Rationality Irrational 2.645751311064590590..._10 ${\displaystyle {\sqrt {7}}}$ ${\displaystyle 2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}$ 10.10100101010011111111..._2 2.A54FF53A5F1D..._16

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]

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

and in exponent form as:

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

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

2.64575131106459059050161575363926042571025918308245018036833.... [2]

which can be rounded up 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

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 thousandth". [9]

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

${\displaystyle [2;1,1,1,4,1,1,1,4,\ldots ]=2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+\dots }}}}}}}}}}.}$ (sequence in the OEIS)

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

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

Their numerators are 2, 3, 5, 8, 37, 45, 82, 127, 590, 717, 1307, 2024, 9403, 11427, 20830, 32257…(sequence in the OEIS) , and their denominators are 1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192,…(sequence in the OEIS).

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

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

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

${\displaystyle x^{2}-7y^{2}=1.}$

When ${\displaystyle {\sqrt {7}}}$ 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:

${\displaystyle x_{1}=3,\quad x_{2}={\frac {8}{3}}=2.66...,\quad x_{3}={\frac {127}{48}}=2.6458...,\quad x_{4}={\frac {32257}{12192}}=2.645751312...,\quad x_{5}={\frac {2081028097}{786554688}}=2.645751311064591...,\quad \dots }$

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 ${\displaystyle x^{2}-7}$. The Newton's method update, ${\displaystyle x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),}$ is equal to ${\displaystyle (x_{n}+7/x_{n})/2}$ when ${\displaystyle f(x)=x^{2}-7}$. The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).

## Geometry

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]

Due to the Pythagorean theorem and Legendre's three-square theorem, ${\displaystyle {\sqrt {7}}}$ 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). ${\displaystyle {\sqrt {15}}}$ is the next smallest such number. [15]

## Outside of mathematics

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. [16]

## References

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; Jerry Bonnell (2008). The square root of 7. Retrieved 25 March 2022 – via gutenberg.org.
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.
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. ISBN  0-7661-7679-7. Dynamic Symmetry root rectangles.
12. ^ Matila Ghyka (1977). . Courier Dover Publications. pp.  126–127. ISBN  978-0-486-23542-4.
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.
14. ^ Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN  978-1-55953-018-7. Retrieved 26 March 2022.
15. ^ Sloane, N. J. A. (ed.). "Sequence A005875". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
16. ^ McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN  978-0-7596-1170-2. Retrieved 26 March 2022.