# Square root of 5

__: Positive real number which when multiplied by itself gives 5__

**Short description**Template:Infobox non-integer number

The **square root of 5** is the positive real number that, when multiplied by itself, gives the prime 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 surd form as:

- [math]\displaystyle{ \sqrt{5}. \, }[/math]

It is an irrational algebraic number.^{[1]} The first sixty significant digits of its decimal expansion are:

which can be rounded down to 2.236 to within 99.99% accuracy. The approximation 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 1/10,000 (approx. 4.3×10^{−5}). As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.^{[2]}

## Rational approximations

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

- [math]\displaystyle{ [2; 4, 4, 4, 4, 4,\ldots] = 2 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + \cfrac 1 {4 + {{} \atop \displaystyle\ddots}}}}}. }[/math] (sequence A040002 in the OEIS)

The successive partial evaluations of the continued fraction, which are called its *convergents*, approach [math]\displaystyle{ \sqrt{5} }[/math]:

- [math]\displaystyle{ \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, … (sequence A001077 in the OEIS), and their denominators are 1, 4, 17, 72, … (sequence A001076 in the OEIS).

Each of these is a best rational approximation of [math]\displaystyle{ \sqrt{5} }[/math]; in other words, it is closer to [math]\displaystyle{ \sqrt{5} }[/math] than any rational with a smaller denominator.

The convergents, expressed as *x*/*y*, satisfy alternately the Pell's equations^{[3]}

- [math]\displaystyle{ x^2 - 5y^2 = -1\quad \text{and} \quad x^2 - 5y^2 = 1 }[/math]

When [math]\displaystyle{ \sqrt{5} }[/math] is approximated with the Babylonian method, starting with *x*_{0} = 2 and using *x*_{n+1} = 1/2(*x*_{n} + 5/*x*_{n}), the *n*th approximant *x*_{n} is equal to the 2^{n}th convergent of the continued fraction:

- [math]\displaystyle{ 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 finding applied to the polynomial [math]\displaystyle{ x^2-5 }[/math]. The Newton's method update, [math]\displaystyle{ x_{n+1} = x_n - f(x_n)/f'(x_n) }[/math], is equal to [math]\displaystyle{ (x_n + 5/x_n)/2 }[/math] when [math]\displaystyle{ f(x) = x^2 - 5 }[/math]. The method therefore converges quadratically.

## Relation to the golden ratio and Fibonacci numbers

The golden ratio φ is the arithmetic mean of 1 and [math]\displaystyle{ \sqrt{5} }[/math].^{[4]} The algebraic relationship between [math]\displaystyle{ \sqrt{5} }[/math], the golden ratio and the conjugate of the golden ratio (Φ = –1/*φ* = 1 − *φ*) is expressed in the following formulae:

- [math]\displaystyle{ \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]\displaystyle{ \sqrt{5} }[/math] rectangle.)

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

- [math]\displaystyle{ F(n) = \frac{\varphi^n-(1-\varphi)^n}{\sqrt 5}. }[/math]

The quotient of [math]\displaystyle{ \sqrt{5} }[/math] and *φ* (or the product of [math]\displaystyle{ \sqrt{5} }[/math] 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:^{[5]}

- [math]\displaystyle{ \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]\displaystyle{ \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 the quotient of the [math]\displaystyle{ n^{th} }[/math] Lucas number [math]\displaystyle{ L_n }[/math] and the [math]\displaystyle{ n^{th} }[/math] Fibonacci number [math]\displaystyle{ F_n }[/math] is directly equal to the square root of [math]\displaystyle{ 5 }[/math]:

- [math]\displaystyle{ \lim_{n\to\infty} \frac{L_n}{F_n}=\sqrt{5}. }[/math]

## Geometry

Geometrically, [math]\displaystyle{ \sqrt{5} }[/math] 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.^{[6]} Together with the algebraic relationship between [math]\displaystyle{ \sqrt{5} }[/math] 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*).

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 [math]\displaystyle{ \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.

A rectangle with side proportions 1:[math]\displaystyle{ \sqrt{5} }[/math] is called a *root-five rectangle* and is part of the series of root rectangles, a subset of dynamic rectangles, which are based on [math]\displaystyle{ \sqrt{1} }[/math] (= 1), [math]\displaystyle{ \sqrt{2} }[/math], [math]\displaystyle{ \sqrt{3} }[/math], [math]\displaystyle{ \sqrt{4} }[/math] (= 2), [math]\displaystyle{ \sqrt{5} }[/math]... and successively constructed using the diagonal of the previous root rectangle, starting from a square.^{[7]} 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*).^{[8]} 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 [math]\displaystyle{ \sqrt{5} }[/math], *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 [math]\displaystyle{ \sqrt{5}/2 }[/math] to both sides.

## Trigonometry

Like [math]\displaystyle{ \sqrt{2} }[/math] and [math]\displaystyle{ \sqrt{3} }[/math], 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.^{[9]} The simplest of these are

- [math]\displaystyle{ \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]\displaystyle{ \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.

## 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

- [math]\displaystyle{ \left|x - \frac{m}{n}\right| \lt \frac{1}{\sqrt{5}\,n^2} }[/math]

and that [math]\displaystyle{ \sqrt{5} }[/math] is best possible, in the sense that for any larger constant than [math]\displaystyle{ \sqrt{5} }[/math], there are some irrational numbers *x* for which only finitely many such approximations exist.^{[10]}

Closely related to this is the theorem^{[11]} 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:

- [math]\displaystyle{ \left|\alpha - {p_i\over q_i}\right| \lt {1\over \sqrt5 q_i^2}, \qquad \left|\alpha - {p_{i+1}\over q_{i+1}}\right| \lt {1\over \sqrt5 q_{i+1}^2}, \qquad \left|\alpha - {p_{i+2}\over q_{i+2}}\right| \lt {1\over \sqrt5 q_{i+2}^2}. }[/math]

And the [math]\displaystyle{ \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.^{[11]}

## Algebra

The ring [math]\displaystyle{ \mathbb{Z}[\sqrt{-5}] }[/math] contains numbers of the form [math]\displaystyle{ a + b\sqrt{-5} }[/math], where *a* and *b* are integers and [math]\displaystyle{ \sqrt{-5} }[/math] is the imaginary number [math]\displaystyle{ i\sqrt{5} }[/math]. This ring is a frequently cited example of an integral domain that is not a unique factorization domain.^{[12]} The number 6 has two inequivalent factorizations within this ring:

- [math]\displaystyle{ 6 = 2 \cdot 3 = (1 - \sqrt{-5})(1 + \sqrt{-5}). \, }[/math]

The field [math]\displaystyle{ \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]\displaystyle{ \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 fractions.^{[13]}^{[14]}

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

- [math]\displaystyle{ \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]

- [math]\displaystyle{ \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]

- [math]\displaystyle{ 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

## References

- ↑ Dauben, Joseph W. (June 1983) Scientific American
*Georg Cantor and the origins of transfinite set theory.*Volume 248; Page 122. - ↑ Yee, Alexander. "Records Set by y-cruncher". http://numberworld.org/y-cruncher/records.html.
- ↑ Conrad, Keith. "Pell's Equation II". https://kconrad.math.uconn.edu/blurbs/ugradnumthy/pelleqn2.pdf.
- ↑ 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). - ↑ Richard K. Guy: "The Strong Law of Small Numbers".
*American Mathematical Monthly*, vol. 95, 1988, pp. 675–712 - ↑ Ivrissimtzis, Ioannis P.; Dodgson, Neil A.; Sabin, Malcolm (2005), "[math]\displaystyle{ \sqrt5 }[/math]-subdivision", in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A.,
*Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003*, Mathematics and Visualization, Berlin: Springer, pp. 285–299, doi:10.1007/3-540-26808-1_16 - ↑ Kimberly Elam (2001),
*Geometry of Design: Studies in Proportion and Composition*, New York: Princeton Architectural Press, ISBN 1-56898-249-6, https://books.google.com/books?id=1KI0JVuWYGkC&q=intitle:%22Geometry+of+Design%22+%22root+5%22&pg=PA41 - ↑ Jay Hambidge (1967),
*The Elements of Dynamic Symmetry*, Courier Dover Publications, ISBN 0-486-21776-0, https://books.google.com/books?id=VYJK2F-dh2oC&q=%22root+five+rectangle%22++section+inauthor:hambidge&pg=PA26 - ↑ Julian D. A. Wiseman, "Sin and cos in surds"
- ↑ LeVeque, William Judson (1956),
*Topics in number theory*, Addison-Wesley Publishing Co., Inc., Reading, Mass. - ↑
^{11.0}^{11.1}Khinchin, Aleksandr Yakovlevich (1964),*Continued Fractions*, University of Chicago Press, Chicago and London - ↑ Chapman, Scott T.; Gotti, Felix; Gotti, Marly (2019), "How do elements really factor in [math]\displaystyle{ \mathbb{Z}[\sqrt{-5}] }[/math]?", in Badawi, Ayman; Coykendall, Jim,
*Advances in Commutative Algebra: Dedicated to David F. Anderson*, Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 171–195, doi:10.1007/978-981-13-7028-1_9, "Most undergraduate level abstract algebra texts use [math]\displaystyle{ \mathbb{Z}[\sqrt{-5}] }[/math] as an example of an integral domain which is not a unique factorization domain" - ↑ Ramanathan, K. G. (1984), "On the Rogers-Ramanujan continued fraction",
*Proceedings of the Indian Academy of Sciences, Section A***93**(2): 67–77, doi:10.1007/BF02840651, ISSN 0253-4142 - ↑ Eric W. Weisstein,
*Ramanujan Continued Fractions*, http://mathworld.wolfram.com/RamanujanContinuedFractions.html at MathWorld

Original source: https://en.wikipedia.org/wiki/Square root of 5.
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