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 × r = 7 {\displaystyle r^{2}=r\times r=7} をみたす実数rであり、冪根形式では 7 {\displaystyle {\sqrt {7}}\,} 、 指数形式では 7 1 2 {\displaystyle 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... 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。 7 {\displaystyle {\sqrt {7}}\,} の小数表示100万桁以上が公開されている。 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 and 1852, 3 in 1835, 6 in 1808, and 7 in 1797. An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth". 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 , … ] = 2 + 1 1 + 1 1 + 1 1 + 1 4 + 1 1 + … . {\displaystyle [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 7 {\displaystyle {\sqrt {7}}} : 2 1 , 3 1 , 5 2 , 8 3 , 37 14 , 45 17 , 82 31 , 127 48 , 590 223 , 717 271 , … {\displaystyle {\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 7 {\displaystyle {\sqrt {7}}} ; in other words, it is closer to 7 {\displaystyle {\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: 2 1 = 2.0 , 3 1 = 3.0 , 5 2 = 2.5 , 8 3 = 2.66 … , 37 14 = 2.6429... , 45 17 = 2.64705... , 82 31 = 2.64516... , 127 48 = 2.645833... , … {\displaystyle {\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 x 2 − 7 y 2 = 1. {\displaystyle x^{2}-7y^{2}=1.} When 7 {\displaystyle {\sqrt {7}}} is approximated with the Babylonian method, starting with x1 = 3 and using xn+1 = 1/2(xn + 7/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction: x 1 = 3 , x 2 = 8 3 = 2.66... , x 3 = 127 48 = 2.6458... , x 4 = 32257 12192 = 2.645751312... , x 5 = 2081028097 786554688 = 2.645751311064591... , … {\displaystyle 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 {\displaystyle x^{2}-7} . The Newton's method update, x n + 1 = x n − f ( x n ) / f ′ ( x n ) , {\displaystyle x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),} is equal to ( x n + 7 / x n ) / 2 {\displaystyle (x_{n}+7/x_{n})/2} when f ( x ) = x 2 − 7 {\displaystyle 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). 平面幾何学において、 7 {\displaystyle {\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。 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 7 {\displaystyle {\sqrt {7}}\,} の対角線を持つ。 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 7 {\displaystyle {\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）。 平方根 2の平方根 3の平方根 5の平方根 6の平方根 10の平方根 ^ Darby, John (1843). The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, .... London: Whittaker & Company. p. 172. https://books.google.com/books?id=lfteAAAAcAAJ&pg=PA172 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. https://books.google.com/books?id=rUvzudeLHbQC&pg=PA104  ^ 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. https://books.google.com/books?id=nY4AAAAAMAAJ&pg=PA132 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. https://books.google.com/books?id=e33H8RjDxocC&pg=PA212 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. https://books.google.com/books?id=8R8AAAAAMAAJ&pg=PA122 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. https://books.google.com/books?id=9eSLbslYbpAC&pg=PA60 27 March 2022閲覧。  ^ George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. https://books.google.com/books?id=sqMXAAAAYAAJ&pg=PA113 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. https://archive.org/details/bub_gb_Qq4gAAAAMAAJ. "Dynamic Symmetry root rectangles."  ^ Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 9780486235424. https://archive.org/details/geometryofartlif00mati  ^ 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. https://infinitemeasure.com/publications/infinite-measure/  ^ Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 9781559530187. https://books.google.com/books?id=AJFZAAAAYAAJ&q=%22square+root+of+seven%22 26 March 2022閲覧。  ^ McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN 9780759611702. https://books.google.com/books?id=1BgpDwAAQBAJ&pg=PA48 26 March 2022閲覧。