7の平方根のソースを表示

{{翻訳中途|1=[https://en.wikipedia.org/w/index.php?title=Square_root_of_7&oldid=1133744606 en:Square root of 7 (08:56, 15 January 2023 (UTC))]|date=2023年1月19日 (木) 03:21 (UTC)}}[[File:Rectangle that bounds equilateral triangle.png|thumb|The rectangle that bounds an equialateral triangle of side 2, or a regular hexagon of side 1, has size [[square root of 3]] by [[square root of 4]], with a diagonal of square root of 7.]]
[[File:Slide rule with square roots of 6 and 7.jpg|thumb|A Logarex system Darmstadt [[slide rule]] with 7 and 6 on A and B scales, and [[square root of 6|square roots of 6]] and of 7 on C and D scales, which can be read as slightly less than 2.45 and somewhat more than 2.64, respectively]]
'''7の平方根'''（ななのへいほうこん、{{lang-en-short|square root of 7}}）は、[[自乗|平方]]して[[7]]となる[[実数]]である。すなわち、<math>r^2 = r \times r = 7</math>
をみたす実数{{mvar|r}}であり、[[冪根]]形式では<ref name="darby">{{cite book | last1=Darby | first1=John | title=The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, ... | date=1843 | publisher=Whittaker & Company | location = London | page=172 | url=https://books.google.com/books?id=lfteAAAAcAAJ&pg=PA172 |access-date=27 March 2022}}</ref><math>\sqrt{7}\, </math>、
[[指数]]形式では<math>7^\frac{1}{2}</math>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
:{{gaps|2.64575|13110|64590|59050|16157|53639|26042|57102|59183|08245|01803|6833...}}<ref name=oeis>{{Cite OEIS |A010465 |Decimal expansion of square root of 7|accessdate=2024-01-21}}</ref>

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{{sfrac|1|4,000}}異なっている。{{sfrac|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{{sfrac|1|12,000}}（33000分の1）未満の差しかない。

<math>\sqrt{7}\, </math>の小数表示100万桁以上が公開されている<ref name="million">{{cite book |authors=Robert Nemiroff and Jerry Bonnell |title=The square root of 7 | website=gutenberg.org | url=https://www.gutenberg.org/ebooks/631 |access-date=25 March 2022 | year=2008}}</ref>。

==有理近似==
[[File:1797 square root of 7.png|thumb|left|Explanation of how to extract the square root of 7 to 7 places and more, from Hawney, 1797]]

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<ref>{{cite book |last1=Ewing |first1=Alexander |title=Institutes of Arithmetic: For the Use of Schools and Academies |date=1773 |publisher=T. Caddell |location=Edinburgh |page=104 |url=https://books.google.com/books?id=rUvzudeLHbQC&pg=PA104}}</ref> and 1852,<ref>{{cite book |last1=Ray |first1=Joseph |title=Ray's Algebra, Part Second: An Analytical Treatise, Designed for High Schools and Academies, Part 2 |date=1852 |publisher=Sargent, Wilson & Hinkle |location=Cincinnati |page=132 |url=https://books.google.com/books?id=nY4AAAAAMAAJ&pg=PA132 |access-date=27 March 2022}}</ref> 3 in 1835,<ref name="bailey">{{cite book |last1=Bailey |first1=Ebenezer |title=First Lessons in Algebra, Being an Easy Introduction to that Science... |date=1835 |publisher=Russell, Shattuck & Company |pages=212–213 |url=https://books.google.com/books?id=e33H8RjDxocC&pg=PA212 |access-date=27 March 2022}}</ref> 6 in 1808,<ref name="thompson">{{cite book |last1=Thompson |first1=James |title=The American Tutor's Guide: Being a Compendium of Arithmetic. In Six Parts |date=1808 |publisher=E. & E. Hosford |location=Albany |page=122 |url=https://books.google.com/books?id=8R8AAAAAMAAJ&pg=PA122 |access-date=27 March 2022}}</ref> and 7 in 1797.<ref name="hawney">{{cite book |last1=Hawney |first1=William |title=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 |year=1797 |location=London |pages=59–60 |url=https://books.google.com/books?id=9eSLbslYbpAC&pg=PA60 |access-date=27 March 2022}}</ref>
An extraction by [[Newton's method]] (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".<ref name="wentworth">{{cite book |authors=George Wentworth, David Eugene Smith, Herbert Druery Harper |title=Fundamentals of Practical Mathematics |date=1922 |publisher=Ginn and Company |page=113 |url=https://books.google.com/books?id=sqMXAAAAYAAJ&pg=PA113 |access-date=27 March 2022}}</ref>

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

: <math> [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}}}}}.</math> {{OEIS|id=A010121}}

The successive partial evaluations of the continued fraction, which are called its [[Continued_fraction#Infinite_continued_fractions_and_convergents|''convergents'']], approach <math>\sqrt{7}</math>:
:<math>\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</math>
Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…{{OEIS|id=A041008}} , and their denominators are 1, 1, 2, 3, 14, 17, 31,  48, 223, 271,  494,  765, 3554, 4319, 7873, 12192,…{{OEIS|id=A041009}}.

Each convergent is a [[Continued fraction#Best rational approximations|best rational approximation]] of <math>\sqrt{7}</math>; in other words, it is closer to <math>\sqrt{7}</math> 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:
:<math>\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</math>

Every fourth convergent, starting with {{math|{{sfrac|8|3}}}}, expressed as {{math|{{sfrac|x|y}}}}, satisfies the [[Pell's equation]]<ref name=conrad>{{cite web |last1=Conrad |first1=Keith |title=Pell's Equation II |url=https://kconrad.math.uconn.edu/blurbs/ugradnumthy/pelleqn2.pdf |website=uconn.edu |access-date=17 March 2022 }}</ref>
:<math>x^2 - 7y^2 = 1 .</math>

When <math>\sqrt{7}</math> is approximated with the [[Methods of computing square roots#Heron's method|Babylonian method]], starting with {{math|''x''<sub>1</sub> {{=}} 3}} and using {{math|''x''<sub>''n''+1</sub> {{=}} {{sfrac|1|2}}{{big|{{big|(}}}}''x''<sub>''n''</sub> + {{sfrac|7|''x''<sub>''n''</sub>}}{{big|{{big|)}}}}}}, the {{math|''n''}}th approximant {{math|''x''<sub>''n''</sub>}} is equal to the {{math|2<sup>''n''</sup>}}th convergent of the continued fraction:
:<math> 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</math>

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 <math>x^2-7</math>.  The Newton's method update, <math>x_{n+1} = x_n - f(x_n)/f'(x_n),</math> is equal to <math>(x_n + 7/x_n)/2</math> when <math>f(x) = x^2 - 7</math>.  The method therefore [[Rate_of_convergence#Q-convergence_definitions|converges quadratically]] (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).
==幾何学==
[[File:Root rectangles up to 6.png|thumb|left|upright=1.35|Root rectangles illustrate a construction of the square root of 7 (the diagonal of the root-6 rectangle).]]
平面幾何学において、<math>\sqrt{7}\, </math>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<ref name=hambidge>
{{cite book
|author=Jay Hambidge
|title=Dynamic Symmetry: The Greek Vase
|year=1920
|edition= Reprint of original Yale University Press 
|publisher=Kessinger Publishing
|location=Whitefish, MT
|orig-year=1920
|pages=[https://archive.org/details/bub_gb_Qq4gAAAAMAAJ/page/n29 19]–29
|isbn=0-7661-7679-7
|url=https://archive.org/details/bub_gb_Qq4gAAAAMAAJ
|quote=Dynamic Symmetry root rectangles.
}}</ref><ref name=ghyka>
{{cite book
|author = Matila Ghyka
|year = 1977
|title = The Geometry of Art and Life
|url = https://archive.org/details/geometryofartlif00mati
|url-access = registration
|publisher=Courier Dover Publications
|pages =[https://archive.org/details/geometryofartlif00mati/page/126 126–127]
|isbn = 9780486235424
}}</ref><ref name="fletcher">{{cite book |last1=Fletcher |first1=Rachel |title=Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature |date=2013 |publisher=George F Thompson Publishing |isbn=978-1-938086-02-1 |url=https://infinitemeasure.com/publications/infinite-measure/}}</ref>。

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<math>\sqrt{7}\, </math>の対角線を持つ<ref name="blackwell">{{cite book |last1=Blackwell |first1=William |title=Geometry in Architecture |date=1984 |publisher=Key Curriculum Press |isbn=9781559530187 |page=25 |url=https://books.google.com/books?id=AJFZAAAAYAAJ&q=%22square+root+of+seven%22 |access-date=26 March 2022}}</ref>。
==数学以外の分野==
[[File:Dollar reverse with root 7 rectangle on large inner rectangle.png|thumb|upright=1.5|left|Scan of US dollar bill reverse with root 7 rectangle annotation]]
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<math>\sqrt{7}\, </math>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）<ref>{{cite book |last1=McGrath |first1=Ken |title=The Secret Geometry of the Dollar |date=2002 |publisher=AuthorHouse |isbn=9780759611702 |pages=47–49 |url=https://books.google.com/books?id=1BgpDwAAQBAJ&pg=PA48 |access-date=26 March 2022 |ref=buck}}</ref>。
==関連項目==
* [[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]]

==脚注==
{{Reflist}}

{{代数的数}}
{{Normdaten}}
{{デフォルトソート:7のへいほうこん}}
[[Category:代数的数]]
[[Category:無理数]]
[[Category:数学に関する記事|/7のへいほうこん]]