Square root of 7

Square root of 7

Rationality	Irrational
Representations
Decimal	2.645751311064590590..._10
Algebraic form	${\displaystyle {\sqrt {7}}}$
Continued fraction	${\displaystyle 2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+\ddots }}}}}}}}}$

The rectangle that bounds an equilateral 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 an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.64575131106459059050161575363926042571025918308245018036833....^[1]

which can be rounded up to 2.646 to within about 99.99% accuracy (about 1 part in 10000).

More than a million decimal digits of the square root of seven have been published.^[2]

Rational approximations

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^[3] and 1852,^[4] 3 in 1835,^[5] 6 in 1808,^[6] and 7 in 1797.^[7] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".^[8]

Geometry

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.^[9][10][11]

The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.^[12]

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.^[13]

Outside of mathematics

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 ${\displaystyle {\sqrt {7}}}$, and a diagonal of 6.0 inches, to within measurement accuracy.^[14]

See also

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

References

1. Sloane, N. J. A. (ed.). "Sequence A010465 (Decimal expansion of square root of 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
2. Robert Nemiroff; Jerry Bonnell (2008). The square root of 7. Retrieved 25 March 2022 – via gutenberg.org.
3. Ewing, Alexander (1773). Institutes of Arithmetic: For the Use of Schools and Academies. Edinburgh: T. Caddell. p. 104.
4. 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.
5. Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science... Russell, Shattuck & Company. pp. 212–213. Retrieved 27 March 2022.
6. 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.
7. 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.
8. George Wentworth; David Eugene Smith; Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. Retrieved 27 March 2022.
9. 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.
10. Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 978-0-486-23542-4.
11. 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.
12. Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 978-1-55953-018-7. Retrieved 26 March 2022.
13. Sloane, N. J. A. (ed.). "Sequence A005875". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
14. McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN 978-0-7596-1170-2. Retrieved 26 March 2022.