Square root of 5


Square root of 5

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. This number appears in the formula for the golden ratio. It can be denoted in surd form as:

:sqrt{5}.

It is an irrational algebraic number. [Dauben, Joseph W. (June 1983) Scientific American "Georg Cantor and the origins of transfinite set theory." Volume 248; Page 122.] The first sixty significant digits of its 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 the arithmetic mean of 1 and the square root of 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 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 numbers, a formula which is usually written in terms of the golden ratio::Fleft(n ight) = varphi^n-(1-varphi)^n} over {sqrt 5

The 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 numbers: [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 numbers 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 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. Together with the algebraic relationship between √5 and 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 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 a cube 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 the square root of three times the edge).Fact|date=August 2007

The number √5 can be algebraically and geometrically related to the square root of 2 and the square root of 3, as it is the length of the hypotenuse 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 rectangles (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
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
] 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.

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 for generating 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 a dodecahedron.Fact|date=August 2007

Diophantine approximations

Hurwitz's theorem in Diophantine approximations states that every irrational number "x" can be approximated by infinitely many rational numbers "m"/"n" in lowest 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 integers and scriptstyle sqrt{-5} is the imaginary number scriptstyle isqrt{5}. This ring is a frequently cited example of an integral domain that is not a unique 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 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:

: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 fractions. [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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