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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 (the result of multiplying the number by itself) is x.

Every non-negative real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 x has a unique non-negative square root, called the principal square root, which is denoted with a radical symbol as , or, using exponent notation, as x1/2.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square
Square (algebra)

In algebra, the square of a number is that number multiplication by itself. To square a quantity is to multiply it by itself.Its notation is a superscripted "2"; a number x squared is written as x?....
 (the result of multiplying the number by itself) is x.

Every non-negative real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
 x has a unique non-negative square root, called the principal square root, which is denoted with a radical symbol as , or, using exponent notation, as x1/2. For example, the principal square root of 9 is 3, denoted = 3, because . If otherwise unqualified, "the square root" of a number refers to the principal square root: the square root of 2 is approximately 1.4142.

Every positive number x has two square roots. One of them is , which is positive, and the other , which is negative. Together, these two roots are denoted . Square roots of negative numbers can be discussed within the framework of complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s. 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
Matrix algebra

Matrix algebra can refer to*Matrix theory, is the branch of mathematics that studies matrix .*A matrix ring thought of as an algebra over a field over a field or a commutative ring....
, endomorphism ring
Endomorphism ring

In abstract algebra, one associates to certain objects a ring , the object's endomorphism ring, which encodes several internal properties of the object....
s, etc).

Square roots of integers that are not perfect square
Square number

In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself....
s are always irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
s: numbers not expressible as a ratio
Ratio

A ratio is an expression which compares quantities relative to each other. The most common examples involve two quantities, but in theory any number of quantities can be compared....
 of two integers. For example, cannot be written exactly as m/n, where n and m are integers. Nonetheless, it is exactly the length of the diagonal
Diagonal

A diagonal can refer to a line joining two nonconsecutive vertices of a polygon or polyhedron, or in informal contexts any upward or downward sloping line....
 of a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 with side length 1. This has been known since ancient times, with the discovery that is irrational attributed to Hippasus
Hippasus

Hippasus of Metapontum , b. c. 500 B.C. in Magna Graecia, was a Ancient Greece philosopher. He was a disciple of Pythagoras. To Hippasus is attributed the discovery of the existence of irrational numbers....
, a disciple of Pythagoras
Pythagoras

Pythagoras of Samos was an Ionians Ancient Greeks mathematician and founder of the religious movement called Pythagoreanism. He is often revered as a great mathematician, mysticism and scientist; however some have questioned the scope of his contributions to mathematics and natural philosophy....
. (See square root of 2
Square root of 2

The square root of 2, also known as Pythagoras' constant,is the positive real number that, when multiplied by itself, gives the number 2 ....
 for proofs of the irrationality of this number and quadratic irrational
Quadratic irrational

In mathematics, a quadratic irrational, also known asa quadratic irrationality or quadratic surd, is an irrational number that is the solution to some quadratic equation with rational coefficients....
 for a proof for all non-square natural numbers)

The term whose root is being considered is known as the radicand. In the expression , ab + 2 is the radicand. The radicand is the number or expression underneath the radical sign.

Properties

The principal square root function (usually just referred to as the "square root function") is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 which maps the set of non-negative real numbers R+ ∪ onto itself, and, like all functions, always returns a unique value. The square root function also maps rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s into algebraic number
Algebraic number

In mathematics, an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational number coefficients....
s (a superset
SuperSet

SuperSet Software was a group founded by friends and former Eyring Research Institute co-workers Drew Major, Dale Neibaur, Kyle Powell and later joined by Mark Hurst....
 of the rational numbers); is rational if and only if x is a rational number which can be represented as a ratio of two perfect square
Perfect square

Perfect square may refer to:...
s. In geometrical
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
 terms, the square root function maps the area of a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 to its side length.

  • For all real numbers x,
    (see absolute value
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
)


  • For all non-negative real numbers x and y,
and


  • The square root function is continuous for all non-negative x and differentiable
    Derivative

    In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point....
     for all positive x. Its derivative is given by


  • The Taylor series
    Taylor series

    In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
     of √1 + x about x = 0 converges for | x | < 1 and is given by


Computation

Many methods of calculating square roots exist today, some meant to be done by hand and some meant to be done by machine.

Many, but not all pocket calculators have a square root key. Computer spreadsheet
Spreadsheet

A spreadsheet is a computer application that simulates a paper worksheet. It displays multiple cells that together make up a grid consisting of rows and columns, each cell containing either alphanumeric text or numeric values....
s and other software are also frequently used to calculate square roots. Computer software programs typically implement good routines to compute the exponential function
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
 and the natural logarithm
Natural logarithm

The natural logarithm, formerly known as the hyperbolic logarithm, is the logarithm to the base e , where e is an irrational number constant approximately equal to 2.718281828....
 or logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
, and then compute the square root of x using the identity or The same identity is exploited when computing square roots with logarithm tables or slide rule
Slide rule

The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division , and also for "scientific" functions such as Nth roots, logarithms and trigonometry, but does not generally perform addition or subtraction....
s.

The most common iterative method
Iterative method

In computational mathematics, an iterative method attempts to solve a problem by finding successive approximations to the solution starting from an initial guess....
 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. It involves a simple algorithm, which results in a number closer to the actual square root each time it is repeated. To find r, the square root of a real number x:
  1. Start with an arbitrary positive start value r (the closer to the square root of x, the better).
  2. Replace r by the average between r and x/r, that is: (It is sufficient to take an approximate value of the average in order to ensure convergence
    Convergence

    In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium point state....
    .)
  3. Repeat step 2 until r and x/r are as close as desired.


The time complexity
Computational Complexity

Computational Complexity may refer to:*Computational complexity theory*Computational Complexity ...
 for computing a square root with n digits of precision is equivalent to that of multiplying two n-digit numbers.

Square roots of negative and complex numbers


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 larger set of numbers, called the complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
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 electricity
Electric current

Electric current is the flow of electric charge. The electric charge may be either electrons or ions.The International System of Units unit of electric current intensity is the ampere....
) and called the imaginary unit
Imaginary unit

In mathematics, physics, and engineering, the imaginary unit is denoted by  or the Latin   or the Greek iota . It allows the real number system, to be extended to the complex number system,   Its precise definition is dependent upon the particular method of extension....
, 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. Similarly to the real numbers, we say the principal square root of −1 is i, or more generally, if x is any positive number, then the principal square root of −x is because

By the argument given above, i can be neither positive nor negative. This creates a problem: for the complex number z, we cannot define √z to be the "positive" square root of z.

For every non-zero complex number z there exist precisely two numbers w such that w2 = z.

Examples

  • the square roots of are:


and

  • The square roots of 4 are in the set .
  • The square roots of 0 are described by the multiset
    Multiset

    In mathematics, a multiset is a generalization of a Set . A Element of a multiset can have more than one Element , while each member of a set has only one membership....
     , because 0 is a root of multiplicity 2 of the polynomial .


Definition

To find a definition for the square root that allows us to consistently choose a single value, called the principal value
Principal value

In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is Single-valued function....
, 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
Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
. The same point may be reinterpreted using polar coordinates as the pair (r, Φ), where r ≥ 0 is the distance of the point from the origin, and Φ is the angle that the line from the origin to the point makes with the positive real (x) axis. In complex analysis, this value is conventionally written r eiΦ. Now, since r is a nonnegative real number, we can take its positive square root; and since squaring always doubles the angle , to undo squaring we halve the angle. The result is what is called in complex analysis a branch cut: if with then we define:

Thus defined, the square root function is holomorphic
Holomorphic function

Holomorphic functions are the central object of study of complex analysis; they are function defined on an open set of the complex number C with values in C that are complex-differentiable at every point....
 everywhere except on the non-positive real numbers (where it isn't even continuous
Continuous function

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be discontinuous....
). The above Taylor series for √1 + x remains valid for complex numbers x with | x | < 1.

Formula

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

where is the absolute value or modulus
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
 of the complex number, unless x = −r and y = 0. Notice that the sign of the imaginary part of the root is the same as the sign of the imaginary part of the original number. The real part of the principal value is always non-negative.

Square roots of matrices and operators

If A is a positive-definite matrix
Positive-definite matrix

In linear algebra, a positive-definite matrix is a Hermitian matrix matrix which in many ways is analogous to a positive real number. The notion is closely related to a Definite bilinear form symmetric bilinear form ....
 or operator, then there exists precisely one positive definite matrix or operator B with B2 = A; we then define √A = B.

More generally, to every normal
Normal operator

In mathematics, especially functional analysis, a 'normal operator' on a complex Hilbert space is a continuous function linear operatorthat commutator with its hermitian adjoint N*:...
 matrix or operator A there exist normal operators B such that B2 = A. In general, there are several such operators B for every A and the square root function cannot be defined for normal operators in a satisfactory manner. Positive definite operators are akin to positive real numbers, and normal operators are akin to complex numbers.

Principal square roots of the positive integers


As decimal expansions

The square roots of the perfect square
Perfect square

Perfect square may refer to:...
s (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots are irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
s, and therefore their decimal representation
Decimal representation

A decimal representation of a non-negative real number r is an expression of the formwhere a0 is a nonnegative integer, and a1,...
s are non-repeating decimal
Repeating decimal

A decimal representation of a real number is called a repeating decimal if at some point it becomes periodicity: there is some finite sequence of digits that is repeated indefinitely....
s.
align="right" style="padding-bottom:5px;"  1
align="right" style="padding-bottom:5px;"  1.4142135623 7309504880 1688724209 6980785696 7187537694 8073176679 7379907324 78462 , , ,
align="right" style="padding-bottom:5px;"  1.7320508075 6887729352 7446341505 8723669428 0525381038 0628055806 9794519330 16909
align="right" style="padding-bottom:5px;"  2
align="right" style="padding-bottom:5px;"  2.2360679774 9978969640 9173668731 2762354406 1835961152 5724270897 2454105209 25638
align="right" style="padding-bottom:5px;"  2.4494897427 8317809819 7284074705 8913919659 4748065667 0128432692 5672509603 77457
align="right" style="padding-bottom:5px;"  2.6457513110 6459059050 1615753639 2604257102 5918308245 0180368334 4592010688 23230
align="right" style="padding-bottom:5px;"  2.8284271247 4619009760 3377448419 3961571393 4375075389 6146353359 4759814649 56924
align="right" style="padding-bottom:5px;"  3
align="right" style="padding-bottom:5px;"  3.1622776601 6837933199 8893544432 7185337195 5513932521 6826857504 8527925944 38639
align="right" style="padding-bottom:5px;"  3.3166247903 5539984911 4932736670 6866839270 8854558935 3597058682 1461164846 42609
align="right" style="padding-bottom:5px;"  3.4641016151 3775458705 4892683011 7447338856 1050762076 1256111613 9589038660 33818
align="right" style="padding-bottom:5px;"  3.6055512754 6398929311 9221267470 4959462512 9657384524 6212710453 0562271669 48293
align="right" style="padding-bottom:5px;"  3.7416573867 7394138558 3748732316 5493017560 1980777872 6946303745 4673200351 56307
align="right" style="padding-bottom:5px;"  3.8729833462 0741688517 9265399782 3996108329 2170529159 0826587573 7661134830 91937
align="right" style="padding-bottom:5px;"  4
align="right" style="padding-bottom:5px;"  4.1231056256 1766054982 1409855974 0770251471 9922537362 0434398633 5730949543 46338
align="right" style="padding-bottom:5px;"  4.2426406871 1928514640 5066172629 0942357090 1562613084 4219530039 2139721974 35386
align="right" style="padding-bottom:5px;"  4.3588989435 4067355223 6981983859 6156591370 0392523244 4936890344 1381595573 28203
align="right" style="padding-bottom:5px;"  4.4721359549 9957939281 8347337462 5524708812 3671922305 1448541794 4908210418 51276

As expansions in other numeral systems

The square roots of the perfect square
Perfect square

Perfect square may refer to:...
s (1, 4, 9, 16, etc.) are integers. In all other cases, the square roots are irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
s, and therefore their representations in any standard positional notation
Positional notation

A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, Geometric progression, called the radix or radix of that numeral system....
 system are non-repeating. Even in non-standard positional numeral systems
Non-standard positional numeral systems

Non-standard positional numeral systems here designates numeral systems that may be denoted Positional notation, but that deviate in one way or another from the following description of standard positional systems:...
, the only notable exception is golden ratio base
Golden ratio base

Golden ratio base is a Non-standard positional numeral systems that uses the golden ratio as its base . It is sometimes referred to as base-f, golden mean base, phi-base, or, colloquially, phinary....
.

The hexadecimal representations of some square roots are used in some SHA hash function
SHA hash functions

The SHA hash functions are a set of cryptographic hash functions designed by the National Security Agency and published by the National Institute of Standards and Technology as a U.S....
 implementations.

As periodic continued fractions


One of the most intriguing results from the study of irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
s as continued fraction
Continued fraction

In mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers....
s was obtained by Joseph Louis Lagrange
Joseph Louis Lagrange

Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia was an Italy mathematician and astronomer, who lived most of his life in Prussia and France, making significant contributions to all fields of mathematical analysis, to number theory, and to classical mechanics and celestial mechanics....
 circa 1780. Lagrange found that the representation of the square root of any non-square positive integer as a continued fraction is periodic
Periodicity

Periodicity is the quality of occurring at regular intervals or periods and can occur in different contexts:In timing devices:* A clock marks time at periodic intervals....
. 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.

align="right"  [1; 2, 2, ...]
align="right"  [1; 1, 2, 1, 2, ...]
align="right"  [2]
align="right"  [2; 4, 4, ...]
align="right"  [2; 2, 4, 2, 4, ...]
align="right"  [2; 1, 1, 1, 4, 1, 1, 1, 4, ...]
align="right"  [2; 1, 4, 1, 4, ...]
align="right"  [3]
align="right"  [3; 6, 6, ...]
align="right"  [3; 3, 6, 3, 6, ...]
align="right"  [3; 2, 6, 2, 6, ...]
align="right"  [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, ...]
align="right"  [3; 1, 2, 1, 6, 1, 2, 1, 6, ...]
align="right"  [3; 1, 6, 1, 6, ...]
align="right"  [4]
align="right"  [4; 8, 8, ...]
align="right"  [4; 4, 8, 4, 8, ...]
align="right"  [4; 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, ...]
align="right"  [4; 2, 8, 2, 8, ...]


The square bracket notation used above is a sort of mathematical shorthand to conserve space. Written in more traditional notation the simple continued fraction
Continued fraction

In mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers....
 for the square root of 11 – [3; 3, 6, 3, 6, ...] – looks like this:

where the two-digit pattern repeats over and over and over again in the partial denominators.

Geometric construction of the square root

A square root can be constructed with a compass and straightedge. In his Elements
Euclid's Elements

Euclid's Elements is a mathematics and geometry treatise consisting of 13 books written by the Greek mathematics Euclid in Alexandria circa 300 BC....
, Euclid
Euclid

Euclid , floruit 300 BC, also known as Euclid of Alexandria, was a Greek mathematics and is often referred to as the Father of Geometry. He was active in Alexandria during the reign of Ptolemy I ....
 (fl. 300 BC) gave the construction of the geometric mean
Geometric mean

The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean, which is what most people think of with the word "average," except that instead of adding the set of numbers and then dividing the sum by the count of numbers in the...
 of two quantities in two different places: and . Since the geometric mean of a and b is √ab, one can construct √a simply by taking b = 1.

The construction is also given by Descartes in his La Géométrie
La Géométrie

La G?om?trie was publishing in 1637 as an appendix to Discours de la m?thode , writing by Ren? Descartes. Descartes was in his own time, and has been since, recognized as a Great Thinker....
, see figure 2 on . However, Descartes made no claim to originality and his audience would have been quite familiar with Euclid.

Another method of geometric construction uses right
Angle

In geometry and trigonometry, an angle is the figure formed by two Ray sharing a common endpoint, called the vertex of the angle . The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept out when one ray is rotated about the vertex to coincide...
 triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
s and induction
Mathematical induction

Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then...
: √1 can, of course, be constructed, and once √x has been constructed, the right triangle with 1 and √x for its legs has a hypotenuse
Hypotenuse

File:Triangle Sides.svgA hypotenuse is the longest side of a right triangle, the side opposite the right angle. The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the Square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides....
 of √x + 1. The Spiral of Theodorus
Spiral of Theodorus

In geometry, the spiral of Theodorus is a spiral composed of contiguous right triangles. It was first constructed by Theodorus of Cyrene....
 is constructed using successive square roots in this manner.

History


The Rhind Mathematical Papyrus
Rhind Mathematical Papyrus

The Rhind Mathematical Papyrus , is named after Alexander Henry Rhind, a Scotland antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum....
 is a copy from 1650 BC of an even earlier work and shows us how the Egyptians extracted square roots.

In Ancient India
History of India

The known history of India begins with the Indus Valley Civilization, which spread and flourished in the north-western part of the Indian subcontinent, from c....
, the knowledge of theoretical and applied aspects of square and square root was at least as old as the Sulba Sutras
Sulba Sutras

The Shulba Sutras or Sulbasutras are sutra texts belonging to the Srauta ritual and containing geometry related to fire-altar construction....
, dated around 800-500 B.C. (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
Aryabhata

Aryabhaa is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya and Arya-Siddhanta....
 in the Aryabhatiya
Aryabhatiya

Aryabhatiya, an astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Aryabhata....
 (section 2.4), has given a method for finding the square root of numbers having many digits.

In the Chinese mathematical work Writings on Reckoning
Suàn shù shu

The Su?n sh? shu , or the Book on Numbers and Computation , is one of the earliest known Chinese mathematics. It was written during the early Western Han Dynasty, sometime between 202 BC and 186 BC....
, written between 202 BC and 186 BC during the early Han Dynasty
Han Dynasty

The Han Dynasty followed the Qin Dynasty and preceded the Three Kingdoms in China. The Han Dynasty was ruled by the family known as the Liu clan who had peasant origins....
, the square root is approximated by using a "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."

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

Pietro di Giacomo Cataneo was an Italian architect.He is principally remembered for his I Quattro Primi Libri di Architettura , a set of four books on architectural theory....
 in 1546.

See also


External links

  • - Professor Fukutaro Kato's method
  • - Takashi Kojima's method
  • - Paul Hsieh's square roots webpage
  • with implementation in Rexx.