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1 Properties
2 Computation
3 Square roots of negative and complex numbers
4 Square roots of matrices and operators
5 Principal square roots of the positive integers
6 Geometric construction of the square root
7 History
8 See also
9 Notes
10 References
11 External links

In mathematics, a square root of a number x is a number r such that r² = x, or, in other words, a number r whose square (the result of multiplying the number by itself) is x.

Every non-negative real number x has a unique non-negative square root, called the principal square root, which is denoted with a radical sign as $\scriptstyle \sqrt{x}$ . The square root can also be written in exponent notation, as x^1/2. For example, the principal square root of 9 is 3, denoted $\scriptstyle \sqrt{9} = 3$ , because 3² = 3 × 3 = 9 and 3 is non-negative. The principal square root of a positive number, however, is only one of its two square roots.

Every positive number x has two square roots. One of them is $\scriptstyle \sqrt{x}$ , which is positive, and the other $\scriptstyle -\sqrt{x}$ , which is negative. Together, these two roots are denoted $\scriptstyle \pm\sqrt{x}$ . 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).

Square roots of integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers. For example, $\scriptstyle \sqrt{2}$ cannot be written exactly as m/n, where n and m are integers. Nonetheless, it is exactly the length of the diagonal of a square with side length 1. This has been known since ancient times, with the discovery that $\scriptstyle \sqrt{2}$ is irrational attributed to Hippasus, a disciple of Pythagoras.

The term whose root is being considered is known as the radicand. In the expression $\scriptstyle \sqrt[n]{ab+2}$ , ab + 2 is the radicand. The radicand is the number or expression underneath the radical sign.

Properties

The graph of the function $\scriptstyle f(x) = \sqrt{x}$ , made up of half a parabola with a vertical directrix.

The principal square root function $\scriptstyle f(x) = \sqrt{x}$ (usually just referred to as the "square root function") is a function which maps the set of non-negative real numbers onto itself, and, like all functions, always returns a unique value. 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 which 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

$\sqrt{x^2} = \left|x\right| = \begin{cases} x, & \mbox{if }x \ge 0 \\ -x, & \mbox{if }x < 0. \end{cases}$ (see absolute value)

For all non-negative 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 non-negative x and differentiable for all positive x. Its derivative is

$f'(x) = \frac{1}{2\sqrt x}.$

The Taylor series of √1 + x about x = 0 converges for | x | < 1 and is given by

$\sqrt{1 + x} = \sum_{n=0}^\infty \frac{(-1)^n(2n)!}{(1-2n)(n!)^2(4^n)}x^n = 1 + \textstyle \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16} x^3 - \frac{5}{128} x^4 + \dots\!$

Computation

Main article: Methods of computing square roots

Most pocket calculators have a square root key. Computer spreadsheets and other software are also frequently used to calculate square roots. Computer software programs typically implement good routines to compute the exponential function and the natural logarithm or logarithm, and then compute the square root of x using the identity

$\sqrt{x} = e^{(\ln x)/2}$ or $\sqrt{x} = 10^{(\log x)/2}.$

The same identity is exploited when computing square roots with logarithm tables or slide rules.

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.[1] It involves a simple algorithm, which results in a number closer to the actual square root each time it is repeated. To find r, the square root of a real number x:

Start with an arbitrary positive start value r (the closer to the square root of x, the better).
Replace r by the average between r and x/r, that is: $\scriptstyle (r + x/r) / 2\,$ (It is sufficient to take an approximate value of the average in order to ensure convergence.)
Repeat step 2 until r and x/r are as close as desired.

The time complexity for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

Square roots of negative and complex numbers

Complex square root

Second leaf of the complex square root

Using the Riemann surface of the square root, one can see how the two leaves fit together

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 i² = −1. Using this notation, we can think of i as the square root of −1, but notice that we also have (−i)² = i² = −1 and so −i is also a square root of −1. By convention, the principal square root of −1 is i, or more generally, if x is any positive number, then the principal square root of −x is

$\sqrt{-x} = i \sqrt x.$

The right side (as well as its negative) is indeed a square root of −x, since

$(i\sqrt x)^2 = i^2(\sqrt x)^2 = (-1)x = -x.$

For every non-zero complex number z there exist precisely two numbers w such that w² = z: the principal square root of z (defined below), and its negative.

Imaginary square root

The square root of $\scriptstyle i \,$ is given by

$\sqrt{i} = \frac{1}{\sqrt{2}}(1+i).$

This result can be obtained algebraically by finding a and b such that

$i = (a+bi)^2\,\!$

or equivalently

$i = a^2 + 2abi - b^2.\,\!$

This gives the two equations

$2ab = 1\,\!$ $a^2 - b^2 = 0,\,\!$

which are easily solved to

$a = b = \pm \frac{1}{\sqrt{2}}.$

The choice of the principal root then gives

$a = b = \frac{1}{\sqrt{2}}.$

The result can also be obtained by using De Moivre's theorem and setting

$i = \cos\left (\frac{\pi}{2}\right ) + i\sin\left (\frac{\pi}{2}\right )$

which produces

$\begin{align} \sqrt{i} & = \left ( \cos\left ( \frac{\pi}{2} \right ) + i\sin \left (\frac{\pi}{2} \right ) \right )^{\frac{1}{2}} \\ & = \cos\left (\frac{\pi}{4} \right ) + i\sin\left ( \frac{\pi}{4} \right ) \\ & = \frac{1}{\sqrt{2}} + i\left ( \frac{1}{\sqrt{2}} \right ) = \frac{1}{\sqrt{2}}(1+i) . \\ \end{align}$

Principal square root of a complex number

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, Φ), where r ≥ 0 is the distance of the point from the origin, and Φ is the angle that the line from the origin to the point makes with the positive real (x) axis. In complex analysis, this value is conventionally written r e^iΦ. If

$z=r e^{\phi i}\,$ with $-\pi < \phi \le \pi \,$

then we define the principal square root of z as follows:

$\sqrt{z} = \sqrt{r} \, e^{i \phi / 2}.$

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 (where it isn't even continuous). The above Taylor series for √1 + x remains valid for complex numbers x with |x| < 1.

Formula

When the number is in rectangular form the following formula can be used for the principal value:

$\sqrt{x+iy} = \sqrt{\frac{r + x}{2}} + i \frac{y}{\sqrt{2 (r + x)}}$

where

$r = |x + iy| = \sqrt{x^2+ y^2}$

is the absolute value or modulus of the complex number, unless x = −r and y = 0. Notice that the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number. The real part of the principal value is always non-negative.

Notes

Note that because of the discontinuous nature of the square root function in the complex plane, the law √zw = √z√w is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √z√w at all, leading to the equality's failure. A similar problem appears with the complex logarithm and the relation log z + log w = log(zw).) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that −1 = 1:

$-1 = i \cdot i = \sqrt{-1} \cdot \sqrt{-1} = \sqrt{-1 \cdot -1} = \sqrt{1} = 1$

The third equality cannot be justified (see invalid proof). It can be made to hold by changing the meaning of √ so that this no longer represents the principal square root (see above) but selects a branch for the square root that contains (√−1)·(√−1). The left hand side becomes either

$\sqrt{-1} \cdot \sqrt{-1}=i \cdot i=-1$

if the branch includes +i or

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