

The ring $\backslash mathbb\{Z\}[\backslash sqrt\{-5\}]$ contains numbers of the form $a\; +\; b\backslash sqrt\{-5\}$, where {{math|a}} and {{math|b}} are integers and $\backslash sqrt\{-5\}$ is the imaginary number $i\backslash sqrt\{5\}$. This ring is a frequently cited example of an integral domain that is not a unique factorization domain.{{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 $\backslash mathbb\{Z\}[\backslash sqrt\{-5\}]$?

| doi = 10.1007/978-981-13-7028-1_9

| location = Singapore

| mr = 3991169

| quote = Most undergraduate level abstract algebra texts use $\backslash mathbb\{Z\}[\backslash sqrt\{-5\}]$ 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

}} The number 6 has two inequivalent factorizations within this ring:

: $6\; =\; 2\; \backslash cdot\; 3\; =\; (1\; -\; \backslash sqrt\{-5\})(1\; +\; \backslash sqrt\{-5\}).\; \backslash ,$

On the other hand, the real quadratic integer ring $\backslash Z[\backslash tfrac\{\backslash sqrt\{5\}+1\}2]$, adjoining the Golden ratio $\backslash phi\; =\; \backslash tfrac\{\backslash sqrt\{5\}+1\}2$, was shown to be Euclidean, and hence a unique factorization domain, by Dedekind.

The field $\backslash mathbb\{Q\}[\backslash sqrt\{-5\}],$ 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:

:$\backslash sqrt5\; =\; e^\{\backslash frac\{2\backslash pi\}\{5\}i\}\; -\; e^\{\backslash frac\{4\backslash pi\}\{5\}i\}\; -\; e^\{\backslash frac\{6\backslash pi\}\{5\}i\}\; +\; e^\{\backslash frac\{8\backslash pi\}\{5\}i\}.\; \backslash ,$

The square root of 5 appears in various identities discovered by Srinivasa Ramanujan involving continued fractions.{{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 }}{{Citation | url=http://mathworld.wolfram.com/RamanujanContinuedFractions.html | author=Eric W. Weisstein | title=Ramanujan Continued Fractions}} at MathWorld

For example, this case of the Rogers–Ramanujan continued fraction:

:$\backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{e^\{-2\backslash pi\}\}\{1\; +\; \backslash cfrac\{e^\{-4\backslash pi\}\}\{1\; +\; \backslash cfrac\{e^\{-6\backslash pi\}\}\{1\; +\; \{\; \{\}\; \backslash atop\; \backslash displaystyle\; \backslash ddots\}\}\}\}\}$

= \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).

:$\backslash cfrac\{1\}\{1\; +\; \backslash cfrac\{e^\{-2\backslash pi\backslash sqrt\{5\}\}\}\{1\; +\; \backslash cfrac\{e^\{-4\backslash pi\backslash sqrt\{5\}\}\}\{1\; +\; \backslash cfrac\{e^\{-6\backslash pi\backslash sqrt\{5\}\}\}\{1\; +\; \{\; \{\}\; \backslash atop\; \backslash displaystyle\; \backslash ddots\}\}\}\}\}$

= \left( {\sqrt{5} \over 1 + \sqrt[5]{5^{\frac34}(\varphi - 1)^{\frac52} - 1}} - \varphi \right)e^{\frac{2\pi}{\sqrt{5}}}.

:$4\backslash int\_0^\backslash infty\backslash frac\{xe^\{-x\backslash sqrt\{5\}\}\}\{\backslash cosh\; x\}\backslash ,dx$

= \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} }}}}}}} \, .

• Golden ratio
• Square root
• Square root of 2
• Square root of 3
• Square root of 6
• Square root of 7

{{Reflist}}

{{Algebraic numbers}}

{{Irrational number}}

Category:Mathematical constants

Category:Quadratic irrational numbers