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Square root of 7
View on Wikipediafrom Wikipedia
| Rationality | Irrational |
|---|---|
| Representations | |
| Decimal | 2.645751311064590590..._10 |
| Algebraic form | |
| Continued fraction | |
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
[edit]
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
[edit]
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, 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). is the next smallest such number.[13]
Outside of mathematics
[edit]
On the reverse of the current US one-dollar bill, the "large inner box" has a length-to-width ratio of , and a diagonal of 6.0 inches, to within measurement accuracy.[14]
See also
[edit]References
[edit]- ^ 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; Jerry Bonnell (2008). The square root of 7. Retrieved 25 March 2022 – via gutenberg.org.
- ^ Ewing, Alexander (1773). Institutes of Arithmetic: For the Use of Schools and Academies. Edinburgh: T. Caddell. p. 104.
- ^ 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.
- ^ Bailey, Ebenezer (1835). First Lessons in Algebra, Being an Easy Introduction to that Science... Russell, Shattuck & Company. pp. 212–213. Retrieved 27 March 2022.
- ^ 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.
- ^ 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.
- ^ George Wentworth; David Eugene Smith; Herbert Druery Harper (1922). Fundamentals of Practical Mathematics. Ginn and Company. p. 113. Retrieved 27 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.
{{cite book}}: ISBN / Date incompatibility (help) - ^ Matila Ghyka (1977). The Geometry of Art and Life. Courier Dover Publications. pp. 126–127. ISBN 978-0-486-23542-4.
- ^ 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.
- ^ Blackwell, William (1984). Geometry in Architecture. Key Curriculum Press. p. 25. ISBN 978-1-55953-018-7. Retrieved 26 March 2022.
- ^ Sloane, N. J. A. (ed.). "Sequence A005875". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ McGrath, Ken (2002). The Secret Geometry of the Dollar. AuthorHouse. pp. 47–49. ISBN 978-0-7596-1170-2. Retrieved 26 March 2022.
Square root of 7
View on Grokipediafrom Grokipedia
The square root of 7, denoted , is the positive real number whose square equals the prime number 7; its approximate numerical value is 2.6457513111.[1] As 7 is not a perfect square, is an irrational number, meaning it cannot be expressed as a ratio of two integers and its decimal expansion is non-terminating and non-repeating.[2]
This irrationality can be proven by contradiction: assume where and are coprime positive integers; then , implying 7 divides and thus (by Euclid's lemma, since 7 is prime), so for some integer ; substituting yields , or , similarly implying 7 divides , contradicting the coprimality of and .[2]
The continued fraction expansion of is periodic with period 4: , providing efficient rational approximations such as 8/3 ≈ 2.666 (error < 0.03) and 37/14 ≈ 2.6429 (error < 0.003).[3] These convergents arise from the theory of quadratic irrationals, where the period length relates to solutions of the Pell equation ; the minimal solution is , .[4]
In algebraic number theory, generates the real quadratic field , whose ring of integers has unit group , where is the fundamental unit (whose coefficients satisfy ).[5] This structure connects to Diophantine approximations and class number computations in number fields.