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# Square root

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 Title: Square root Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Square root

In mathematics, a square root of a number a is a number y such that y2 = a, in other words, a number y whose square (the result of multiplying the number by itself, or y × y) is a. For example, 4 and −4 are square roots of 16 because 42 = (−4)2 = 16.

Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by a, where √ is called the radical sign or radix. For example, the principal square root of 9 is 3, denoted 9 = 3, because 32 = 3 × 3 = 9 and 3 is non-negative. The term whose 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: a, which is positive, and −a, which is negative. Together, these two roots are denoted ± a (see ± 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.

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.)

## Contents

• History 1
• Properties and uses 2
• Computation 3
• Square roots of negative and complex numbers 4
• Square root of an imaginary number 4.1
• Principal square root of a complex number 4.2
• Algebraic formula 4.3
• Notes 4.4
• Square roots of matrices and operators 5
• In integral domains, including fields 6
• In rings in general 7
• Principal square roots of the positive integers 8
• As decimal expansions 8.1
• As expansions in other numeral systems 8.2
• As periodic continued fractions 8.3
• Geometric construction of the square root 9
• See also 10
• Notes 11
• References 12
• External links 13

## History

The Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 BC, showing 2 and 302 as 1;24,51,10 and 42;25,35 base 60 numbers on a square crossed by two diagonals.

The Rhind Mathematical Papyrus is a copy from 1650 BC of an even earlier work and shows how the Egyptians extracted square roots.

In Ancient India, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras, dated around 800–500 BC (possibly much earlier). A method for finding very good approximations to the square roots of 2 and 3 are given in the Baudhayana Sulba Sutra. Aryabhata in the Aryabhatiya (section 2.4), has given a method for finding the square root of numbers having many digits.

It was known to the ancient Greeks that square roots of positive whole numbers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is to say they cannot be written exactly as m/n, where m and n are integers). This is the theorem Euclid X, 9 almost certainly due to Theaetetus dating back to circa 380 BC. The particular case 2 is assumed to date back earlier to the Pythagoreans and is traditionally attributed to Hippasus. It is exactly the length of the diagonal of a square with side length 1.

In the Chinese mathematical work Writings on Reckoning, written between 202 BC and 186 BC during the early Han Dynasty, the square root is approximated by using an "excess and deficiency" method, which says to "...combine the excess and deficiency as the divisor; (taking) the deficiency numerator multiplied by the excess denominator and the excess numerator times the deficiency denominator, combine them as the dividend."

Mahāvīra, a 9th-century Indian mathematician, was the first to state that square roots of negative numbers do not exist.

A symbol for square roots, written as an elaborate R, was invented by Regiomontanus (1436–1476). An R was also used for Radix to indicate square roots in Giralamo Cardano's Ars Magna.

According to historian of mathematics D.E. Smith, Aryabhata's method for finding the square root was first introduced in Europe by Cataneo in 1546.

The symbol '√' for the square root was first used in print in 1525 in Christoph Rudolff's Coss, which was also the first to use the then-new signs '+' and '−'.

## Properties and uses

The principal square root function f(x) = x (usually just referred to as the "square root function") is a function that maps the set of non-negative 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|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. If f denotes the square-root function, its derivative is given by:

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,\!

Square root of a non-negative 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.

## Computation

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. When computing square roots with logarithm tables or slide rules, one can exploit the identity

\sqrt{a} = e^{(\ln a)/2} or \sqrt{a} = 10^{(\log_{10} a)/2}.

where \ln and \log_{10} are the natural and base-10 logarithms.

By trial-and-error, one can square an estimate for a and raise or lower the estimate until it agrees to sufficient accuracy. For this technique it's 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 \approx x^2+2xc when c is close to 0, because the tangent line to the graph of x^2+2xc+c^2 at c=0, as a function of c alone, is y=2xc+x^2. Thus, small adjustments to x can be planned out by setting 2xc to a, or c=\frac{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. The method uses the same iterative scheme as the Newton–Raphson method yields when applied to the function y = f(x)=x2a, using the fact that its slope at any point is dy/dx=f'(x)=2x, but predates it by many centuries. 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 non-negative 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 a is x_0, and xn+1 = (xn + a/xn)/2, then each xn is an approximation of 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 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 [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 n = 2.

## Square roots of negative and complex numbers

Using the Riemann surface of the square root, it is shown 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 non-negative 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.

### Square root of an imaginary number

The square root of i is given by

\sqrt{i} = \frac{1}{2}\sqrt{2} + i\frac{1}{2}\sqrt{2} = \frac{\sqrt{2}}{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 simultaneous equations

\begin{cases} 2ab = 1\! \\ a^2 - b^2 = 0\! \end{cases}

with solutions

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 formula 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 re. If

z=r e^{\varphi i} \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 1 + x remains valid for complex numbers x with |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

When the number is expressed using Cartesian coordinates the following formula can be used for the principal square root:

\sqrt{z} = \sqrt{\frac{|z| + \operatorname{Re}(z)}{2}} + i\ \sgn(\operatorname{Im}(z))\ \sqrt{\frac{|z| - \operatorname{Re}(z)}{2}}

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. The real part of the principal value is always non-negative.

As the other square root is simply −1 times the principal square root, both roots can be written as

\pm\left(\sqrt{\frac{|z| + \operatorname{Re}(z)}{2}} + i\ \sgn(\operatorname{Im}(z))\ \sqrt{\frac{|z| - \operatorname{Re}(z)}{2}}\right)

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