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square root
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{{Redirect|Square roots|other uses|Square Roots (disambiguation)}}(File:Nuvola apps edu mathematics blue-p.svg|thumb|right|168px|The mathematical expression "The (principal) square root of {{mvar|x}}")thumb|right|168px|For example, {{math|{{sqrt|25}} {{=}} 5}}, since {{math|25 {{=}} 5â€¯⋅â€¯5}}, or {{math|52}} (5 squared).In mathematics, a square root of a number a is a number y such that {{nowrap|1=y2 = a}}; in other words, a number y whose square (the result of multiplying the number by itself, or yâ€¯⋅â€¯y) is a.Gel'fand, p. 120 {{webarchive|url=https://web.archive.org/web/20160902151740weblink |date=2016-09-02 }}
For example, 4 and âˆ’4 are square roots of 16 because {{nowrap|1=42 = (âˆ’4)2 = 16}}.
Every nonnegative real number a has a unique nonnegative square root, called the principal square root, which is denoted by {{sqrt|a}}, where âˆš is called the radical sign or radix. For example, the principal square root of 9 is 3, which is denoted by {{sqrt|9}} = 3, because {{nowrap|1=32 = 3â€¯⋅â€¯3 = 9}} and 3 is nonnegative. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this example 9.Every positive number a has two square roots: {{sqrt|a}}, which is positive, and âˆ’{{sqrt|a}}, which is negative. Together, these two roots are denoted as Â±{{sqrt|a}} (see Â± shorthand). Although the principal square root of a positive number is only one of its two square roots, the designation "the square root" is often used to refer to the principal square root. For positive a, the principal square root can also be written in exponent notation, as a1/2.BOOK, A First Course in Complex Analysis With Applications, 2nd, Dennis G., Zill, Patrick, Shanahan, Jones & Bartlett Learning, 2008, 0-7637-5772-1, 78,weblink no,weblink 2016-09-01, Extract of page 78 {{webarchive|url=https://web.archive.org/web/20160901091148weblink |date=2016-09-01 }}Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of "squaring" of some mathematical objects is defined (including algebras of matrices, endomorphism rings, etc.)

Properties and uses

Image:Square root 0 25.svg|thumb|400px|The graph of the function f(x) = {{sqrt|x}}, made up of half a parabola with a vertical directrix ]]The principal square root function f(x) = {{sqrt|x}} (usually just referred to as the "square root function") is a function that maps the set of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the area of a square to its side length.The square root of x is rational if and only if x is a rational number that can be represented as a ratio of two perfect squares. (See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.) The square root function maps rational numbers into algebraic numbers (a superset of the rational numbers).For all real numbers x,
sqrt{x^2} = left|xright| = begin{cases}
x, & mbox{if }x ge 0
-x, & mbox{if }x < 0.
end{cases}
(see absolute value)
For all nonnegative real numbers x and y,
sqrt{xy} = sqrt x sqrt y
and
sqrt x = x^{1/2}.
The square root function is continuous for all nonnegative x and differentiable for all positive x. If f denotes the square root function, its derivative is given by:
f'(x) = frac{1}{2sqrt x}.
The Taylor series of {{sqrt|1 + x}} about x = 0 converges for {{abs|x}} â‰¤ 1 and is given by
sqrt{1 + x} = sum_{n=0}^infty frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + frac{1}{2}x - frac{1}{8}x^2 + frac{1}{16} x^3 - frac{5}{128} x^4 + cdots,
The square root of a nonnegative number is used in the definition of Euclidean norm (and distance), as well as in generalizations such as Hilbert spaces. It defines an important concept of standard deviation used in probability theory and statistics. It has a major use in the formula for roots of a quadratic equation; quadratic fields and rings of quadratic integers, which are based on square roots, are important in algebra and have uses in geometry. Square roots frequently appear in mathematical formulas elsewhere, as well as in many physical laws.

Square roots of positive integers

As a positive number has two square roots, one positive, and one negative, which are opposite one to the other. So, when talking of the square root of a positive integer, this is the positive square root that is meant. The square roots of an integer are algebraic integers and, more specifically, quadratic integers.The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since {{math|1={{sqrt|p2k}} = pk}}, only roots of those primes having an odd power in the factorization are necessary. More precisely, the square root of a prime factorization is
sqrt{p_1^{2e_1+1}cdots p_k^{2e_k+1}p_{k+1}^{2e_{k+1}}dots p_n^{2e_n}}=p_1^{e_1}dots p_n^{e_n}sqrt{p_1dots p_k}.

As decimal expansions

The square roots of the perfect squares (0, 1, 4, 9, 16, etc.) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore their decimal representations are non-repeating decimals. Decimal approximations of the square roots of the first few natural numbers are given in the following table.
{|class="wikitable"
! {{mvar|n}} !! {{math|{{sqrt|n}}}}, truncated to 50 decimal places
0 0
1 1
2 Square root of 21.414213562316887242097187537694}}
3 Square root of 31.732050807574463415050525381038}}
4 2
5 Square root of 52.236067977491736687311835961152}}
6 {{gaps83178098198913919659|4748065667}}
7 {{gaps64590590502604257102|5918308245}}
8 {{gaps46190097603961571393|4375075389}}
9 3
10 {{gaps68379331997185337195|5513932521}}

As expansions in other numeral systems

The square roots of the perfect squares (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots of positive integers are irrational numbers, and therefore their representations in any standard positional notation system are non-repeating.The square roots of small integers are used in both the SHA-1 and SHA-2 hash function designs to provide nothing up my sleeve numbers.

As periodic continued fractions

One of the most intriguing results from the study of irrational numbers as continued fractions was obtained by Joseph Louis Lagrange {{circa}} 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic. That is, a certain pattern of partial denominators repeats indefinitely in the continued fraction. In a sense these square roots are the very simplest irrational numbers, because they can be represented with a simple repeating pattern of integers.
{|
{{sqrt| = [1; 2, 2, ...]
{{sqrt| = [1; 1, 2, 1, 2, ...]
{{sqrt| = [2]
{{sqrt| = [2; 4, 4, ...]
{{sqrt| = [2; 2, 4, 2, 4, ...]
{{sqrt| = [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
{{sqrt|= [2; 1, 4, 1, 4, ...]
{{sqrt| = [3]
{{sqrt| = [3; 6, 6, ...]
{{sqrt| = [3; 3, 6, 3, 6, ...]
{{sqrt| = [3; 2, 6, 2, 6, ...]
{{sqrt| = [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
{{sqrt| = [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
{{sqrt| = [3; 1, 6, 1, 6, ...]
{{sqrt| = [4]
{{sqrt| = [4; 8, 8, ...]
{{sqrt| = [4; 4, 8, 4, 8, ...]
{{sqrt| = [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
{{sqrt| = [4; 2, 8, 2, 8, ...]
The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in more traditional notation the simple continued fraction for the square root of 11, [3; 3, 6, 3, 6, ...], looks like this:
sqrt{11} = 3 + cfrac{1}{3 + cfrac{1}{6 + cfrac{1}{3 + cfrac{1}{6 + cfrac{1}{3 + ddots}}}}}where the two-digit pattern {3, 6} repeats over and over again in the partial denominators. Since {{nowrap|1=11 = 32 + 2}}, the above is also identical to the following generalized continued fractions:
sqrt{11} = 3 + cfrac{2}{6 + cfrac{2}{6 + cfrac{2}{6 + cfrac{2}{6 + cfrac{2}{6 + ddots}}}}} = 3 + cfrac{6}{20 - 1 - cfrac{1}{20 - cfrac{1}{20 - cfrac{1}{20 - cfrac{1}{20 - ddots}}}}}.

Computation

Square roots of positive numbers are not in general rational numbers, and so cannot be written as a terminating or recurring decimal expression. Therefore in general any attempt to compute a square root expressed in decimal form can only yield an approximation, though a sequence of increasingly accurate approximations can be obtained.Most pocket calculators have a square root key. Computer spreadsheets and other software are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as the Newton's method (frequently with an initial guess of 1), to compute the square root of a positive real number.BOOK, Parkhurst, David F., Introduction to Applied Mathematics for Environmental Science, 2006, Springer, 9780387342283, 241, BOOK, Solow, Anita E., Learning by Discovery: A Lab Manual for Calculus, 1993, Cambridge University Press, 9780883850831, 48, When computing square roots with logarithm tables or slide rules, one can exploit the identities
sqrt{a} = e^{(ln a)/2} = 10^{(log_{10} a)/2},
where {{math|ln}} and {{math|log}}10 are the natural and base-10 logarithms.By trial-and-error,BOOK, Mathematics for Biological Scientists, Mike, Aitken, Bill, Broadhurst, Stephen, Hladky, Garland Science, 2009, 978-1-136-84393-8, 41,weblink no,weblink 2017-03-01, Extract of page 41 {{webarchive|url=https://web.archive.org/web/20170301100516weblink |date=2017-03-01 }} one can square an estimate for {{sqrt|a}} and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it is prudent to use the identity
(x + c)^2 = x^2 + 2xc + c^2,
as it allows one to adjust the estimate x by some amount c and measure the square of the adjustment in terms of the original estimate and its square. Furthermore, (x + c)2 â‰ˆ x2 + 2xc when c is close to 0, because the tangent line to the graph of x2 + 2xc + c2 at c = 0, as a function of c alone, is y = 2xc + x2. Thus, small adjustments to x can be planned out by setting 2xc to a, or c = a/(2x).The most common iterative method of square root calculation by hand is known as the "Babylonian method" or "Heron's method" after the first-century Greek philosopher Heron of Alexandria, who first described it.BOOK
, Heath
, Sir Thomas L.
,
,
, A History of Greek Mathematics, Vol. 2
, Clarendon Press
, 1921
, Oxford
, 323â€“324
,
,
,
The method uses the same iterative scheme as the Newtonâ€“Raphson method yields when applied to the function y = f(x) = x2 âˆ’ a, using the fact that its slope at any point is dy/dx = {{prime|f}}(x) = 2x, but predates it by many centuries.BOOK
, Elementary functions: algorithms and implementation
, Jean-Mic
, Muller
, Springer
, 2006
, 0-8176-4372-9
, 92â€“93
,
, , Chapter 5, p 92 {{webarchive|url=https://web.archive.org/web/20160901091516weblink |date=2016-09-01 }}The algorithm is to repeat a simple calculation that results in a number closer to the actual square root each time it is repeated with its result as the new input. The motivation is that if x is an overestimate to the square root of a nonnegative real number a then a/x will be an underestimate and so the average of these two numbers is a better approximation than either of them. However, the inequality of arithmetic and geometric means shows this average is always an overestimate of the square root (as noted below), and so it can serve as a new overestimate with which to repeat the process, which converges as a consequence of the successive overestimates and underestimates being closer to each other after each iteration. To find x:
1. Start with an arbitrary positive start value x. The closer to the square root of a, the fewer the iterations that will be needed to achieve the desired precision.
2. Replace x by the average (x + a/x) / 2 between x and a/x.
3. Repeat from step 2, using this average as the new value of x.
That is, if an arbitrary guess for {{sqrt|a}} is x0, and {{nowrap|1 = xn + 1 = (xn + a/xn) / 2}}, then each xn is an approximation of {{sqrt|a}} which is better for large n than for small n. If a is positive, the convergence is quadratic, which means that in approaching the limit, the number of correct digits roughly doubles in each next iteration. If {{nowrap|1 =a = 0}}, the convergence is only linear.Using the identity
sqrt{a} = 2^{-n}sqrt{4^n a},
the computation of the square root of a positive number can be reduced to that of a number in the range {{closed-open|1,4}}. This simplifies finding a start value for the iterative method that is close to the square root, for which a polynomial or piecewise-linear approximation can be used.The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.Another useful method for calculating the square root is the shifting nth root algorithm, applied for {{nowrap|1= n = 2}}.The name of the square root function varies from programming language to programming language, with sqrtWEB, Function sqrt, CPlusPlus.com, 2016, The C++ Resources Network,weblink June 24, 2016, no,weblink" title="web.archive.org/web/20121122050619weblink">weblink November 22, 2012, (often pronounced "squirt" BOOK, C++ for the Impatient, Brian, Overland, 338, Addison-Wesley, 2013, 9780133257120, 850705706,weblink June 24, 2016, no,weblink September 1, 2016, ) being common, used in C, C++, and derived languages like JavaScript, PHP, and Python.

Square roots of negative and complex numbers

{{multiple image |align=left |direction=horizontal
|image1=Complex sqrt leaf1.jpg |caption1=First leaf of the complex square root
|image2=Complex sqrt leaf2.jpg |caption2=Second leaf of the complex square root
|image3=Riemann surface sqrt.svg |caption3=Using the Riemann surface of the square root, it is shown how the two leaves fit together
}}{{clear}}The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work with a more inclusive set of numbers, called the complex numbers, that does contain solutions to the square root of a negative number. This is done by introducing a new number, denoted by i (sometimes j, especially in the context of electricity where "i" traditionally represents electric current) and called the imaginary unit, which is defined such that {{nowrap|1=i2 = âˆ’1}}. Using this notation, we can think of i as the square root of âˆ’1, but we also have {{nowrap|1=(âˆ’i)2 = i2 = âˆ’1}} and so âˆ’i is also a square root of âˆ’1. By convention, the principal square root of âˆ’1 is i, or more generally, if x is any nonnegative number, then the principal square root of âˆ’x is
sqrt{-x} = i sqrt x.
The right side (as well as its negative) is indeed a square root of âˆ’x, since
(isqrt x)^2 = i^2(sqrt x)^2 = (-1)x = -x.
For every non-zero complex number z there exist precisely two numbers w such that {{nowrap|1=w2 = z}}: the principal square root of z (defined below), and its negative.

Principal square root of a complex number

{{Visualisation complex number roots}}To find a definition for the square root that allows us to consistently choose a single value, called the principal value, we start by observing that any complex number x + iy can be viewed as a point in the plane, (x, y), expressed using Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair (r, varphi), where r â‰¥ 0 is the distance of the point from the origin, and varphi is the angle that the line from the origin to the point makes with the positive real (x) axis. In complex analysis, the location of this point is conventionally written re^{ivarphi}. If
z=r e^{i varphi} text{ with } -pi < varphi le pi,
then we define the principal square root of z as follows:
sqrt{z} = sqrt{r} e^{i varphi / 2}.
The principal square root function is thus defined using the nonpositive real axis as a branch cut. The principal square root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it isn't even continuous). The above Taylor series for {{sqrt|1 + x}} remains valid for complex numbers x with {{nowrap|{{abs|x}} < 1}}.The above can also be expressed in terms of trigonometric functions:
sqrt{r left(cos varphi + i sin varphi right)} = sqrt{r} left ( cos frac{varphi}{2} + i sin frac{varphi}{2} right ) .

Algebraic formula

right|thumb|The square roots of {{mvar|i}}When the number is expressed using Cartesian coordinates the following formula can be used for the principal square root:BOOK
, Handbook of mathematical functions with formulas, graphs, and mathematical tables
,
, Milton
, Abramowitz
, Irene A.
, Stegun
, Courier Dover Publications
, 1964
, 0-486-61272-4
, 17
, no
, 2016-04-23
,
, , Section 3.7.27, p. 17 {{webarchive|url=https://web.archive.org/web/20090910094533weblink |date=2009-09-10 }}BOOK
, Classical algebra: its nature, origins, and uses
, Roger
, Cooke
, John Wiley and Sons
, 2008
, 0-470-25952-3
, 59
, no
, 2016-04-23
,
,
sqrt{x+iy} = sqrt{frac{sqrt{x^2+y^2} + x}{2}} pm isqrt{frac{sqrt{x^2+y^2} - x}{2}},
where the sign of the imaginary part of the root is taken to be the same as the sign of the imaginary part of the original number, or positive when zero. The real part of the principal value is always nonnegative.For example, the principal square roots of {{math|±i}} are given by:
begin{align}
sqrt{i} &= frac{1}{sqrt{2}} + ifrac{1}{sqrt{2}} = frac{sqrt{2}}{2}(1+i),
sqrt{-i} &= frac{1}{sqrt{2}} - ifrac{1}{sqrt{2}} = frac{sqrt{2}}{2}(1-i).
end{align}

Notes

In the following, the complex z and w may be expressed as:
• z=|z|e^{i theta_z}
• w=|w|e^{i theta_w}
where -pi

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