TheInfoList In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a square root of a number is a number such that ; in other words, a number whose ''
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...
'' (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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
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 In mathematics, the radical sign, radical symbol, root symbol, radix, or surd is a symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, Object (philosophy), object, or wikt:relationshi ...
'' or ''radix''. For example, the principal square root of 9 is 3, which is denoted by $\sqrt = 3,$ because 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 case 9. Every
positive number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
has two square roots: $\sqrt,$ which is positive, and $-\sqrt,$ which is negative. Together, these two roots are denoted as $\plusmn\sqrt$ (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 , the principal square root can also be written in
exponent Exponentiation is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europ ...
notation, as . Square roots of negative numbers can be discussed within the framework of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... s. More generally, square roots can be considered in any context in which a notion of the "
square In Euclidean geometry, a square is a regular The term regular can mean normal or in accordance with rules. It may refer to: People * Moses Regular (born 1971), America football player Arts, entertainment, and media Music * Regular (Badfinger ...
" of a mathematical object is defined. These include
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s and
square matrices In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, among other
mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s.

# History

The
Yale Babylonian Collection The collection holds Babylonian clay tablet YBC 7289 (c. 1800–1600 BC). The tablet displays an approximation of the square root of 2. Comprising some 45,000 items, the Yale Babylonian Collection is an independent branch of the Yale University ...
YBC 7289 YBC 7289 is a Babylonian clay tablet notable for containing an accurate sexagesimal approximation to the square root of 2, the length of the diagonal of a unit square. This number is given to the equivalent of six decimal digits, "the greatest kno ...
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 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 The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scotland, Scottish antiquarian, who p ... is a copy from 1650 BC of an earlier Berlin Papyrus and other textspossibly the
Kahun Papyrus The Kahun Papyri (KP; also Petrie Papyri or Lahun Papyri) are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie Sir William Matthew Flind ...
that 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 mod ...
, the knowledge of theoretical and applied aspects of square and square root was at least as old as the ''
Sulba Sutras The ''Shulba Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to vedi (altar), fire-altar construction. Purpose an ...
'', 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 are a group of Vedic Sanskrit Vedic Sanskrit, or Vedic, is the name given by modern scholarship to the oldest, attested form of the Proto-Indo-Aryan language belonging to the Indo-Aryan subgroup of the Indo-European language The Indo ...
''.
Aryabhata Aryabhata (, ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such ...
, in the ''
Aryabhatiya ''Aryabhatiya'' (IAST The International Alphabet of Sanskrit Transliteration (IAST) is a transliteration scheme that allows the lossless romanisation of Brahmic family, Indic scripts as employed by Sanskrit and related Indic languages. It is ...
'' (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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
that are not perfect squares are always
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s: numbers not expressible as a
ratio In mathematics, a ratio indicates how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8∶6, which is equivalent to ... of two integers (that is, 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 of the
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
is assumed to date back earlier to the
Pythagoreans Pythagoreanism originated in the 6th century BC, based on the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans. Pythagoras established the first Pythagorean community in Crotone, Italy. Early Pythagorean communities spr ...
, and is traditionally attributed to
Hippasus Hippasus of Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Taranto, Tarentum, between the river Bradanus and the Casuentus (modern Basento). It wa ...
. It is exactly the length of the
diagonal In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ... 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#REDIRECT Han dynasty The Han dynasty () was the second Dynasties in Chinese history, imperial dynasty of China (202 BC – 220 AD), established by the rebel leader Liu Bang and ruled by the House of Liu. Preceded by the short-lived Qin dynas ... , 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 Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) inc ... (1436–1476). An R was also used for radix to indicate square roots in
Gerolamo Cardano Gerolamo (also Girolamo or Geronimo) Cardano (; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501 (O. S.)– 21 September 1576 (O. S.)) was an Italian polymath A polymath ( el, πολυμαθής, ', "having learn ... '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 $f\left(x\right) = \sqrt$ (usually just referred to as the "square root function") is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that maps the set of nonnegative real numbers onto itself. In
geometrical Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related ... terms, the square root function maps the
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ... of a square to its side length. The square root of ''x'' is rational if and only if ''x'' is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
that can be represented as a ratio of two perfect squares. (See
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they ...
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 any complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
s, the latter being a
superset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 , the absolute value or modulus of a  , denoted , is the value of  without regard to its . Namely, if is , and if is (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of − ... ) For all nonnegative real numbers ''x'' and ''y'', :$\sqrt = \sqrt x \sqrt y$ and :$\sqrt x = x^.$ The square root function is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
for all nonnegative ''x'', and
differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domain of a function, domain. In other words, the Graph of a function, graph of a differen ... for all positive ''x''. If ''f'' denotes the square root function, whose derivative is given by: :$f\text{'}\left(x\right) = \frac.$ The
Taylor series In , the Taylor series of a is an of terms that are expressed in terms of the function's s at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after ...
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 Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called pa ...
(and
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
), as well as in generalizations such as
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. It defines an important concept of
standard deviation In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, 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 v ... used in
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ... . It has a major use in the formula for roots of a
quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ... ;
quadratic field In algebraic number theory, a quadratic field is an algebraic number field ''K'' of Degree of a field extension, degree two over Q, the rational numbers. The map ''d'' ↦ Q() is a bijection from the Set (mathematics), set of all square-f ...
s and rings of
quadratic integer In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are ...
s, 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 Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (album), ''Physical'' (album), a 1981 album by Olivia Newton-John **Physical (Olivia Newton-John song), "Physical" (Olivia Newton-John song) *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 Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s—more specifically
quadratic integer In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are ...
s. 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 (mathematics), 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 ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
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 An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ... . In all other cases, the square roots of positive integers are
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, and hence have non-
repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It c ...
s in their
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
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 of the (or ). More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number ...
system. The square roots of small integers are used in both the
SHA-1 In cryptography, SHA-1 (Secure Hash Algorithm 1) is a cryptographic hash function which takes an input and produces a 160-bit (20-byte) hash value known as a message digest – typically rendered as a hexadecimal number, 40 digits long. It was de ... and
SHA-2 SHA-2 (Secure Hash Algorithm 2) is a set of cryptographic hash functions designed by the United States National Security Agency (NSA) and first published in 2001. They are built using the Merkle–Damgård construction, from a one-way compression ... hash function designs to provide
nothing up my sleeve number In cryptography, nothing-up-my-sleeve numbers are any numbers which, by their construction, are above suspicion of hidden properties. They are used in creating cryptographic functions such as cryptographic hash, hashes and ciphers. These algorithms ...
s.

## As periodic continued fractions

One of the most intriguing results from the study of
irrational number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s as
continued fraction In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s was obtained by
Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
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. : The
square bracket In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...
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 number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s, 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 Electronics, electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first Solid-state electronics, solid-state electronic calculator was created ... s have a square root key. Computer
spreadsheet A spreadsheet is a computer application for organization, analysis, and storage of data in table (information), tabular form. Spreadsheets were developed as computerized analogs of paper accounting Worksheet#Accounting, worksheets. The program op ... s and other
software Software is a collection of instructions Instruction or instructions may refer to: Computing * Instruction, one operation of a processor within a computer architecture instruction set * Computer program, a collection of instructions Music * I ... are also frequently used to calculate square roots. Pocket calculators typically implement efficient routines, such as the
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physic ... (frequently with an initial guess of 1), to compute the square root of a positive real number. When computing square roots with
logarithm table In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s or
slide rule The slide rule is a mechanical . The slide rule is used primarily for and and for functions such as , , s, and . They are not designed for addition or subtraction which was usually performed manually, with used to keep track of the magnitude ... s, one can exploit the identities :$\sqrt = e^ = 10^,$ where and 10 are the
natural Nature, in the broadest sense, is the natural, physical, material world or universe The universe ( la, universus) is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and ...
and
base-10 logarithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. 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 :$\left(x + c\right)^2 = x^2 + 2xc + c^2,$ 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ... 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 Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ... of square root calculation by hand is known as the "
Babylonian method Methods of computing square roots are numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... ... " or "Heron's method" after the first-century Greek philosopher
Heron of Alexandria The herons are long-legged, long-necked, freshwater and coastal bird Birds are a group of warm-blooded vertebrates constituting the class (biology), class Aves , characterised by feathers, toothless beaked jaws, the Oviparity, laying of ... , who first described it. The method uses the same iterative scheme as the
Newton–Raphson method In numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Numerical analysis is the study o ... 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 In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and f ...
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 *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853� ...
), 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
or
piecewise-linear approximation An approximation is anything that is intentionally similar but not exactly equal Equal or equals may refer to: Arts and entertainment * Equals (film), ''Equals'' (film), a 2015 American science fiction film * Equals (game), ''Equals'' (game), a ...
can be used. The
time complexity In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of com ...
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 The shifting ''n''th root algorithm is an algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The algorithm proceeds by successive subtrac ...
, applied for . The name of the square root Function (programming), function varies from programming language to programming language, with sqrt (often pronounced "squirt" ) being common, used in C (programming language), C, C++, and derived languages like JavaScript, PHP, and Python (programming language), Python.

# 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 number, real square root. However, it is possible to work with a more inclusive set of numbers, called the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ... s, 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 electric current, electricity where "''i''" traditionally represents electric current) and called the imaginary unit, 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 :$\left(i\sqrt x\right)^2 = i^2\left(\sqrt x\right)^2 = \left(-1\right)x = -x.$ 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, $\left(x, y\right),$ expressed using Cartesian coordinate system, Cartesian coordinates. The same point may be reinterpreted using polar coordinates as the pair $\left(r, \varphi\right),$ 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. 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 function, holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ga ...
). 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: :$\sqrt = \sqrt +i\sgn\left(y\right) \sqrt,$ where is the Sign function, 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: :$\begin \sqrt &= \frac + i\frac = \frac\left(1+i\right),\\ \sqrt &= \frac - i\frac = \frac\left(1-i\right). \end$

## 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 $-\pi<\theta_w-\theta_z\le\pi$ *$\sqrt = \left\left( \sqrt z \right\right)^*$ (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 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). 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 :$\sqrt \cdot \sqrt=\left(-i\right) \cdot \left(-i\right)=-1$ 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 of $x$ is a number $y$ such that $y^3 = x$; it is denoted $\sqrt\left[3\right]x.$ If is an integer greater than two, a Nth root, th root of $x$ is a number $y$ such that $y^n = x$; it is denoted $\sqrt\left[n\right]x.$ Given any polynomial , a polynomial root, 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 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 has no more than 2 square roots. The difference of two squares identity is proved using the commutative ring, 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 inverses 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 (algebra), 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 field, finite of characteristic 2 then every element has a unique square root. In a field (mathematics), field 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 (mathematics), group 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 modular arithmetic, 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 quaternion#Square roots of −1, 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 Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ... equal to the given number. But the square shape is not necessary for it: if one of two similarity (geometry), similar Euclidean plane, 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 Euclid's Elements, Elements, Euclid (floruit, fl. 300 BC) gave the construction of the geometric mean 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#Similar triangles, 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 Pythagorean theorem#Proof using similar triangles, 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 inequality of arithmetic and geometric means, arithmetic–geometric mean inequality for two variables and, as noted Square root#Computation, above, is the basis of the Greek Mathematics, Ancient Greek understanding of "Heron's method". Another method of geometric construction uses right triangles and Mathematical induction, induction: $\sqrt$ can be constructed, and once $\sqrt$ has been constructed, the right triangle with legs 1 and $\sqrt$ has a hypotenuse of $\sqrt$. Constructing successive square roots in this manner yields the Spiral of Theodorus depicted above.

* Apotome (mathematics) * Cube root * Functional square root * Integer square root * Nested radical * Nth root * Root of unity * Solving quadratic equations with continued fractions * Square root principle *

* * * * * .