Square root of 7

Square root of 7
Rationality	Irrational
Representations
Decimal	2.645751311064590590..._10
Algebraic form	7
Continued fraction	2+11+11+11+14+⋱
Binary	10.10100101010011111111..._2
Hexadecimal	2.A54FF53A5F1D..._16

File:Rectangle that bounds equilateral triangle.png

The rectangle that bounds an equialateral triangle of side 2, or a regular hexagon of side 1, has a diagonal of square root of 7.

File:Slide rule with square roots of 6 and 7.jpg

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]

\sqrt{7},

and in exponent form as:

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

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

2.64575131106459059050161575363926042571025918308245018036833....^[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

File:1797 square root of 7.png

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, \dots] = 2 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{4 + \frac{1}{1 + \dots}}}}}

(sequence A010121 in the OEIS).

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…(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 $\sqrt{7}$ ; 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, \frac{3}{1} = 3.0, \frac{5}{2} = 2.5, \frac{8}{3} = 2.66 \dots, \frac{37}{14} = 2.6429 . . ., \frac{45}{17} = 2.64705 . . ., \frac{82}{31} = 2.64516 . . ., \frac{127}{48} = 2.645833 . . ., \dots

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

x^{2} - 7 y^{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, x_{2} = \frac{8}{3} = 2.66 . . ., x_{3} = \frac{127}{48} = 2.6458 . . ., x_{4} = \frac{32257}{12192} = 2.645751312 . . ., x_{5} = \frac{2081028097}{786554688} = 2.645751311064591 . . ., \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).

Geometry

File:Root rectangles up to 6.png

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]

References

↑ Darby, John (1843). The Practical Arithmetic, with Notes and Demonstrations to the Principal Rules, ... London: Whittaker & Company. p. 172. Retrieved 27 March 2022. Search this book on
↑ Sloane, N. J. A. (ed.). "Sequence A010465 (Decimal expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ Robert Nemiroff and Jerry Bonnell (2008). "The square root of 7". gutenberg.org. Retrieved 25 March 2022.CS1 maint: Uses authors parameter (link)
↑ Ewing, Alexander (1773). Institutes of Arithmetic: For the Use of Schools and Academies. Edinburgh: T. Caddell. p. 104. Search this book on
↑ 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. Search this book on
↑ Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science... Russell, Shattuck & Company. pp. 212–213. Retrieved 27 March 2022. Search this book on
↑ 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. Search this book on
↑ Hawney, William. 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. Search this book on
↑ George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. Retrieved 27 March 2022.CS1 maint: Uses authors parameter (link) Search this book on
↑ Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022.
↑ Jay Hambidge (1920) [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. Search this book on
↑ Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 9780486235424. Search this book on
↑ 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. Search this book on
↑ Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 9781559530187. Retrieved 26 March 2022. Search this book on

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

[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.

[million-3] Robert Nemiroff and Jerry Bonnell (2008). "The square root of 7". gutenberg.org. Retrieved 25 March 2022.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. Search this book on

[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. Search this book on

[bailey-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. Search this book on

[thompson-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. Search this book on

[hawney-8] Hawney, William. 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. Search this book on

[wentworth-9] George Wentworth, David Eugene Smith, Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. Retrieved 27 March 2022.CS1 maint: Uses authors parameter (link) Search this book on

[conrad-10] Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022.

[hambidge-11] Jay Hambidge (1920) [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. Search this book on

[ghyka-12] Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 9780486235424. Search this book on

[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. Search this book on

[blackwell-14] Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 9781559530187. Retrieved 26 March 2022. Search this book on

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v t e Irrational numbers
Chaitin's ( $Ω$ ) Liouville Prime ( $ρ$ ) Logarithm of 2 Gauss's ( $G$ ) Twelfth root of 2 Apéry's ( $ζ (3)$ ) Plastic ( $ρ$ ) Square root of 2 Super-golden ratio ( $ψ$ ) Erdős–Borwein ( $E$ ) Golden ratio ( $φ$ ) Square root of 3 Square root of 5 Silver ratio ( $δ S$ ) Euler's ( $e$ ) Pi ( $π$ )		File:Gold, square root of 2, and square root of 3 rectangles.png
Schizophrenic Transcendental Trigonometric

Square root of 7

Contents

Rational approximations

Geometry

See also

References

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