User:Dicklyon/Square root of 4



The square root of 4 is the unique positive real number that, when multiplied by itself, gives 4. 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.

It is a rational and integer number, found through a process known as evolution. 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. The exact same value can be denoted by this repeating decimal:


 * 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. 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 ($Φ = 1⁄φ = φ − 1$), 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


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.



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.

Root rectangles


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. 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.

In trigonometry


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. 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 square root of 4 is often used in statistics when computing statistical deviation across the span of four samples.

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.