Rationality | Irrational |
---|---|

Representations | |

Decimal | 2.23606797749978969... |

Algebraic form | |

Continued fraction | |

Binary | 10.0011110001101110... |

Hexadecimal | 2.3C6EF372FE94F82C... |

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:

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]}

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

The successive partial evaluations of the continued fraction, which are called its
*convergents*, approach :

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 ; in other words, it is closer to than any rational number with a smaller denominator.

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

When 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:

The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial . The Newton's method update, , is equal to when . The method therefore converges quadratically.

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

(See the section below for their geometrical interpretation as decompositions of a rectangle.)

then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio:

The quotient of and *φ* (or the product of 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]}

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:

In fact, the limit of the quotient of the Lucas number and the Fibonacci number is directly equal to the square root of :

Geometrically, 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 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 . 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.^{[
citation needed]}

A rectangle with side proportions 1: is called a *root-five rectangle* and is part of the series of root rectangles, a subset of
dynamic rectangles, which are based on (= 1), , , (= 2), ... 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 , *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 to both sides.

Like and , 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

As such the computation of its value is important for generating
trigonometric tables.^{[
citation needed]} Since 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.^{[
citation needed]}

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

and that is best possible, in the sense that for any larger constant than , 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:

And the 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]}

The
ring contains numbers of the form , where *a* and *b* are
integers and is the
imaginary number . 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:

The field 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:

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:

**^**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".**^**Conrad, Keith. "Pell's Equation II" (PDF).*uconn.edu*. Retrieved 17 March 2022.**^**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), "-subdivision", in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A. (eds.),*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, MR 2112357**^**Kimberly Elam (2001),*Geometry of Design: Studies in Proportion and Composition*, New York: Princeton Architectural Press, ISBN 1-56898-249-6**^**Jay Hambidge (1967),*The Elements of Dynamic Symmetry*, Courier Dover Publications, ISBN 0-486-21776-0**^**Julian D. A. Wiseman, "Sin and cos in surds"**^**LeVeque, William Judson (1956),*Topics in number theory*, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR 0080682- ^
^{a}^{b}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 ?", in Badawi, Ayman; Coykendall, Jim (eds.),*Advances in Commutative Algebra: Dedicated to David F. Anderson*, Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 171–195, arXiv: 1711.10842, doi: 10.1007/978-981-13-7028-1_9, MR 3991169, S2CID 119142526,Most undergraduate level abstract algebra texts use 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, MR 0813071, S2CID 121808904**^**Eric W. Weisstein,*Ramanujan Continued Fractions*at MathWorld