Square root of 6

Square root of 6
Rationality	Irrational
Representations
Decimal	2.449489742783178098...
Algebraic form
Continued fraction

Last updated February 03, 2024

The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as:^[1]

which can be rounded up to 2.45 to within about 99.98% accuracy (about 1 part in 4800); that is, it differs from the correct value by about 1/2,000. It takes two more digits (2.4495) to reduce the error by about half. The approximation 218/89 (≈ 2.449438...) is nearly ten times better: despite having a denominator of only 89, it differs from the correct value by less than 1/20,000, or less than one part in 47,000.

Since 6 is the product of 2 and 3, the square root of 6 is the geometric mean of 2 and 3, and is the product of the square root of 2 and the square root of 3, both of which are irrational algebraic numbers.

NASA has published more than a million decimal digits of the square root of six.^[4]

Rational approximations

The square root of 6 can be expressed as the continued fraction

[2;2,4,2,4,2,\ldots ]=2+{\cfrac {1}{2+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{4+\dots }}}}}}}}.

(sequence A040003 in the OEIS )

The successive partial evaluations of the continued fraction, which are called its convergents, approach ${\sqrt {6}}$ :

{\frac {2}{1}},{\frac {5}{2}},{\frac {22}{9}},{\frac {49}{20}},{\frac {218}{89}},{\frac {485}{198}},{\frac {2158}{881}},{\frac {4801}{1960}},\dots

Their numerators are 2, 5, 22, 49, 218, 485, 2158, 4801, 21362, 47525, 211462, …(sequence A041006 in the OEIS ), and their denominators are 1, 2, 9, 20, 89, 198, 881, 1960, 8721, 19402, 86329, …(sequence A041007 in the OEIS ).^[5]

Each convergent is a best rational approximation of ${\sqrt {6}}$ ; in other words, it is closer to ${\sqrt {6}}$ than any rational with a smaller denominator. Decimal equivalents improve linearly, at a rate of nearly one digit per convergent:

{\frac {2}{1}}=2.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {22}{9}}=2.4444\dots ,\quad {\frac {49}{20}}=2.45,\quad {\frac {218}{89}}=2.44943...,\quad {\frac {485}{198}}=2.449494...,\quad \ldots

The convergents, expressed as $x / y$ , satisfy alternately the Pell's equations ^[5]

x^{2}-6y^{2}=-2\quad \mathrm {and} \quad x^{2}-6y^{2}=1

When ${\sqrt {6}}$ is approximated with the Babylonian method, starting with $x 0 = 2$ and using $x n +1 = 1 / 2 (x n + 6 / x n)$ , the $n$ th approximant $x n$ is equal to the $2 n$ th convergent of the continued fraction:

x_{0}=2,\quad x_{1}={\frac {5}{2}}=2.5,\quad x_{2}={\frac {49}{20}}=2.45,\quad x_{3}={\frac {4801}{1960}}=2.449489796...,\quad x_{4}={\frac {46099201}{18819920}}=2.449489742783179...,\quad \dots

The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial $x^{2}-6$ . The Newton's method update, $x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),$ is equal to $(x_{n}+6/x_{n})/2$ when $f(x)=x^{2}-6$ . The method therefore converges quadratically.

Geometry

In plane geometry, the square root of 6 can be constructed via a sequence of dynamic rectangles, as illustrated here.^[6]^[7]^[8]

In solid geometry, the square root of 6 appears as the longest distances between corners (vertices) of the double cube, as illustrated above. The square roots of all lower natural numbers appear as the distances between other vertex pairs in the double cube (including the vertices of the included two cubes).^[8]

The edge length of a cube with total surface area of 1 is ${\frac {\sqrt {6}}{6}}$ or the reciprocal square root of 6. The edge lengths of a regular tetrahedron ( $t$ ), a regular octahedron ( $o$ ), and a cube ( $c$ ) of equal total surface areas satisfy ${\frac {t\cdot o}{c^{2}}}={\sqrt {6}}$ .^[3]^[9]

The edge length of a regular octahedron is the square root of 6 times the radius of an inscribed sphere (that is, the distance from the center of the solid to the center of each face).^[10]

The square root of 6 appears in various other geometry contexts, such as the side length ${\frac {{\sqrt {6}}+{\sqrt {2}}}{2}}$ for the square enclosing an equilateral triangle of side 2 (see figure).

Trigonometry

The square root of 6, with the square root of 2 added or subtracted, appears in several exact trigonometric values for angles at multiples of 15 degrees ( $\pi /12$ radians).^[11]

:{| class="wikitable" style="text-align: center;"

!Radians!!Degrees!! $sin$ !! $cos$ !! $tan$ !! $cot$ !! $sec$ !! $csc$ |- ! ${\frac {\pi }{12}}$ !! $15^{\circ }$ | ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ || ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ || $2-{\sqrt {3}}$ || $2+{\sqrt {3}}$ || ${\sqrt {6}}-{\sqrt {2}}$ || ${\sqrt {6}}+{\sqrt {2}}$ |- ! ${\frac {5\pi }{12}}$ !! $75^{\circ }$ | ${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$ || ${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$ || $2+{\sqrt {3}}$ || $2-{\sqrt {3}}$ || ${\sqrt {6}}+{\sqrt {2}}$ || ${\sqrt {6}}-{\sqrt {2}}$ |}

In culture

Villard de Honnecourt's 13th century construction of a Gothic "fifth-point arch" with circular arcs of radius 5 has a height of twice the square root of 6, as illustrated here.^[12]^[13]

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References

↑ Ray, Joseph (1842). Ray's Eclectic Arithmetic on the Inductive and Analytic Methods of Instruction. Cincinnati: Truman and Smith. p. 217. Retrieved 20 March 2022.
↑ O'Sullivan, Daniel (1872). The Principles of Arithmetic: A Comprehensive Text-Book. Dublin: Alexander Thom. p. 234. Retrieved 17 March 2022.
1 2 Sloane, N. J. A. (ed.). "SequenceA010464(Decimal expansion of square root of 6)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
↑ Robert Nemiroff; Jerry Bonnell. "the first 1 million digits of the square root of 6". nasa.gov. Retrieved 17 March 2022.
1 2 Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022. The continued fraction of √6 is [2; 2, 4], and the table of convergents below suggests (and it is true) that every other convergent provides a solution to $x 2 - 6 y 2 = 1$ .
↑ Jay Hambidge (1920) [1920]. Dynamic Symmetry: The Greek Vase (Reprint of original Yale University Press ed.). Whitefish, MT: Kessinger Publishing. pp. 19–29. ISBN 0-7661-7679-7. Dynamic Symmetry root rectangles.
↑ Matila Ghyka (1977). The Geometry of Art and Life . Courier Dover Publications. pp. 126–127. ISBN 9780486235424.
1 2 Fletcher, Rachel (2013). Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature. George F Thompson Publishing. ISBN 978-1-938086-02-1.
↑ Rechtman, Ana. "Un défi par semaine Avril 2016, 3e défi (Solution du 2e défi d'Avril)". Images des Mathématiques. Retrieved 23 March 2022.
↑ S. C. & L. M. Gould (1890). The Bizarre Notes and Queries in History, Folk-lore, Mathematics, Mysticism, Art, Science, Etc, Volumes 7-8. Manchester, N. H. p. 342. Retrieved 19 March 2022. In the octahedron whose diameter is 2, the linear edge equals the square root of 6.{{cite book}}: CS1 maint: location missing publisher (link)
↑ Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. p. 74. ISBN 978-0-486-61272-0.
↑ Branner, Robert (1960). "Villard de Honnecourt, Archimedes, and Chartres". Journal of the Society of Architectural Historians. 19 (3): 91–96. doi:10.2307/988023. JSTOR 988023 . Retrieved 25 March 2022.
↑ Shelby, Lon R. (1969). "Setting Out the Keystones of Pointed Arches: A Note on Medieval 'Baugeometrie'". Technology and Culture. 10 (4): 537–548. doi:10.2307/3101574. JSTOR 3101574 . Retrieved 25 March 2022.

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[1] Ray, Joseph (1842). Ray's Eclectic Arithmetic on the Inductive and Analytic Methods of Instruction. Cincinnati: Truman and Smith. p. 217. Retrieved 20 March 2022.

[2] O'Sullivan, Daniel (1872). The Principles of Arithmetic: A Comprehensive Text-Book. Dublin: Alexander Thom. p. 234. Retrieved 17 March 2022.

[oeis-3] 1 2 Sloane, N. J. A. (ed.). "SequenceA010464(Decimal expansion of square root of 6)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation.

[nasa-4] Robert Nemiroff; Jerry Bonnell. "the first 1 million digits of the square root of 6". nasa.gov. Retrieved 17 March 2022.

[conrad-5] 1 2 Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022. The continued fraction of √6 is [2; 2, 4], and the table of convergents below suggests (and it is true) that every other convergent provides a solution to $x 2 - 6 y 2 = 1$ .

[hambidge-6] Jay Hambidge (1920) [1920]. Dynamic Symmetry: The Greek Vase (Reprint of original Yale University Press ed.). Whitefish, MT: Kessinger Publishing. pp. 19–29. ISBN 0-7661-7679-7. Dynamic Symmetry root rectangles.

[ghyka-7] Matila Ghyka (1977). The Geometry of Art and Life . Courier Dover Publications. pp. 126–127. ISBN 9780486235424.

[fletcher-8] 1 2 Fletcher, Rachel (2013). Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature. George F Thompson Publishing. ISBN 978-1-938086-02-1.

[rechtman-9] Rechtman, Ana. "Un défi par semaine Avril 2016, 3e défi (Solution du 2e défi d'Avril)". Images des Mathématiques. Retrieved 23 March 2022.

[gould-10] S. C. & L. M. Gould (1890). The Bizarre Notes and Queries in History, Folk-lore, Mathematics, Mysticism, Art, Science, Etc, Volumes 7-8. Manchester, N. H. p. 342. Retrieved 19 March 2022. In the octahedron whose diameter is 2, the linear edge equals the square root of 6.{{cite book}}: CS1 maint: location missing publisher (link)

[Abramowitz_and_Stegun-11] Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover Publications. p. 74. ISBN 978-0-486-61272-0.

[12] Branner, Robert (1960). "Villard de Honnecourt, Archimedes, and Chartres". Journal of the Society of Architectural Historians. 19 (3): 91–96. doi:10.2307/988023. JSTOR 988023 . Retrieved 25 March 2022.

[13] Shelby, Lon R. (1969). "Setting Out the Keystones of Pointed Arches: A Note on Medieval 'Baugeometrie'". Technology and Culture. 10 (4): 537–548. doi:10.2307/3101574. JSTOR 3101574 . Retrieved 25 March 2022.

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