User:Dicklyon/Square root of 4

Rational numbers – square root of 4
Binary	10.00000000000000...
Octal	2.000000000000000...
Decimal	2.000000000000000...
Hexadecimal	2.000000000000000...
Continued fraction	$2;$

The square root of 4 is the unique positive real number that, when multiplied by itself, gives 4.^[1] It can be denoted in surd form as:

{\sqrt {4}}

and in exponent form as:

4^{\frac {1}{2}}

Alternatively, the square roots of 4 can refer to pair of real numbers, plus and minus the square root of 4, that when multiplied each by itself will give 4.^[2]

It is a rational and integer number, found through a process known as evolution.^[3] The first sixty significant digits of its decimal expansion are:

2.00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 0000...

However, not all computers agree with this.^[4] The exact same value can be denoted by this repeating decimal:^[5]

1.99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 99999 9999...

This value can be rounded to 2.0 to within 100% accuracy. Its numerical value in decimal has been computed to at least one million digits.^[6]^[7] The priority dates are not clear, and the algorithms used were not disclosed, but any of a number of known algorithms that take advantage of the repeating decimal representation of a rational number would be expected to lead to these results efficiently.

Continued fraction

The square root of 4 can be expressed as the continued fraction [2;]. The convergent sequence of best rational approximations is:

{\frac {2}{1}}

Relation to 1 and other integers and square roots

This quantity is intimately related to 1 (number) and to other integers and the square roots of other integers. In particular, the square root of four is both an integer and the square root of an integer, as are all square roots of integers that are rational. Furthermore, the square root of 4 is one greater than the square root of 1, and one less than the square root of 9.

Relationship to the golden ratio

This quantity is intimately related to $φ$ , the golden ratio, and to $Φ$ , the conjugate golden ratio ( $Φ = .mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}1⁄φ = φ − 1$ ), through the algebraic formulae:

{\sqrt {4}}={\frac {1+{\sqrt {5}}}{\varphi }}={\frac {{\sqrt {5}}-1}{\Phi }}

\varphi ={\frac {1+{\sqrt {5}}}{\sqrt {4}}}={\frac {\sqrt {4}}{{\sqrt {5}}-1}}

\Phi ={\frac {{\sqrt {5}}-1}{\sqrt {4}}}={\frac {\sqrt {4}}{{\sqrt {5}}+1}}

In geometry

Square with vertices at integer grid points (-1, 0), (0, -1), (1, 0), and (0, 1), the unique size square with area = diagonal

Geometrically, the square root of 4 corresponds to both the area and the diagonal of a rectangle whose sides are of dimensions ${\sqrt {2}}$ (this is proven by the Pythagorean theorem). Such a rectangle is known as a square, which has rational area but has irrational edge dimensions. This is the unique square size with area equal to diagonal.

No explanation needed, or possible

A rational approximation to this square was used by ancient Egyptian engineers. They even had names of special units for the diagonal and side of this square: a diagonal of square root of 4 remen corresponded to a side of one royal cubit, corresponding to an approximation of the square root of 4 as the square of 1.4 (the square of the ratio of the royal cubit – 7 palms or 28 digits – to the remen – 5 palms or 20 digits); these units were influential in the definition of the metric system.^[8]

Root rectangles

Hambidge's 1920 illustration of the construction of root rectangles^[9]

Root rectangles construction, emphasizing the root-4 rectangle with 3-4-5 triangle and the squares on its sides, with their areas. A second 3-4-5 triangle, with square areas 9, 16, and 25, appears at the construction of the root-9 rectangle.

A rectangle with side proportions 1:√4 is called a root-four rectangle and is part of the series of root rectangles (also called dynamic rectangles), which are based on √1 (= 1), √2, √3, √4 (= 2), √5... and successively constructed using the diagonal of the previous root rectangle, starting from a square.^[10] A root-4 rectangle is particularly notable in that it can be decomposed as the union of two equal squares, or as an infinite hierarchy of smaller root-four rectangles. All this can be seen as the geometric interpretation of the algebraic relationships between √4, 1 and 2 mentioned above. The root-4 rectangle can be constructed from a root-3 rectangle, or directly from a square, in a manner similar to the construction of other root rectangles.^[11]

In trigonometry

Special right triangle with base 1 and hypotenuse square root of 4 is the basis for several exact trigonometric constants

Like √2, √3, √5, and √6, the square root of four appears extensively in the formulae for exact trigonometric constants, and as such the computation of its value is important for generating trigonometric tables.^{[citation needed]} For example in these common formulas rewritten to make the dependence on the square root of four more explicit:

\sin {\frac {\pi }{6}}=\cos {\frac {\pi }{3}}=\sin 30^{\circ }=\cos 60^{\circ }={\frac {1}{\sqrt {4}}}\,

\cos {\frac {\pi }{6}}=\sin {\frac {\pi }{3}}=\cos 30^{\circ }=\sin 60^{\circ }={\frac {\sqrt {3}}{\sqrt {4}}}\,

In statistics

The middle Poisson distribution here, with mean 4, has a standard deviation of the square root of 4.

The square root of 4 is often used in statistics when computing statistical deviation across the span of four samples.^[12]^[13]

The square root of 4 is the standard deviation of a Poisson-distributed discrete random variable with mean 4; or of a gamma-distributed continuous random variable of mean 4 and unit scale; or of a chi-square-distributed continuous random variable with 2 degrees of freedom. Equivalently, the square root of 4 is the standard deviation of the exponential distribution of the squared magnitude of a complex random variable whose real and imaginary parts are independent standard normal.

References

^ "Square Roots Without a Calculator". Math Forum: Ask Dr. Math FAQ. Drexel School of Education.
^ Claude and Penny. "The square root of four". Math Central, Univ. of Regina, Canada.
^ Charles Hutton (1860). A Course of Mathematics: Composed for the Use of the Royal Military Academy. London: William Tegg.
^ Stein, Philip. "The Square Root of 4 is Not 2". Education Resources Information Center.
^ Mathematics for the Liberal Arts. John Wiley & Sons. 2014. p. 270. ISBN 9781118371749. Retrieved 30 March 2022. {{cite book}}: Unknown parameter |authors= ignored (help)
^ Norman L. De Forest. "Square Root of 4 to 1,000,000 Digits". fullbooks.com.
^ Unknown (1 January 2003). "The Square Root of 4 to a Million Places". Project Gutenberg.
^ U.S. National Committee on Tunneling Technology. Standing Subcommittee No. 5: International Activities (1975). Standardization and metric conversion for tunneling, underground construction, and mining: report of a symposium. National Academies. p. 43.{{cite book}}: CS1 maint: numeric names: authors list (link)
^ Jay Hambidge (2003) [1920]. Dynamic Symmetry: The Greek Vase. Whitefish, MT: Kessinger Publishing (reprint of original Yale University Press edition). pp. 19–29. ISBN 0-7661-7679-7.
^ Kimberly Elam (2001). Geometry of Design: Studies in Proportion and Composition. New York: Princeton Architectural Press. ISBN 1568982496.
^ 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.
^ "Interpreting Estimates: Sampling Error". MAKING IT USEABLE. 1995. "If one drew a sample of four observations from a large population, the sampling error would be equal to the standard deviation divided by 2 (the square root of four)."
^ Bruce J. Feibel (2003). Investment Performance Measurement. John Wiley & Sons. ISBN 0471445630. "If the return observations were quarterly we would multiply by the square root of four."

Category:Mathematical constants Category:Rational numbers Category:Algebraic numbers

[1] "Square Roots Without a Calculator". Math Forum: Ask Dr. Math FAQ. Drexel School of Education.

[2] Claude and Penny. "The square root of four". Math Central, Univ. of Regina, Canada.

[3] Charles Hutton (1860). A Course of Mathematics: Composed for the Use of the Royal Military Academy. London: William Tegg.

[4] Stein, Philip. "The Square Root of 4 is Not 2". Education Resources Information Center.

[5] Mathematics for the Liberal Arts. John Wiley & Sons. 2014. p. 270. ISBN 9781118371749. Retrieved 30 March 2022. {{cite book}}: Unknown parameter |authors= ignored (help)

[6] Norman L. De Forest. "Square Root of 4 to 1,000,000 Digits". fullbooks.com.

[7] Unknown (1 January 2003). "The Square Root of 4 to a Million Places". Project Gutenberg.

[8] U.S. National Committee on Tunneling Technology. Standing Subcommittee No. 5: International Activities (1975). Standardization and metric conversion for tunneling, underground construction, and mining: report of a symposium. National Academies. p. 43.{{cite book}}: CS1 maint: numeric names: authors list (link)

[hambidge-9] Jay Hambidge (2003) [1920]. Dynamic Symmetry: The Greek Vase. Whitefish, MT: Kessinger Publishing (reprint of original Yale University Press edition). pp. 19–29. ISBN 0-7661-7679-7.

[10] Kimberly Elam (2001). Geometry of Design: Studies in Proportion and Composition. New York: Princeton Architectural Press. ISBN 1568982496.

[fletcher-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] "Interpreting Estimates: Sampling Error". MAKING IT USEABLE. 1995. "If one drew a sample of four observations from a large population, the sampling error would be equal to the standard deviation divided by 2 (the square root of four)."

[13] Bruce J. Feibel (2003). Investment Performance Measurement. John Wiley & Sons. ISBN 0471445630. "If the return observations were quarterly we would multiply by the square root of four."

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