# Square root of 5

Template:Irrational numbers | |

Binary | 10.0011110001101111... |

Decimal | 2.23606797749978969... |

Hexadecimal | 2.3C6EF372FE94F82C... |

Continued fraction |

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:

- 2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089... (sequence A002163 in OEIS).

which can be rounded down to 2.236 to within 99.99% accuracy. As of December 2013, its numerical value in decimal has been computed to at least ten billion digits.^{[2]}

## Proof of irrationality

This irrationality proof for the square root of 5 uses Fermat's method of infinite descent:

Suppose that √5 is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as for natural numbers *m* and *n*. Then √5 can be expressed in lower terms as , which is a contradiction.^{[3]} (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives and hence , which is true by the premise. The second fractional expression for √5 is in lower terms since, comparing denominators, since since since . And both the numerator and the denominator of the second fractional expression are positive since and .)

## Continued fraction

It can be expressed as the continued fraction [2; 4, 4, 4, 4, 4...] (sequence A040002 in OEIS). The sequence of best rational approximations is:

Convergents of the continued fraction are colored; their numerators are 2, 9, 38, 161, ... (sequence A001077 in OEIS), and their denominators are 1, 4, 17, 72, ... (sequence A001076 in OEIS). The other (non-colored) terms are semiconvergents.

## Babylonian method

When 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:

## Relation to the golden ratio and Fibonacci numbers

This golden ratio is the arithmetic mean of 1 and the square root of 5.^{[4]} The algebraic relationship between the square root of 5, the golden ratio and the conjugate of the golden ratio () are expressed in the following formulae:

(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:

The quotient of √5 and (or the product of √5 and ), 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:

## 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).{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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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, a subset of 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.^{[6]} 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).^{[7]} 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 Template:Fraction to both sides.

## Trigonometry

Like √2 and √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.^{[8]} The simplest of these are

As such the computation of its value is important for generating trigonometric tables.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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## 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

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.^{[9]}

Closely related to this is the theorem^{[10]} 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 √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.^{[10]}

## Algebra

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.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B=
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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:

## Identities of Ramanujan

The square root of 5 appears in various identities of Ramanujan involving continued fractions.^{[11]}^{[12]}

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

## See also

## References

- ↑ Dauben, Joseph W. (June 1983) Scientific American
*Georg Cantor and the origins of transfinite set theory.*Volume 248; Page 122. - ↑ Lukasz Komsta:
*Computations page* - ↑ Grant, Mike, and Perella, Malcolm, "Descending to the irrational",
*Mathematical Gazette*83, July 1999, pp.263-267. - ↑ 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 - ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}
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- ↑ Julian D. A. Wiseman, "Sin and cos in surds"
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- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }} at MathWorld