Skip to content Skip to sidebar Skip to footer
Home / How to Write Sqrt 7 as a Continued Fraction

How to Write Sqrt 7 as a Continued Fraction

Post a Comment

Positive real number which when multiplied by itself gives 7

Square root of 7
Rationality Irrational
Representations
Decimal 2.6457513110 64590 590..._10
Algebraic form 7 {\displaystyle {\sqrt {7}}}
Continued fraction 2 + 1 1 + 1 1 + 1 1 + 1 4 + ⋱ {\displaystyle 2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}
Binary 10.10100101 0100 1111 1111..._2
Hexadecimal 2.A54FF53A 5F1D..._16

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.

A Logarex system Darmstadt slide rule with 7 and 6 on A and B scales, and 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

The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7. It is more precisely called the principal square root of 7, to distinguish it from the negative number with the same property. This number appears in various geometric and number-theoretic contexts. It can be denoted in surd form as:[1]

7 , {\displaystyle {\sqrt {7}}\,,}

and in exponent form as:

7 1 2 . {\displaystyle 7^{\frac {1}{2}}.}

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.6457513110 64590 59050 16157 53639 26042 57102 59183 08245 01803 6833... .[2]

which can be rounded down to 2.646 to within about 99.99% accuracy (about 1 part in 10000); that is, it differs from the correct value by about 1 / 4,000 . The approximation 127 / 48 (≈ 2.645833...) is better: despite having a denominator of only 48, it differs from the correct value by less than 1 / 12,000 , or less than one part in 33,000.

More than a million decimal digits of the square root of seven have been published.[3]

Rational approximations [edit]

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[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 thousanth".[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 , … ] = 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 }}}}}}}}}}.} (sequence A010121 in the OEIS)

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…(sequence A041008 in the OEIS) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequence A041009 in the OEIS).

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[10]

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 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 , 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).

Geometry [edit]

Root rectangles illustrate a construction of the square root of 7 (the diagonal of the root-6 rectangle).

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.[11] [12] [13]

The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.[14]

In culture [edit]

Scan of US dollar bill reverse with root 7 rectangle annotation

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.[15]

The square root of 7 has been featured in a humorous condom advertisement.[16]

See also [edit]

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

References [edit]

  1. ^ Darby, John (1843). The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, ... London: Whittaker & Company. p. 172. Retrieved 27 March 2022.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A010465 (Decimal expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Robert Nemiroff and Jerry Bonnell (2008). "The square root of 7". gutenberg.org . Retrieved 25 March 2022. {{cite web}}: CS1 maint: uses authors parameter (link)
  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. Retrieved 27 March 2022.
  6. ^ Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science... Russell, Shattuck & Company. pp. 212–213. Retrieved 27 March 2022.
  7. ^ Thompson, James (1808). The American Tutor's Guide: Being a Compendium of Arithmetic. In Six Parts. Albany: E. & E. Hosford. p. 122. Retrieved 27 March 2022.
  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. Retrieved 27 March 2022.
  9. ^ George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. Retrieved 27 March 2022. {{cite book}}: CS1 maint: uses authors parameter (link)
  10. ^ Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu . Retrieved 17 March 2022.
  11. ^ Jay Hambidge (1920) [1920]. Dynamic Symmetry: The Greek Vase (Reprint of original Yale University Press ed.). Whitefish, MT: Kessinger Publishing. pp. 19–29. ISBN0-7661-7679-7. Dynamic Symmetry root rectangles.
  12. ^ Matila Ghyka (1977). The Geometry of Art and Life . Courier Dover Publications. pp. 126–127. ISBN9780486235424.
  13. ^ Fletcher, Rachel (2013). Infinite Measure: Learning to Design in Geometric Harmony with Art, Architecture, and Nature. George F Thompson Publishing. ISBN978-1-938086-02-1.
  14. ^ Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN9781559530187 . Retrieved 26 March 2022.
  15. ^ McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN9780759611702 . Retrieved 26 March 2022.
  16. ^ copyranter (15 September 2013). "The Best Condom Ads From Around The World: Safe Sex Tour 2013 starts now!". BuzzFeed . Retrieved 27 March 2022. Interspersed between photos of a woman flagrante delicto are shots of unsexy thoughts (to a man). They include: gnomes, cockroaches, toenail fungus, the square root of 7, plumber's ass crack, etc. – It's the best condom I've seen in years.

sawyerlaure1962.blogspot.com

Source: https://en.wikipedia.org/wiki/Square_root_of_7

Post a Comment for "How to Write Sqrt 7 as a Continued Fraction"