7の平方根

7の平方根（ななのへいほうこん、英: square root of 7）は、平方して7となる実数である。すなわち、 $r^{2}=r\times r=7$ をみたす実数 $r$ であり、冪根形式では^[1] ${\sqrt {7}}\,$ 、指数形式では $7^{\frac {1}{2}}$ と表される。無理数かつ代数的数である。

最初の60桁の有効数字は

2.64575131106459059050161575363926042571025918308245018036833...^[2]

これは約99.99%の精度（1000分の1）以内で2.646に切り上げることができるが、正確な値とは約1/4,000異なっている。127/48（≈ 2.645833...）の方がより良い近似値である。分母がわずか48しかないにもかかわらず、正確な値とは1/12,000（33000分の1）未満の差しかない。

${\sqrt {7}}\,$ の小数表示100万桁以上が公開されている^[3]。

有理近似

The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773^[4] and 1852,^[5] 3 in 1835,^[6] 6 in 1808,^[7] and 7 in 1797.^[8] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".^[9]

For a family of good rational approximations, the square root of 7 can be expressed as the continued fraction

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

オンライン整数列大辞典の数列 A010121

The successive partial evaluations of the continued fraction, which are called its convergents, approach ${\sqrt {7}}$ :

{\frac {2}{1}},{\frac {3}{1}},{\frac {5}{2}},{\frac {8}{3}},{\frac {37}{14}},{\frac {45}{17}},{\frac {82}{31}},{\frac {127}{48}},{\frac {590}{223}},{\frac {717}{271}},\dots

Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…オンライン整数列大辞典の数列 A041008 , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…オンライン整数列大辞典の数列 A041009.

Each convergent is a best rational approximation of ${\sqrt {7}}$ ; in other words, it is closer to ${\sqrt {7}}$ than any rational with a smaller denominator. Approximate decimal equivalents improve linearly (number of digits proportional to convergent number) at a rate of less than one digit per step:

{\frac {2}{1}}=2.0,\quad {\frac {3}{1}}=3.0,\quad {\frac {5}{2}}=2.5,\quad {\frac {8}{3}}=2.66\dots ,\quad {\frac {37}{14}}=2.6429...,\quad {\frac {45}{17}}=2.64705...,\quad {\frac {82}{31}}=2.64516...,\quad {\frac {127}{48}}=2.645833...,\quad \ldots

Every fourth convergent, starting with $8 / 3$ , expressed as $x / y$ , satisfies the Pell's equation^[10]

x^{2}-7y^{2}=1.

When ${\sqrt {7}}$ is approximated with the Babylonian method, starting with $x 1 = 3$ and using $x n +1 = 1 / 2 (x n + 7 / x n)$ , the $n$ th approximant $x n$ is equal to the $2 n$ th convergent of the continued fraction:

x_{1}=3,\quad x_{2}={\frac {8}{3}}=2.66...,\quad x_{3}={\frac {127}{48}}=2.6458...,\quad x_{4}={\frac {32257}{12192}}=2.645751312...,\quad x_{5}={\frac {2081028097}{786554688}}=2.645751311064591...,\quad \dots

All but the first of these satisfy the Pell's equation above.

The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial $x^{2}-7$ . The Newton's method update, $x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),$ is equal to $(x_{n}+7/x_{n})/2$ when $f(x)=x^{2}-7$ . The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).

幾何学

平面幾何学において、 ${\sqrt {7}}\,$ は一連の動的な長方形により、すなわち上図の長方形の最大の対角線として表される^[11]^[12]^[13]。

辺の長さが2の正三角形に外接する最小の長方形は長さ ${\sqrt {7}}\,$ の対角線を持つ^[14]。

数学以外の分野

現行のアメリカ合衆国1ドル紙幣の裏にある大きな内箱は長さと幅の比が ${\sqrt {7}}\,$ で、対角線の長さが6.0インチである（測定精度の範囲内で）^[15]。

脚注

^ Darby, John (1843). The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, .... London: Whittaker & Company. p. 172 27 March 2022閲覧。
^ Sloane, N.J.A. (ed.). "Sequence A010465 (Decimal expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2024年1月21日閲覧。
^ Robert Nemiroff and Jerry Bonnell (2008). The square root of 7. https://www.gutenberg.org/ebooks/631 25 March 2022閲覧。
^ Ewing, Alexander (1773). Institutes of Arithmetic: For the Use of Schools and Academies. Edinburgh: T. Caddell. p. 104
^ Ray, Joseph (1852). Ray's Algebra, Part Second: An Analytical Treatise, Designed for High Schools and Academies, Part 2. Cincinnati: Sargent, Wilson & Hinkle. p. 132 27 March 2022閲覧。
^ Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science.... Russell, Shattuck & Company. pp. 212–213 27 March 2022閲覧。
^ Thompson, James (1808). The American Tutor's Guide: Being a Compendium of Arithmetic. In Six Parts. Albany: E. & E. Hosford. p. 122 27 March 2022閲覧。
^ Hawney, William (1797). The Complete Measurer: Or, the Whole Art of Measuring. In Two Parts. Part I. Teaching Decimal Arithmetic ... Part II. Teaching to Measure All Sorts of Superficies and Solids ... Thirteenth Edition. To which is Added an Appendix. 1. Of Gaging. 2. Of Land-measuring. London. pp. 59–60 27 March 2022閲覧。
^ George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113 27 March 2022閲覧。
^ “Pell's Equation II”. uconn.edu. 17 March 2022閲覧。
^ Jay Hambidge (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
^ 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
^ Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 9781559530187 26 March 2022閲覧。
^ McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN 9780759611702 26 March 2022閲覧。

[darby-1] Darby, John (1843). The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, .... London: Whittaker & Company. p. 172 27 March 2022閲覧。

[oeis-2] Sloane, N.J.A. (ed.). "Sequence A010465 (Decimal expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 2024年1月21日閲覧。

[million-3] Robert Nemiroff and Jerry Bonnell (2008). The square root of 7. https://www.gutenberg.org/ebooks/631 25 March 2022閲覧。

[4] Ewing, Alexander (1773). Institutes of Arithmetic: For the Use of Schools and Academies. Edinburgh: T. Caddell. p. 104

[5] Ray, Joseph (1852). Ray's Algebra, Part Second: An Analytical Treatise, Designed for High Schools and Academies, Part 2. Cincinnati: Sargent, Wilson & Hinkle. p. 132 27 March 2022閲覧。

[bailey-6] Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science.... Russell, Shattuck & Company. pp. 212–213 27 March 2022閲覧。

[thompson-7] Thompson, James (1808). The American Tutor's Guide: Being a Compendium of Arithmetic. In Six Parts. Albany: E. & E. Hosford. p. 122 27 March 2022閲覧。

[hawney-8] Hawney, William (1797). The Complete Measurer: Or, the Whole Art of Measuring. In Two Parts. Part I. Teaching Decimal Arithmetic ... Part II. Teaching to Measure All Sorts of Superficies and Solids ... Thirteenth Edition. To which is Added an Appendix. 1. Of Gaging. 2. Of Land-measuring. London. pp. 59–60 27 March 2022閲覧。

[wentworth-9] George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113 27 March 2022閲覧。

[conrad-10] “Pell's Equation II”. uconn.edu. 17 March 2022閲覧。

[hambidge-11] Jay Hambidge (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-12] Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 9780486235424

[fletcher-13] 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

[blackwell-14] Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 9781559530187 26 March 2022閲覧。

[15] McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN 9780759611702 26 March 2022閲覧。

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表話編歴代数的数
代数的整数チェビシェフ分点（英語版）作図可能数コンウェイの定数（英語版） ( $λ$ ) アイゼンシュタイン整数ガウス整数黄金数 ( $φ$ ) クンマー環ペロン数（英語版）ピゾ＝ヴィジャヤラガヴァン数（英語版）二次の無理数（英語版）有理数 ( $\mathbb {Q}$ ) 1の冪根サレム数白銀比 ( $δ S$ ) プラスチック数 ( $ρ$ ) $\sqrt 2$ $\sqrt 3$ $\sqrt 5$ $\sqrt 6$ $\sqrt 7$ $\sqrt 10$ $3 \sqrt 2$ $12 \sqrt 2$		$ℚ$
カテゴリ

有理近似

幾何学

数学以外の分野

関連項目

脚注