square root of 7

File:1797 square root of 7.png

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{{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}} and 1852,{{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}} 3 in 1835,{{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}} 6 in 1808,{{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}} and 7 in 1797.{{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}}

An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".{{cite book |author=George Wentworth |author2=David Eugene Smith |author3=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}}

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

: $[2;\; 1,\; 1,\; 1,\; 4,\; 1,\; 1,\; 1,\; 4,\backslash ldots]\; =\; 2\; +\; \backslash cfrac\; 1\; \{1\; +\; \backslash cfrac\; 1\; \{1\; +\; \backslash cfrac\; 1\; \{1\; +\; \backslash cfrac\; 1\; \{4\; +\; \backslash cfrac\; 1\; \{1\; +\; \backslash dots\}\}\}\}\}.$ {{OEIS|id=A010121}}

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

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

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 best rational approximation of $\backslash sqrt\{7\}$; in other words, it is closer to $\backslash sqrt\{7\}$ 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:

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

Every fourth convergent, starting with {{math|{{sfrac|8|3}}}}, expressed as {{math|{{sfrac|x|y}}}}, satisfies the Pell's equation{{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 }}

:$x^2\; -\; 7y^2\; =\; 1\; .$

When $\backslash sqrt\{7\}$ is approximated with the Babylonian method, starting with {{math|x₁ {{=}} 3}} and using {{math|x_n+1 {{=}} {{sfrac|1|2}}{{big|{{big|(}}}}x_n + {{sfrac|7|x_n}}{{big|{{big|)}}}}}}, the {{math|n}}th approximant {{math|x_n}} is equal to the {{math|2ⁿ}}th convergent of the continued fraction:

:$x\_1\; =\; 3,\; \backslash quad\; x\_2\; =\; \backslash frac\{8\}\{3\}\; =\; 2.66...,\; \backslash quad\; x\_3\; =\; \backslash frac\{127\}\{48\}\; =2.6458...,\; \backslash quad\; x\_4\; =\; \backslash frac\{32257\}\{12192\}\; =\; 2.645751312...,\; \backslash quad\; x\_5\; =\; \backslash frac\{2081028097\}\{786554688\}\; =\; 2.645751311064591...,\; \backslash quad\; \backslash dots$

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, $x\_\{n+1\}\; =\; x\_n\; -\; f(x\_n)/f\text{'}(x\_n),$ 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).

File:Root rectangles up to 6.png

In plane geometry, the square root of 7 can be constructed via a sequence of dynamic rectangles, that is, as the largest diagonal of those rectangles illustrated here.

{{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-date=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.

}}

{{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 = 978-0-486-23542-4

}}{{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/}}

The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.{{cite book |last1=Blackwell |first1=William |title=Geometry in Architecture |date=1984 |publisher=Key Curriculum Press |isbn=978-1-55953-018-7 |page=25 |url=https://books.google.com/books?id=AJFZAAAAYAAJ&q=%22square+root+of+seven%22 |access-date=26 March 2022}}

Due to the Pythagorean theorem and Legendre's three-square theorem, $\backslash sqrt\{7\}$ is the smallest square root of a natural number that cannot be the distance between any two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). $\backslash sqrt\{15\}$ is the next smallest such number.{{cite OEIS|A005875}}

File:Dollar reverse with root 7 rectangle on large inner rectangle.png

On the reverse of the current US one-dollar bill, the "large inner box" has a length-to-width ratio of the square root of 7, and a diagonal of 6.0 inches, to within measurement accuracy.{{cite book |last1=McGrath |first1=Ken |title=The Secret Geometry of the Dollar |date=2002 |publisher=AuthorHouse |isbn=978-0-7596-1170-2 |pages=47–49 |url=https://books.google.com/books?id=1BgpDwAAQBAJ&pg=PA48 |access-date=26 March 2022 |ref=buck}}

• Square root
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6

{{Reflist}}

{{Algebraic numbers}}

{{Irrational number}}

Category:Mathematical constants

Category:Quadratic irrational numbers