Quadratwurzel aus 5

Vorlage:Irrational numbers
Binary	Vorlage:Gaps
Decimal	Vorlage:Gaps
Hexadecimal	Vorlage:Gaps
Continued fraction	$2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+\ddots }}}}}}}}$

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:

${\sqrt {5}}.\,$

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

Vorlage:Gaps Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS.

which can be rounded down to 2.236 to within 99.99% accuracy. The approximation Vorlage:Sfrac (≈ 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 Vorlage:Sfrac (approx. Vorlage:Val). As of December 2013, its numerical value in decimal has been computed to at least ten billion digits.

Proofs of irrationality

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

Suppose that Vorlage:Sqrt is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as Vorlage:Math for natural numbers Vorlage:Math and Vorlage:Math. Then Vorlage:Sqrt can be expressed in lower terms as Vorlage:Math, which is a contradiction. (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives Vorlage:Math and Vorlage:Math, which is true by the premise. The second fractional expression for Vorlage:Sqrt is in lower terms since, comparing denominators, Vorlage:Math since Vorlage:Math since Vorlage:Math since Vorlage:Math. And both the numerator and the denominator of the second fractional expression are positive since Vorlage:Math and Vorlage:Math.)

2. This irrationality proof is also a proof by contradiction:

Suppose that Vorlage:Math where Vorlage:Math is in reduced form.

Thus Vorlage:Math and Vorlage:Math. If Vorlage:Math were even, Vorlage:Math, Vorlage:Math, and Vorlage:Math would be even making the fraction Vorlage:Math not in reduced form. Thus Vorlage:Math is odd, and by following a similar process, Vorlage:Math is odd.

Now, let Vorlage:Math and Vorlage:Math where Vorlage:Math and Vorlage:Math are integers.

Substituting into Vorlage:Math we get:

$5(2n+1)^{2}=(2m+1)^{2}$

which simplifies to:

$5(4n^{2}+4n+1)=4m^{2}+4m+1$

making:

$20n^{2}+20n+5=4m^{2}+4m+1$

By subtracting 1 from both sides, we get:

$20n^{2}+20n+4=4m^{2}+4m$

which reduces to:

$5n^{2}+5n+1=m^{2}+m$

In other words:

$5n(n+1)+1=m(m+1)$

The expression Vorlage:Math is even for any integer Vorlage:Math (since either Vorlage:Math or Vorlage:Math is even). So this says that 5 × even + 1 = even, or odd = even. Since there is no integer that is both even and odd, we have reached a contradiction and Vorlage:Sqrt is irrational.

Continued fraction

It can be expressed as the continued fraction

$[2;4,4,4,4,4,\ldots ]=2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+\ddots }}}}}}}}.$ Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS

The convergents and semiconvergents of this continued fraction are as follows (the black terms are the semiconvergents):

${\color {red}{\frac {2}{1}}},{\frac {7}{3}},{\color {red}{\frac {9}{4}}},{\frac {20}{9}},{\frac {29}{13}},{\color {red}{\frac {38}{17}}},{\frac {123}{55}},{\color {red}{\frac {161}{72}}},{\frac {360}{161}},{\frac {521}{233}},{\color {red}{\frac {682}{305}}},{\frac {2207}{987}},{\color {red}{\frac {2889}{1292}}},\dots$

Convergents of the continued fraction are colored red; their numerators are 2, 9, 38, 161, ... Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS, and their denominators are 1, 4, 17, 72, ... Folge [[:OEIS:{{{1}}}|{{{1}}}]] in OEIS.

Each of these is the best rational approximation of Vorlage:Sqrt that has as small a denominator as it has

Babylonian method

When Vorlage:Sqrt is computed with the Babylonian method, starting with Vorlage:Math and using Vorlage:Math, the Vorlage:Mathth approximant Vorlage:Math is equal to the Vorlage:Mathth convergent of the convergent sequence:

${\frac {2}{1}}=2.0,\quad {\frac {9}{4}}=2.25,\quad {\frac {161}{72}}=2.23611\dots ,\quad {\frac {51841}{23184}}=2.2360679779\ldots$

Nested square expansions

The following nested square expressions converge to ${\sqrt {5}}$:

${\begin{aligned}{\sqrt {5}}&=3-10\left({\frac {1}{5}}+\left({\frac {1}{5}}+\left({\frac {1}{5}}+\left({\frac {1}{5}}+\cdots \right)^{2}\right)^{2}\right)^{2}\right)^{2}\\&={\frac {9}{4}}-4\left({\frac {1}{16}}-\left({\frac {1}{16}}-\left({\frac {1}{16}}-\left({\frac {1}{16}}-\cdots \right)^{2}\right)^{2}\right)^{2}\right)^{2}\\&={\frac {9}{4}}-5\left({\frac {1}{20}}+\left({\frac {1}{20}}+\left({\frac {1}{20}}+\left({\frac {1}{20}}+\cdots \right)^{2}\right)^{2}\right)^{2}\right)^{2}\end{aligned}}$

Relation to the golden ratio and Fibonacci numbers

The Vorlage:Sfrac diagonal of a half square forms the basis for the geometrical construction of a golden rectangle.

This golden ratio Vorlage:Math is the arithmetic mean of 1 and Vorlage:Sqrt. The algebraic relationship between Vorlage:Sqrt, the golden ratio and the conjugate of the golden ratio (Vorlage:Math) are expressed in the following formulae:

${\begin{aligned}{\sqrt {5}}&=\varphi +\Phi =2\varphi -1=2\Phi +1\\[5pt]\varphi &={\frac {{\sqrt {5}}+1}{2}}\\[5pt]\Phi &={\frac {{\sqrt {5}}-1}{2}}.\end{aligned}}$

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

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

$F(n)={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}.$

The quotient of Vorlage:Sqrt and Vorlage:Math (or the product of Vorlage:Sqrt and Vorlage:Math), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the Lucas numbers:

${\begin{aligned}{\frac {\sqrt {5}}{\varphi }}=\Phi \cdot {\sqrt {5}}={\frac {5-{\sqrt {5}}}{2}}&=1.3819660112501051518\dots \\&=[1;2,1,1,1,1,1,1,1,\ldots ]\\[5pt]{\frac {\varphi }{\sqrt {5}}}={\frac {1}{\Phi \cdot {\sqrt {5}}}}={\frac {5+{\sqrt {5}}}{10}}&=0.72360679774997896964\ldots \\&=[0;1,2,1,1,1,1,1,1,\ldots ].\end{aligned}}$

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:

${\begin{aligned}&{1,{\frac {3}{2}},{\frac {4}{3}},{\frac {7}{5}},{\frac {11}{8}},{\frac {18}{13}},{\frac {29}{21}},{\frac {47}{34}},{\frac {76}{55}},{\frac {123}{89}}},\ldots \ldots [1;2,1,1,1,1,1,1,1,\ldots ]\\[8pt]&{1,{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{7}},{\frac {8}{11}},{\frac {13}{18}},{\frac {21}{29}},{\frac {34}{47}},{\frac {55}{76}},{\frac {89}{123}}},\dots \dots [0;1,2,1,1,1,1,1,1,\dots ].\end{aligned}}$

Geometry

Conway triangle decomposition into homothetic smaller triangles.

Geometrically, Vorlage:Sqrt 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 Vorlage:Sqrt and Vorlage:Math, 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 Vorlage:Math).

Forming a dihedral right angle with the two equal squares that halve a 1:2 rectangle, it can be seen that Vorlage:Sqrt 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).Vorlage:Citation needed

The number Vorlage:Sqrt can be algebraically and geometrically related to [[square root of 2|Vorlage:Sqrt]] and [[square root of 3|Vorlage:Sqrt]], as it is the length of the hypotenuse of a right triangle with catheti measuring Vorlage:Sqrt and Vorlage:Sqrt (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 Vorlage:Sqrt:Vorlage:Sqrt:Vorlage:Sqrt. This follows from the geometrical relationships between a cube and the quantities Vorlage:Sqrt (edge-to-face-diagonal ratio, or distance between opposite edges), Vorlage:Sqrt (edge-to-cube-diagonal ratio) and Vorlage:Sqrt (the relationship just mentioned above).

A rectangle with side proportions 1:Vorlage:Sqrt is called a root-five rectangle and is part of the series of root rectangles, a subset of dynamic rectangles, which are based on Vorlage:Sqrt (= 1), Vorlage:Sqrt, Vorlage:Sqrt, Vorlage:Sqrt (= 2), Vorlage:Sqrt… and successively constructed using the diagonal of the previous root rectangle, starting from a square. A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Vorlage:Math × 1), or into two golden rectangles of different sizes (of dimensions Vorlage:Math × 1 and 1 × Vorlage:Math). It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × Vorlage:Math) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between Vorlage:Sqrt, Vorlage:Math and Vorlage:Math 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 Vorlage:Sfrac to both sides.

Trigonometry

Like Vorlage:Sqrt and Vorlage:Sqrt, 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. The simplest of these are

${\begin{aligned}\sin {\frac {\pi }{10}}=\sin 18^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}-1)={\frac {1}{{\sqrt {5}}+1}},\\[5pt]\sin {\frac {\pi }{5}}=\sin 36^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5-{\sqrt {5}})}},\\[5pt]\sin {\frac {3\pi }{10}}=\sin 54^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}+1)={\frac {1}{{\sqrt {5}}-1}},\\[5pt]\sin {\frac {2\pi }{5}}=\sin 72^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5+{\sqrt {5}})}}\,.\end{aligned}}$

As such the computation of its value is important for generating trigonometric tables.Vorlage:Citation needed Since Vorlage:Sqrt 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.Vorlage:Citation needed

Diophantine approximations

Hurwitz's theorem in Diophantine approximations states that every irrational number Vorlage:Math can be approximated by infinitely many rational numbers Vorlage:Math in lowest terms in such a way that

$\left|x-{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}$

and that Vorlage:Sqrt is best possible, in the sense that for any larger constant than Vorlage:Sqrt, there are some irrational numbers Vorlage:Math for which only finitely many such approximations exist.

Closely related to this is the theorem that of any three consecutive convergents Vorlage:Math, Vorlage:Math, Vorlage:Math, of a number Vorlage:Math, at least one of the three inequalities holds:

$\left|\alpha -{p_{i} \over q_{i}}\right|<{1 \over {\sqrt {5}}q_{i}^{2}},\qquad \left|\alpha -{p_{i+1} \over q_{i+1}}\right|<{1 \over {\sqrt {5}}q_{i+1}^{2}},\qquad \left|\alpha -{p_{i+2} \over q_{i+2}}\right|<{1 \over {\sqrt {5}}q_{i+2}^{2}}.$

And the Vorlage:Sqrt 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.

Algebra

The ring Vorlage:Math contains numbers of the form Vorlage:Math, where Vorlage:Math and Vorlage:Math are integers and Vorlage:Math is the imaginary number Vorlage:Math. This ring is a frequently cited example of an integral domain that is not a unique factorization domain.Vorlage:Citation needed The number 6 has two inequivalent factorizations within this ring:

$6=2\cdot 3=(1-{\sqrt {-5}})(1+{\sqrt {-5}}).\,$

The field Vorlage: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:

${\sqrt {5}}=e^{{\frac {2\pi }{5}}i}-e^{{\frac {4\pi }{5}}i}-e^{{\frac {6\pi }{5}}i}+e^{{\frac {8\pi }{5}}i}.\,$

Identities of Ramanujan

The square root of 5 appears in various identities discovered by Srinivasa Ramanujan involving continued fractions.

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

${\cfrac {1}{1+{\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-4\pi }}{1+{\cfrac {e^{-6\pi }}{1+\ddots }}}}}}}}=\left({\sqrt {\frac {5+{\sqrt {5}}}{2}}}-{\frac {{\sqrt {5}}+1}{2}}\right)e^{\frac {2\pi }{5}}=e^{\frac {2\pi }{5}}\left({\sqrt {\varphi {\sqrt {5}}}}-\varphi \right).$

${\cfrac {1}{1+{\cfrac {e^{-2\pi {\sqrt {5}}}}{1+{\cfrac {e^{-4\pi {\sqrt {5}}}}{1+{\cfrac {e^{-6\pi {\sqrt {5}}}}{1+\ddots }}}}}}}}=\left({{\sqrt {5}} \over 1+{\sqrt[{5}]{5^{\frac {3}{4}}(\varphi -1)^{\frac {5}{2}}-1}}}-\varphi \right)e^{\frac {2\pi }{\sqrt {5}}}.$

$4\int _{0}^{\infty }{\frac {xe^{-x{\sqrt {5}}}}{\cosh x}}\,dx={\cfrac {1}{1+{\cfrac {1^{2}}{1+{\cfrac {1^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {3^{2}}{1+{\cfrac {3^{2}}{1+\ddots }}}}}}}}}}}}}}.$

See also

• Golden ratio
• Square root
• Square root of 2
• Square root of 3