- Square root of 5
The

**square root of 5**is the positivereal number that, when multiplied by itself, gives the prime number 5. This number appears in the formula for thegolden ratio . It can be denoted in surd form as::$sqrt\{5\}.$

It is an irrational

algebraic number . [*Dauben, Joseph W. (June 1983)*] The first sixty significant digits of itsScientific American "Georg Cantor and the origins of transfinite set theory." Volume 248; Page 122.decimal expansion are::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. As of April 1994, its numerical value in decimal had been computed to at least one million digits. [

*R. Nemiroff and J. Bonnell: " [*]*http://antwrp.gsfc.nasa.gov/htmltest/gifcity/sqrt5.1mil The first 1 million digits of the square root of 5*] "**Continued fraction**It can be expressed as the

continued fraction [2; 4, 4, 4, 4, 4...] OEIS|id=A040002. The sequence of best rational approximations is::$\{color\{OliveGreen\}frac\{2\}\{1,\; frac\{7\}\{3\}\; ,\; \{color\{OliveGreen\}frac\{9\}\{4\; ,\; frac\{20\}\{9\}\; ,\; frac\{29\}\{13\}\; ,\; \{color\{OliveGreen\}frac\{38\}\{17\; ,\; frac\{123\}\{55\}\; ,\; \{color\{OliveGreen\}frac\{161\}\{72\; ,\; frac\{360\}\{161\}\; ,\; frac\{521\}\{233\}\; ,\; \{color\{OliveGreen\}frac\{682\}\{305\; ,\; frac\{2207\}\{987\}\; ,\; \{color\{OliveGreen\}frac\{2889\}\{1292,\; dots$

Convergents of the continued fraction are colored; their numerators are sequence OEIS url|id=A001077, and their denominators are sequence OEIS url|id=A001076. The other (non-colored) terms are semiconvergents.

When $sqrt\{5\}$ is computed with the Babylonian method, starting with "r"

_{0}= 2 and using "r"_{"n"+1}= ("r"_{"n"}+ 5/"r"_{"n"}) / 2, the "n"th approximant "r"_{"n"}is equal to the 2^{"n"}-th convergent of the convergent sequence::$frac\{2\}\{1\}\; =\; 2.0,quad\; frac\{9\}\{4\}\; =\; 2.25,quad\; frac\{161\}\{72\}\; =\; 2.23611dots,quad\; frac\{51841\}\{23184\}\; =\; 2.2360679779\; ldots$

**Relation to the golden ratio and Fibonacci numbers**This

golden ratio φ is thearithmetic mean of 1 and the square root of 5. [*Browne, Malcolm W. (July 30, 1985)*] TheNew York Times "Puzzling Crystals Plunge Scientists into Uncertainty." Section: C; Page 1. (Note - this is a widely cited article).algebra ic relationship between the square root of 5, the golden ratio and the conjugate golden ratio (Φ = fraction|1|φ = φ − 1) are expressed in the following formulae:Fact|date=August 2007:$sqrt\{5\}\; =\; varphi\; +\; Phi\; =\; 2varphi\; -\; 1\; =\; 2Phi\; +\; 1$

:$varphi\; =\; frac\{1\; +\; sqrt\{5\{2\}$

:$Phi\; =\; frac\{sqrt\{5\}\; -\; 1\}\{2\}$

(See section below for their geometrical interpretation as decompositions of a root-5 rectangle.)

The square root of 5 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::$Fleft(n\; ight)\; =$varphi^n-(1-varphi)^n} over {sqrt 5The quotient of √5 and φ (or the product of √5 and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the

Lucas number s: [*Richard K. Guy: "The Strong Law of Small Numbers". The American Mathematical Monthly, vol. 95, 1988, pp. 675-712*]:$frac\{sqrt\{5\{varphi\}\; =\; Phi\; cdot\; sqrt\{5\}\; =\; frac\{5\; -\; sqrt\{5\{2\}\; =\; 1.3819660112501051518dots\; =\; [1;\; 2,\; 1,\; 1,\; 1,\; 1,\; 1,\; 1,\; 1,\; dots]$

:$frac\{varphi\}\{sqrt\{5\; =\; frac\{1\}\{Phi\; cdot\; sqrt\{5\; =\; frac\{2\}\{5\; -\; sqrt\{5\; =\; 0.72360679774997896964dots\; =\; [0;\; 1,\; 2,\; 1,\; 1,\; 1,\; 1,\; 1,\; 1,\; dots]$

The series of convergents to these values feature the series of Fibonacci numbers and the series of

Lucas number s as numerators and denominators, and viceversa, respectively::$\{1,\; frac\{3\}\{2\},\; frac\{4\}\{3\},\; frac\{7\}\{5\},\; frac\{11\}\{8\},\; frac\{18\}\{13\},\; frac\{29\}\{21\},\; frac\{47\}\{34\},\; frac\{76\}\{55\},\; frac\{123\}\{89,\; dots\; dots\; [1;\; 2,\; 1,\; 1,\; 1,\; 1,\; 1,\; 1,\; 1,\; dots]$

:$\{1,\; frac\{2\}\{3\},\; frac\{3\}\{4\},\; frac\{5\}\{7\},\; frac\{8\}\{11\},\; frac\{13\}\{18\},\; frac\{21\}\{29\},\; frac\{34\}\{47\},\; frac\{55\}\{76\},\; frac\{89\}\{123,\; dots\; dots\; [0;\; 1,\; 2,\; 1,\; 1,\; 1,\; 1,\; 1,\; 1,dots]$

**Geometry**Geometrically, the square root of 5 corresponds to the

diagonal of arectangle whose sides are of length 1 and 2, as is evident from thePythagorean theorem . Such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. Together with the algebraic relationship between √5 and φ, this forms the basis for the geometrical construction of agolden rectangle from a square, and for the construction of a regularpentagon given its side (since the side-to-diagonal ratio in a regular pentagon is φ).Forming a dihedral

right angle with the two equal squares that halve a 1:2 rectangle, it can be seen that √5 corresponds also to the ratio between the length of acube edge and the shortest distance from one of its vertices to the opposite one, when traversing the cube "surface" (the shortest distance when traversing through the "inside" of the cube corresponds to the length of the cube diagonal, which is thesquare root of three times the edge).Fact|date=August 2007The number √5 can be algebraically and geometrically related to the

square root of 2 and thesquare root of 3 , as it is the length of thehypotenuse of a right triangle with catheti measuring √2 and √3 (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the centre point of a cube, one of its vertices, and the middle point of a side located on one the faces containing that vertex and opposite to it, are in the ratio √2:√3:√5. This follows from the geometrical relationships between a cube and the quantities √2 (edge-to-face-diagonal ratio, or distance between opposite edges), √3 (edge-to-cube-diagonal ratio) and √5 (the relationship just mentioned above).A rectangle with side proportions 1:√5 is called a "root-five rectangle" and is part of the series of

root rectangle s (also called "dynamic rectangles"), which are based on √1 (= 1), √2, √3, √4 (= 2), √5... and successively constructed using the diagonal of the previous root rectangle, starting from a square. [*cite book | url = http://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&ots=8ZNc5ZKfTG&dq=intitle:%22Geometry+of+Design%22+%22root+5%22&sig=YitS7tv3b4_r87coR4s7EcjL4kk | author = Kimberly Elam | title = Geometry of Design: Studies in Proportion and Composition | place = New York | publisher = Princeton Architectural Press | year = 2001 | isbn = 1568982496*] A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ). [*cite book | title = The Elements of Dynamic Symmetry*] It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between √5, φ and 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 fraction|√5|2 to both sides.

author = Jay Hambidge | publisher = Courier Dover Publications | year = 1967 | isbn = 0486217760 | url = http://books.google.com/books?id=VYJK2F-dh2oC&pg=PA26&ots=MqxrsVLmIH&dq=%22root+five+rectangle%22++section+inauthor:hambidge&sig=meu0juFja5gpsjHKk_gG1stMbYo#PPA27,M1**Trigonometry**Like √2 and √3, the square root of five appears extensively in the formulae for

exact trigonometric constants , and as such the computation of its value is important forgenerating trigonometric tables .Fact|date=August 2007 Since √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 adodecahedron .Fact|date=August 2007**Diophantine approximations**Hurwitz's theorem inDiophantine approximations states that everyirrational number "x" can be approximated by infinitely manyrational number s "m"/"n" inlowest terms in such a way that:$left|x\; -\; frac\{m\}\{n\}\; ight|\; <\; frac\{1\}\{sqrt\{5\},n^2\}$

and that √5 is best possible, in the sense that for any larger constant than √5, there are some irrational numbers "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. | id=MathSciNet | id = 0080682 | year=1956*]Closely related to this is the theorem that of any three consecutive convergents "p"

_{"i"}/"q"_{"i"},"p"_{"i"+1}/"q"_{"i"+1},"p"_{"i"+2}/"q"_{"i"+2},of a number α, at least one of the three inequalities holds::$left|alpha\; -\; \{p\_iover\; q\_i\}\; ight|\; <\; \{1over\; sqrt5\; q\_i^2\},\; qquadleft|alpha\; -\; \{p\_\{i+1\}over\; q\_\{i+1\; ight|\; <\; \{1over\; sqrt5\; q\_\{i+1\}^2\},\; qquadleft|alpha\; -\; \{p\_\{i+2\}over\; q\_\{i+2\; ight|\; <\; \{1over\; sqrt5\; q\_\{i+2\}^2\}$

And the √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 | first1=Aleksandr Yakovlevich | title=Continued Fractions | publisher = University of Chicago Press, Chicago and London | year = 1964]**Algebra**The ring $scriptstylemathbb\{Z\}left\; [,sqrt\{-5\},\; ight]$ contains numbers of the form $scriptstyle\; a,\; +,\; bsqrt\{-5\}$, where "a" and "b" are

integer s and $scriptstyle\; sqrt\{-5\}$ is theimaginary number $scriptstyle\; isqrt\{5\}$. This ring is a frequently cited example of anintegral domain that is not aunique factorization domain .Fact|date=August 2007 The number 6 has two inequivalent factorizations within this ring:: $6\; =\; 2\; cdot\; 3\; =\; (1\; -\; sqrt\{-5\})(1\; +\; sqrt\{-5\}).$

The field $scriptstylemathbb\{Q\}left\; [,sqrt\{5\},\; ight]$, like any other

quadratic field , is anabelian extension of the rational numbers. TheKronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination ofroots of unity ::$sqrt5\; =\; e^\{2pi\; i/5\}\; -\; e^\{4pi\; i/5\}\; -\; e^\{6pi\; i/5\}\; +\; e^\{8pi\; i/5\}.$

**Identities of Ramanujan**The square root of 5 appears in various identities of Ramanujan involving

continued fraction s. [*Citation | last1=Ramanathan | first1=K. G. | title=On the Rogers-Ramanujan continued fraction | id=MathSciNet | id = 813071 | year=1984 | journal=Indian Academy of Sciences. Proceedings. Mathematical Sciences | issn=0253-4142 | volume=93 | issue=2 | pages=67--77*] [*Citation | url=http://mathworld.wolfram.com/RamanujanContinuedFractions.html | author=Eric W. Weisstein | title=Ramanujan Continued Fractions at*]MathWorld For example:

:$cfrac\{1\}$}quad 1 + cfrac{e^{-2pi{1 + cfrac{e^{-4pi{1 + egin{matrix} \ ddotsend{matrix} qquadqquad{}quad{= left( sqrt{frac{5 + sqrt{5{2 - frac{sqrt{5} + 1}{2} ight)e^{2pi/5} = e^{2pi/5}left( sqrt{varphisqrt{5 - varphi ight).

:$cfrac\{1\}$}quad 1 + cfrac{e^{-2pisqrt{5}{1 + cfrac{e^{-4pisqrt{5}{1 + egin{matrix} \ ddotsend{matrix} qquadqquad{}quad{= left( {sqrt{5} over 1 + left [5^{3/4}(varphi - 1)^{5/2} - 1 ight] ^{1/5 - varphi ight)e^{2pi/sqrt{5.

:$4int\_0^inftyfrac\{xe^\{-xsqrt\{5\}\{cosh\; x\},dx=\; cfrac\{1\}$}quad 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 + egin{matrix} \ ddotsend{matrix} qquadqquad{} quad{.

**ee also***

Golden ratio

*Square root

*Square root of 2

*Square root of 3 **References**

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