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In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every
nonnegative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or i ...
real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. 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 case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sqrt. 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. 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 the " square" of a mathematical object is defined. These include function spaces and square matrices, among other
mathematical structure In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
s.


History

The Yale Babylonian Collection YBC 7289 clay tablet was created between 1800 BC and 1600 BC, showing \sqrt and \frac = \frac respectively as 1;24,51,10 and 0;42,25,35
base 60 Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form� ...
numbers on a square crossed by two diagonals. (1;24,51,10) base 60 corresponds to 1.41421296, which is a correct value to 5 decimal points (1.41421356...). The Rhind Mathematical Papyrus is a copy from 1650 BC of an earlier
Berlin Papyrus Berlin Papyrus may refer to several papyri kept in the Egyptian Museum of Berlin, including: * Berlin Papyrus 3033 or the Westcar Papyrus, a storytelling papyrus * Berlin Papyrus 3038 or the Brugsch Papyrus, a medical papyrus * Berlin Papyrus 6619 ...
and other textspossibly the Kahun Papyrusthat shows how the Egyptians extracted square roots by an inverse proportion method. In
Ancient India According to consensus in modern genetics, anatomically modern humans first arrived on the Indian subcontinent from Africa between 73,000 and 55,000 years ago. Quote: "Y-Chromosome and Mt-DNA data support the colonization of South Asia by m ...
, the knowledge of theoretical and applied aspects of square and square root was at least as old as the ''
Sulba Sutras The ''Shulva Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction. Purpose and origins The ...
'', 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 The (Sanskrit: बौधायन) are a group of Vedic Sanskrit texts which cover dharma, daily ritual, mathematics and is one of the oldest Dharma-related texts of Hinduism that have survived into the modern age from the 1st-millennium BCE. Th ...
''.
Aryabhata Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (which ...
, in the ''
Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the ''magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that th ...
'' (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 integers that are not perfect squares are always
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s: numbers not expressible as a
ratio In mathematics, a ratio shows how many times one number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lan ...
of two integers (that is, they cannot be written exactly as \frac, 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 of the square root of 2 is assumed to date back earlier to the Pythagoreans, and is traditionally attributed to
Hippasus Hippasus of Metapontum (; grc-gre, Ἵππασος ὁ Μεταποντῖνος, ''Híppasos''; c. 530 – c. 450 BC) was a Greek philosopher and early follower of Pythagoras. Little is known about his life or his beliefs, but he is sometimes ...
. 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." 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 Gerolamo 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. According to Jeffrey A. Oaks, Arabs used the letter '' jīm/ĝīm'' (), the first letter of the word "" (variously transliterated as ''jaḏr'', ''jiḏr'', ''ǧaḏr'' or ''ǧiḏr'', "root"), placed in its initial form () over a number to indicate its square root. The letter ''jīm'' resembles the present square root shape. Its usage goes as far as the end of the twelfth century in the works of the Moroccan mathematician Ibn al-Yasamin. The symbol "√" for the square root was first used in print in 1525, in Christoph Rudolff's ''Coss''.


Properties and uses

The principal square root function f(x) = \sqrt (usually just referred to as the "square root function") is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
that maps the
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of nonnegative real numbers onto itself. In geometrical terms, the square root function maps the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
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 number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
s, the latter being a
superset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of the rational numbers). For all real numbers ''x'', : \sqrt = \left, x\ = \begin x, & \mboxx \ge 0 \\ -x, & \mboxx < 0. \end     (see
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
) For all nonnegative real numbers ''x'' and ''y'', :\sqrt = \sqrt x \sqrt y and :\sqrt x = x^. The square root function is continuous for all nonnegative ''x'', and differentiable for all positive ''x''. If ''f'' denotes the square root function, whose derivative is given by: :f'(x) = \frac. The Taylor series of \sqrt about ''x'' = 0 converges for ≤ 1, and is given by :\sqrt = \sum_^\infty \fracx^n = 1 + \fracx - \fracx^2 + \frac x^3 - \frac x^4 + \cdots, The square root of a nonnegative number is used in the definition of Euclidean norm (and
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
), as well as in generalizations such as Hilbert spaces. It defines an important concept of
standard deviation In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
used in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
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

A positive number has two square roots, one positive, and one negative, which are opposite to each other. When talking of ''the'' square root of a positive integer, it is usually the positive square root that is meant. The square roots of an integer are
algebraic integer In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients ...
s—more specifically quadratic integers. The square root of a positive integer is the product of the roots of its
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
factors, because the square root of a product is the product of the square roots of the factors. Since \sqrt = p^k, 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^\dots p_n^\sqrt.


As decimal expansions

The square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s, and hence have non-
repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if a ...
s in their
decimal representation A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\ldots b_0.a_1a_2\ldots Here is the decimal separator, i ...
s. Decimal approximations of the square roots of the first few natural numbers are given in the following table. :


As expansions in other numeral systems

As with before, the square roots of the perfect squares (e.g., 0, 1, 4, 9, 16) are integers. In all other cases, the square roots of positive integers are
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s, and therefore have non-repeating digits in any standard
positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...
system. 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 number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s as
continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s was obtained by
Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaperiodic. 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. : The square bracket notation used above is a short form for a continued fraction. Written in the more suggestive algebraic form, the simple continued fraction for the square root of 11, ; 3, 6, 3, 6, ... looks like this: : \sqrt = 3 + \cfrac where the two-digit pattern repeats over and over again in the partial denominators. Since , the above is also identical to the following generalized continued fractions: : \sqrt = 3 + \cfrac = 3 + \cfrac.


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 calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-size ...
s 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 identities :\sqrt = e^ = 10^, where and 10 are the natural and base-10 logarithms. By trial-and-error, one can square an estimate for \sqrt 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 ≈ ''x''2 + 2''xc'' when ''c'' is close to 0, because the tangent line to the graph of ''x''2 + 2''xc'' + ''c''2 at ''c'' = 0, as a function of ''c'' alone, is ''y'' = 2''xc'' + ''x''2. Thus, small adjustments to ''x'' can be planned out by setting 2''xc'' to ''a'', or ''c'' = ''a''/(2''x''). The most common
iterative method In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pre ...
of square root calculation by hand is known as the "
Babylonian method Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted \sqrt, \sqrt /math>, or S^) of a real number. Arithmetically, it means given S, a procedure for fin ...
" 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'') = ''x''2 − ''a'', using the fact that its slope at any point is ''dy''/''dx'' = '(''x'') = 2''x'', 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 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 Below may refer to: *Earth * Ground (disambiguation) *Soil *Floor * Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Ernst von Below (1863–1955), German World War I general *Fred 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'': # 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. # Replace ''x'' by the average (''x'' + ''a''/''x'') / 2 between ''x'' and ''a''/''x''. # Repeat from step 2, using this average as the new value of ''x''. That is, if an arbitrary guess for \sqrt is ''x''0, and , then each xn is an approximation of \sqrt 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 , the convergence is only linear. Using the identity :\sqrt = 2^\sqrt, the computation of the square root of a positive number can be reduced to that of a number in the range . 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 An approximation is anything that is intentionally similar but not exactly equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix '' ...
can be used. The
time complexity In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
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 . The name of the square root
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
varies from programming language to programming language, with sqrt (often pronounced "squirt" ) being common, used in C,
C++ C, or c, is the third letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''cee'' (pronounced ), plural ''cees''. History "C" ...
, and derived languages like JavaScript, PHP, and
Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (pr ...
.


Square roots of negative and complex numbers

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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
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 written as ''j'', especially in the context of
electricity Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described ...
where "''i''" traditionally represents electric current) and called the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and ...
, which is ''defined'' such that . Using this notation, we can think of ''i'' as the square root of −1, but we also have 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 nonnegative number, then the principal square root of −''x'' is :\sqrt = 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 : the principal square root of ''z'' (defined below), and its negative.


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 + i y can be viewed as a point in the plane, (x, y), expressed using
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
. The same point may be reinterpreted using
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
as the pair (r, \varphi), where r \geq 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 r e^. If z = r e^ \text -\pi < \varphi \leq \pi, then the of z is defined to be the following: \sqrt = \sqrt e^. The principal square root function is thus defined using the nonpositive real axis as a
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, ...
. If z is a non-negative real number (which happens if and only if \varphi = 0) then the principal square root of z is \sqrt e^ = \sqrt; in other words, the principal square root of a non-negative real number is just the usual non-negative square root. It is important that -\pi < \varphi \leq \pi because if, for example, z = - 2 i (so \varphi = -\pi/2) then the principal square root is \sqrt = \sqrt = \sqrt e^ = \sqrt e^ = 1 - i but using \tilde := \varphi + 2 \pi = 3\pi/2 would instead produce the other square root \sqrt e^ = \sqrt e^ = -1 + i = - \sqrt. The principal square root function is
holomorphic In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above Taylor series for \sqrt remains valid for complex numbers x with , x, < 1. The above can also be expressed in terms of trigonometric functions: \sqrt = \sqrt \left(\cos \frac + i \sin \frac \right).


Algebraic formula

When the number is expressed using its real and imaginary parts, the following formula can be used for the principal square root: :\sqrt = \sqrt +i\sgn(y) \sqrt, where is the sign of (except that, here, sgn(0) = 1). In particular, the imaginary parts of the original number and the principal value of its square root have the same sign. The real part of the principal value of the square root is always nonnegative. For example, the principal square roots of are given by: :\begin \sqrt &= \frac + i\frac = \frac(1+i),\\ \sqrt &= \frac - i\frac = \frac(1-i). \end


Notes

In the following, the complex ''z'' and ''w'' may be expressed as: * z=, z, e^ * w=, w, e^ where -\pi<\theta_z\le\pi and -\pi<\theta_w\le\pi. Because of the discontinuous nature of the square root function in the complex plane, the following laws are not true in general. * \sqrt = \sqrt \sqrt
Counterexample for the principal square root: and
This equality is valid only when -\pi<\theta_z+\theta_w\le\pi * \frac = \sqrt
Counterexample for the principal square root: and
This equality is valid only when -\pi<\theta_w-\theta_z\le\pi *\sqrt = \left( \sqrt z \right)^*
Counterexample for the principal square root: )
This equality is valid only when \theta_z\ne\pi A similar problem appears with other complex functions with branch cuts, e.g., the
complex logarithm In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: * A complex logarithm of a nonzero complex number z, defined to ...
and the relations or which are not true in general. Wrongly assuming one of these laws underlies several faulty "proofs", for instance the following one showing that : : \begin -1 &= i \cdot i \\ &= \sqrt \cdot \sqrt \\ &= \sqrt \\ &= \sqrt \\ &= 1. \end The third equality cannot be justified (see
invalid proof In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a 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 \sqrt\cdot\sqrt. The left-hand side becomes either :\sqrt \cdot \sqrt=i \cdot i=-1 if the branch includes +''i'' or :\sqrt \cdot \sqrt=(-i) \cdot (-i)=-1 if the branch includes −''i'', while the right-hand side becomes :\sqrt=\sqrt=-1, where the last equality, \sqrt = -1, is a consequence of the choice of branch in the redefinition of √.


''N''th roots and polynomial roots

The definition of a square root of x as a number y such that y^2 = x has been generalized in the following way. A
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
of x is a number y such that y^3 = x; it is denoted \sqrt . If is an integer greater than two, a th root of x is a number y such that y^n = x; it is denoted \sqrt . Given any polynomial , a root of is a number such that . For example, the th roots of are the roots of the polynomial (in ) y^n-x.
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means t ...
states that, in general, the roots of a polynomial of degree five or higher cannot be expressed in terms of th roots.


Square roots of matrices and operators

If ''A'' is a positive-definite matrix or operator, then there exists precisely one positive definite matrix or operator ''B'' with ; we then define . In general matrices may have multiple square roots or even an infinitude of them. For example, the identity matrix has an infinity of square roots,Mitchell, Douglas W., "Using Pythagorean triples to generate square roots of I2", ''Mathematical Gazette'' 87, November 2003, 499–500. though only one of them is positive definite.


In integral domains, including fields

Each element of an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
has no more than 2 square roots. The difference of two squares identity is proved using the commutativity of multiplication. If and are square roots of the same element, then . Because there are no zero divisors this implies or , where the latter means that two roots are
additive inverse In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opp ...
s of each other. In other words if an element a square root of an element exists, then the only square roots of are and . The only square root of 0 in an integral domain is 0 itself. In a field of characteristic 2, an element either has one square root or does not have any at all, because each element is its own additive inverse, so that . If the field is
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past particip ...
of characteristic 2 then every element has a unique square root. In a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of any other characteristic, any non-zero element either has two square roots, as explained above, or does not have any. Given an odd prime number , let for some positive integer . A non-zero element of the field with elements is a quadratic residue if it has a square root in . Otherwise, it is a quadratic non-residue. There are quadratic residues and quadratic non-residues; zero is not counted in either class. The quadratic residues form a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under multiplication. The properties of quadratic residues are widely used in number theory.


In rings in general

Unlike in an integral domain, a square root in an arbitrary (unital) ring need not be unique up to sign. For example, in the ring \mathbb/8\mathbb of integers modulo 8 (which is commutative, but has zero divisors), the element 1 has four distinct square roots: ±1 and ±3. Another example is provided by the ring of quaternions \mathbb, which has no zero divisors, but is not commutative. Here, the element −1 has infinitely many square roots, including , , and . In fact, the set of square roots of −1 is exactly :\ . A square root of 0 is either 0 or a zero divisor. Thus in rings where zero divisors do not exist, it is uniquely 0. However, rings with zero divisors may have multiple square roots of 0. For example, in \mathbb/n^2\mathbb, any multiple of is a square root of 0.


Geometric construction of the square root

The square root of a positive number is usually defined as the side length of a square with the
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open s ...
equal to the given number. But the square shape is not necessary for it: if one of two similar planar Euclidean objects has the area ''a'' times greater than another, then the ratio of their linear sizes is \sqrt. A square root can be constructed with a compass and straightedge. In his Elements,
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Elements'' treatise, which established the foundations of ge ...
( fl. 300 BC) gave the construction of the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
of two quantities in two different places
Proposition II.14
an

Since the geometric mean of ''a'' and ''b'' is \sqrt, one can construct \sqrt simply by taking . The construction is also given by Descartes in his '' La Géométrie'', see figure 2 o
page 2
However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid. Euclid's second proof in Book VI depends on the theory of similar triangles. Let AHB be a line segment of length with and . Construct the circle with AB as diameter and let C be one of the two intersections of the perpendicular chord at H with the circle and denote the length CH as ''h''. Then, using Thales' theorem and, as in the proof of Pythagoras' theorem by similar triangles, triangle AHC is similar to triangle CHB (as indeed both are to triangle ACB, though we don't need that, but it is the essence of the proof of Pythagoras' theorem) so that AH:CH is as HC:HB, i.e. , from which we conclude by cross-multiplication that , and finally that h = \sqrt. When marking the midpoint O of the line segment AB and drawing the radius OC of length , then clearly OC > CH, i.e. \frac \ge \sqrt (with equality if and only if ), which is the arithmetic–geometric mean inequality for two variables and, as noted above, is the basis of the Ancient Greek understanding of "Heron's method". Another method of geometric construction uses
right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
s and induction: \sqrt can be constructed, and once \sqrt has been constructed, the right triangle with legs 1 and \sqrt has a
hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equ ...
of \sqrt. Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above.


See also

* Apotome (mathematics) *
Cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. ...
* Functional square root * Integer square root * Nested radical * Nth root * Root of unity * Solving quadratic equations with continued fractions * Square root principle *


Notes


References

* * * * * .


External links


Algorithms, implementations, and more
aul Hsieh's square roots webpage


AMS Featured Column, Galileo's Arithmetic by Tony Philips
ncludes a section on how Galileo found square roots {{DEFAULTSORT:Square Root Elementary special functions Elementary mathematics Unary operations