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Encyclopedia > Complex number
A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram

In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:[1] The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

$i^2=-1.,$

Every complex number can be written in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Complex numbers were first discovered by the Italian mathematician Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations [2]. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher. Gerolamo Cardano or Jerome Cardan (September 24, 1501 - September 21, 1576) was a celebrated Renaissance mathematician, physician, astrologer, and gambler. ... A cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

The rules for addition, subtraction, multiplication, and division of complex numbers were first developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions. Raphael Bombelli (1526-1572) was an Italian mathematician. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra. Engineering is the discipline and profession of applying scientific knowledge and utilizing natural laws and physical resources in order to design and implement materials, structures, machines, devices, systems, and processes that realize a desired objective and meet specified criteria. ... This box:      Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Fig. ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... For other uses, see Chaos Theory (disambiguation). ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...

### Notation

The set of all complex numbers is usually denoted by C, or in blackboard bold by $mathbb{C}$. In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... An example of blackboard bold letters. ...

Although other notations can be used, complex numbers are very often written in the form

$a + bi ,$

where a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + ib, the real part (a) is denoted Re(z) or ℜ(z), and the imaginary part (b) is denoted Im(z) or ℑ(z).

The real numbers, R, may be regarded as a subset of C by considering every real number a complex number with an imaginary part of zero; that is, the real number a is identified with the complex number a + 0i. Complex numbers with a real part which is zero are called imaginary numbers; instead of writing 0 + bi, that imaginary number is usually denoted as just bi. If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i. Superset redirects here. ...

In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + jb. Electrical Engineers design power systems. ... This box:      Electric current is the flow (movement) of electric charge. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

Domain Coloring plot of the function
f(x)=(x²-1)(x-2-i)²/
(x²+2+2i). The hue represents the function argument, while the saturation represents the magnitude.

Image File history File links Size of this preview: 600 Ã— 600 pixelsFull resolution (800 Ã— 800 pixel, file size: 59 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Size of this preview: 600 Ã— 600 pixelsFull resolution (800 Ã— 800 pixel, file size: 59 KB, MIME type: image/jpeg) File historyClick on a date/time to view the file as it appeared at that time. ...

### Equality

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if the two complex numbers are written as a + bi and c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d. IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...

### Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1: In mathematics, associativity is a property that a binary operation can have. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

• Addition: $,(a + bi) + (c + di) = (a + c) + (b + d)i$
• Subtraction: $,(a + bi) - (c + di) = (a - c) + (b - d)i$
• Multiplication: $,(a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac - bd) + (bc + ad)i$
• Division: $,frac{(a + bi)}{(c + di)} = left({ac + bd over c^2 + d^2}right) + left( {bc - ad over c^2 + d^2} right)i,,$

where c and d are not both zero.

It is also possible to represent complex numbers as ordered pairs of real numbers, so that the complex number a + ib corresponds to (a, b). In this representation, the algebraic operations have the following formulas: In mathematics, an ordered pair is a collection of two not necessarily distinct objects, one of which is distinguished as the first coordinate (or first entry or left projection) and the other as the second coordinate (second entry, right projection). ...

(a,b) + (c,d) = (a + c, b+ d)

Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. This complex plane is described below. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

## The field of complex numbers

A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers are a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess: In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ...

• An additive identity ("zero"), 0 + 0i.
• A multiplicative identity ("one"), 1+ 0i.
• An additive inverse of every complex number. The additive inverse of a+bi is -a-bi.
• A multiplicative inverse (reciprocal) of every nonzero complex number. The multiplicative inverse of a+bi is ${aover a^2+b^2}+ left( {-bover a^2+b^2}right)i.$

Other fields include the real numbers and the rational numbers. When each real number a is identified with the complex number a+ 0i, the field of real numbers R becomes a subfield of C. The reciprocal function: y = 1/x. ...

The complex numbers C can also be characterized as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below. In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

### The complex plane

Geometric representation of z and its conjugate $bar{z}$ in the complex plane.

A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001) named after Jean-Robert Argand. The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number. Image File history File links Complex_conjugate_picture. ... Image File history File links Complex_conjugate_picture. ... This article is about vectors that have a particular relation to the spatial coordinates. ... Fig. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... Jean-Robert Argand was an accountant and bookkeeper in Paris who was only an amateur mathematician. ...

### Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = reiφ is defined as | z | = r. Algebraically, if z = x + yi, then $|z|=sqrt{x^2+y^2}.$

The absolute value has three important properties:

$| z | geq 0, ,$ where $| z | = 0 ,$ if and only if $z = 0 ,$
$| z + w | leq | z | + | w | ,$ (triangle inequality)
$| z cdot w | = | z | cdot | w | ,$

for all complex numbers z and w. These imply that | 1 | = 1 and | z / w | = | z | / | w | . By defining the distance function d(z,w) = | zw | , we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity. â†” â‡” â‰¡ logical symbols representing iff. ... In mathematics, triangle inequality is the theorem stating that for any triangle, the measure of a given side must be less than the sum of the other two sides but greater than the difference between the two sides. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

The complex conjugate of the complex number z = x + yi is defined to be xyi, written as $bar{z}$ or $z^*,$. As seen in the figure, $bar{z}$ is the "reflection" of z about the real axis, and so both $z+bar{z}$ and $zcdotbar{z}$ are real numbers. Many identities relate complex numbers and their conjugates: In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

$overline{z+w} = bar{z} + bar{w}$
$overline{zcdot w} = bar{z}cdotbar{w}$
$overline{(z/w)} = bar{z}/bar{w}$
$bar{bar{z}}=z$
$bar{z}=z$   if and only if z is real
$bar{z}=-z$   if and only if z is purely imaginary
$operatorname{Re},z = tfrac{1}{2}(z+bar{z})$
$operatorname{Im},z = tfrac{1}{2i}(z-bar{z})$
$|z|=|bar{z}|$
$|z|^2 = zcdotbar{z}$
$z^{-1} = frac{bar{z}}{|z|^{2}}$   if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation commutes with all the algebraic operations (and many functions; e.g. $sinbar z=overline{sin z}$) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function $f(z) = bar{z}$ is not complex-differentiable (see holomorphic function). Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...

### Geometric interpretation of the operations on complex numbers

X = A + B
X = AB
X = A*

The operations of addition, multiplication, and complex conjugation in the complex plane admit natural geometrical interpretations. Image File history File links Complex_numbers_addition. ... Image File history File links Complex_numbers_addition. ... Image File history File links Complex_numbers_multiplication. ... Image File history File links Complex_numbers_multiplication. ... Image File history File links Complex_numbers_conjugation. ... Image File history File links Complex_numbers_conjugation. ...

• The sum of two points A and B of the complex plane is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent.
• The product of two points A and B is the point X = AB such that the triangles with vertices 0, 1, A, and 0, B, X, are similar.
• The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other.

These geometric interpretations allow problems of geometry to be translated into algebra. And, conversely, geometric problems can be examined algebraically. For example, the problem of the geometric construction of the 17-gon is thus translated into the analysis of the algebraic equation x17 = 1. For other uses, see Triangle (disambiguation). ... An example of congruence. ... // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ... A mirror image is a mirror based duplicate of a single image. ... (Redirected from 17-gon) In geometry, a heptadecagon is a seventeen-sided polygon. ...

## Polar form

Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. The polar coordinates are r =  |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. For r = 0 any value of φ describes the same number. To get a unique representation, a conventional choice is to set arg(0) = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is called the polar form of the complex number. A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... Not to be confused with Natural number. ...

### Conversion from the polar form to the Cartesian form

$x = r cos varphi$
$y = r sin varphi$

### Conversion from the Cartesian form to the polar form

$r = sqrt{x^2+y^2}$
$varphi = arg(z) = operatorname{atan2}(y,x)$

(See arg function and atan2.) Atan2 is a two-parameter function for computing the arctangent in the C programming language. ...

The resulting value for φ is in the range (−π, +π]; it is negative for negative values of y. If instead non-negative values in the range [0, 2π) are desired, add 2π to negative results.

### Notation of the polar form

The notation of the polar form as

$z = r,(cos varphi + isin varphi ),$

is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

$z = r,mathrm{e}^{i varphi},$

which is called exponential form.

### Multiplication, division, exponentiation, and root extraction in the polar form

Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.

Using sum and difference identities it follows that In mathematics, trigonometric identities are equalities that involve trigonometric functions that are true for all values of the occurring variables. ...

$r_1,e^{ivarphi_1} cdot r_2,e^{ivarphi_2} = r_1,r_2,e^{i(varphi_1 + varphi_2)} ,$

and that

$frac{r_1,e^{ivarphi_1}}{r_2,e^{ivarphi_2}} = frac{r_1}{r_2},e^{i (varphi_1 - varphi_2)}. ,$

Exponentiation with integer exponents; according to De Moivre's formula, de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

$(cosvarphi + isinvarphi)^n = cos(nvarphi) + isin(nvarphi),,$

from which it follows that

$(r(cosvarphi + isinvarphi))^n = (r,e^{ivarphi})^n = r^n,e^{invarphi} = r^n,(cos nvarphi + mathrm{i} sin n varphi).,$

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation. â€œExponentâ€ redirects here. ...

The addition of two complex numbers is just the vector addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...

Multiplication by i corresponds to a counter-clockwise rotation by 90 degrees (π/2 radians). The geometric content of the equation i 2 = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns. This article describes the unit of angle. ... For the musical group, see Radian (band). ...

If c is a complex number and n a positive integer, then any complex number z satisfying zn = c is called an n-th root of c. If c is nonzero, there are exactly n distinct n-th roots of c, which can be found as follows. Write $c=re^{ivarphi}$ with real numbers r > 0 and φ, then the set of n-th roots of c is

${ sqrt[n]r,e^{i(frac{varphi+2kpi}{n})} mid kin{0,1,ldots,n-1} , },$

where $sqrt[n]{r}$ represents the usual (positive) n-th root of the positive real number r. If c = 0, then the only n-th root of c is 0 itself, which as n-th root of 0 is considered to have multiplicity n. In mathematics, the multiplicity of a member of a multiset is how many memberships in the multiset it has. ...

## Some properties

### Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

$begin{bmatrix} a & -b b & ;; a end{bmatrix}$

where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. Every such matrix can be written as In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

$begin{bmatrix} a & -b b & ;; a end{bmatrix} = a begin{bmatrix} 1 & ;; 0 0 & ;; 1 end{bmatrix} + b begin{bmatrix} 0 & -1 1 & ;; 0 end{bmatrix}$

which suggests that we should identify the real number 1 with the identity matrix

$begin{bmatrix} 1 & ;; 0 0 & ;; 1 end{bmatrix},$

and the imaginary unit i with

$begin{bmatrix} 0 & -1 1 & ;; 0 end{bmatrix},$

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix. In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every nÃ—n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

$|z|^2 = begin{vmatrix} a & -b b & a end{vmatrix} = (a^2) - ((-b)(b)) = a^2 + b^2.$

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z. In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A...

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ...

It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate. In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ...

### Real vector space

C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. More generally, no field containing a square root of −1 can be ordered. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics and set theory, a total order, linear order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix â‰¤) on some set X. The relation is transitive, antisymmetric, and total. ... In mathematics, an ordered field is a field together with an ordering of its elements. ...

R-linear maps CC have the general form In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

$f(z)=az+boverline{z}$

with complex coefficients a and b. Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations. Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ...

The function

$f(z)=az,$

corresponds to rotations combined with scaling, while the function

$f(z)=boverline{z}$

corresponds to reflections combined with scaling.

### Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... This article is about the mathematical term; Multiplicity is also the title of a 1996 film. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ...

Indeed, the complex number field C is the algebraic closure of the real number field, and Cauchy constructed the field of complex numbers in this way. It can also be characterized as the quotient ring of the polynomial ring R[X] over the ideal generated by the polynomial X2 + 1: In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

$mathbb{C} = mathbb{R}[ X ] / ( X^2 + 1). ,$

This is indeed a field because X2 + 1 is irreducible over the real numbers, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring is the imaginary unit i. In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ...

### Algebraic characterization

The field C is (up to field isomorphism) characterized by the following three facts: Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In the jargon of mathematics, the statement that Property P characterizes object X means, not simply that X has property P, but that X is the only thing that has property P. It is also common to find statements such as Property Q characterises Y up to isomorphism. The first...

Consequently, C contains many proper subfields which are isomorphic to C. Another consequence of this characterization is that the Galois group of C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In abstract algebra, the transcendence degree of a field extension L / K is a certain rather coarse measure of the size of the extension. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero in F. In that case, every such polynomial splits into linear factors. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ... In set theory and other branches of mathematics, ‭ב‬2 (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. ...

### Characterization as a topological field

As noted above, the algebraic characterization of C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the topology of C. The following properties characterize C as a topological field: Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... For other uses, see Topology (disambiguation). ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ...

• C is a field.
• C contains a subset P of nonzero elements satisfying:
• P is closed under addition, multiplication and taking inverses.
• If x and y are distinct elements of P, then either x-y or y-x is in P
• If S is any nonempty subset of P, then S+P=x+P for some x in C.
• C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.

Given these properties, one can then define a topology on C by taking the sets

• $B(x,p) = {y | p - (y-x)(y-x)^*in P}$

as a base, where x ranges over C, and p ranges over P. In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases...

To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization. In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... In mathematics, a Dedekind cut in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for any element x in S, if a is in A and x ≤ a, then x is in A as well) and B is... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not. Lev Semenovich Pontryagin. ... Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... Connected and disconnected subspaces of RÂ². The space A at top is connected; the shaded space B at bottom is not. ...

## Complex analysis

For more details on this topic, see Complex analysis.

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane. Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...

## Applications

The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most real descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.

### Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. For control theory in psychology and sociology, see control theory (sociology). ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... In the branch of mathematics called functional analysis, the Laplace transform, , is a linear operator on a function f(t) (original) with a real argument t (t â‰¥ 0) that transforms it to a function F(s) (image) with a complex argument s. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... In control theory, the root locus is the locus of the poles of a transfer function as the system gain K is varied on some interval. ... A Nyquist plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ... A Nichols plot is a graph used in signal processing in which the magnitude and phase of a frequency response are plotted on orthogonal axes. ...

In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ...

If a system has zeros in the right half plane, it is a nonminimum phase system. Instability in systems is generally characterized by some of the outputs or internal states growing without bounds. ... In electrical engineering, specifically signal processing and control theory, BIBO Stability is a form of stability for signals and systems. ... In the theory of dynamical systems, a linear time-invariant system is marginally stable if every eigenvalue in the systems transfer-function is non-positive, and all eigenvalues with zero real value are simple roots. ... In control theory, a nonminimum phase system is one with zeros in the right half plane. ...

### Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg(z) the phase. Signal analysis is the extraction of information from a signal. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C is the... For other uses, see Frequency (disambiguation). ... For quantum-mechanical amplitude, see probability amplitude. ... This article is about a portion of a periodic process. ...

If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ...

$f ( t ) = z e^{iomega t} ,$

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. It has been suggested that this article or section be merged into Angular velocity. ...

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and Wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals. Electrical Engineers design power systems. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as high voltage symbol. ... This box:      Electric current is the flow (movement) of electric charge. ... Resistor symbols (American) Resistor symbols (Europe, IEC) Axial-lead resistors on tape. ... See Capacitor (component) for a discussion of specific types. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. ... Sine waves are commonly found in physics and engineering, such as in the study of alternating current (AC) circuits. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... Digital image processing is the use of computer algorithms to perform image processing on digital images. ... A wavelet is a kind of mathematical function used to divide a given function into different frequency components and study each component with a resolution that matches its scale. ... Source coding redirects here. ... For other uses, see Digital (disambiguation). ... This article is about audible acoustic waves. ... For other uses, see Video (disambiguation). ...

### Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration. It is recommended that the reader be familiar with antiderivatives, integrals, and limits. ... In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. ...

### Quantum mechanics

The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... SchrÃ¶dinger in 1933, when he was awarded the Nobel Prize in Physics Bust of SchrÃ¶dinger, in the courtyard arcade of the main building, University of Vienna, Austria. ... This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... Werner Karl Heisenberg (December 5, 1901 â€“ February 1, 1976) was a celebrated German physicist and Nobel laureate, one of the founders of quantum mechanics and acknowledged to be one of the most important physicists of the twentieth century. ... Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. ...

### Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity. For a generally accessible and less technical introduction to the topic, see Introduction to special relativity. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... For other uses of this term, see Spacetime (disambiguation). ... In physics, a Wick rotation is the process by which a theory in Euclidean space is analytically continued into one in Minkowski space and vice versa. ... Quantum field theory (QFT) is the quantum theory of fields. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

### Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = ert. In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In linear algebra, the characteristic equation of a square matrix A is the equation in one variable λ where I is the identity matrix. ... In mathematics, a linear differential equation is a differential equation of the form Ly = f, where the differential operator L is a linear operator, y is the unknown function, and the right hand side f is a given function. ...

### Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions. --68. ... In fluid dynamics, potential flow in two dimensions is simple to analyse using complex numbers. ...

### Fractals

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set. The boundary of the Mandelbrot set is a famous example of a fractal. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. ... In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of can change drastically under arbitrarily small perturbations. ...

## History

The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid,[3] though negative numbers were not conceived in the Hellenistic world. In mathematics, a square root (âˆš) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... A negative number is a number that is less than zero, such as −3. ... Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... Hero (or Heron) of Alexandria (Greek: Î‰ÏÏ‰Î½ Î¿ Î‘Î»ÎµÎ¾Î±Î½Î´ÏÎµÏÏ‚) (c. ... The 1st century was that century that lasted from 1 to 100 according the Gregorian calendar. ... Look up AD, ad-, and ad in Wiktionary, the free dictionary. ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... For other meanings, see pyramid (disambiguation). ... The term Hellenistic (derived from HÃ©llÄ“n, the Greeks traditional self-described ethnic name) was established by the German historian Johann Gustav Droysen to refer to the spreading of Greek culture over the non-Greek people that were conquered by Alexander the Great. ...

Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0: (15th century - 16th century - 17th century - more centuries) As a means of recording the passage of time, the 16th century was that century which lasted from 1501 to 1600. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, a quartic equation is the result of setting a quartic function equal to zero. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... Niccolo Fontana Tartaglia. ... Gerolamo Cardano. ...

$frac{1}{sqrt{3}}left(sqrt{-1}^{1/3}+frac{1}{sqrt{-1}^{1/3}}right).$

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions –i, ${scriptstylefrac{sqrt{3}}{2}}+{scriptstylefrac{1}{2}}i$ and ${scriptstylefrac{-sqrt{3}}{2}}+{scriptstylefrac{1}{2}}i$. Substituting these in turn for ${scriptstylesqrt{-1}^{1/3}}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory[citation needed] (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation $sqrt{-1}^2=sqrt{-1}sqrt{-1}=-1$ seemed to be capriciously inconsistent with the algebraic identity $sqrt{a}sqrt{b}=sqrt{ab}$, which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity $scriptstyle 1/sqrt{a}=sqrt{1/a}$) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of $sqrt{-1}$ to guard against this mistake. RenÃ© Descartes (French IPA:  Latin:Renatus Cartesius) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ... Leonhard Euler aged 49 (oil painting by Emanuel Handmann, 1756) Leonhard Euler (April 15, 1707 - September 18, 1783) (pronounced oiler) was a Swiss mathematician and physicist. ...

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To de Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula: (17th century - 18th century - 19th century - more centuries) As a means of recording the passage of time, the 18th century refers to the century that lasted from 1701 through 1800. ... Abraham de Moivre. ... Euler redirects here. ... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ...

$(cos theta + isin theta)^{n} = cos n theta + isin n theta ,$

and to Euler (1748) Euler's formula of complex analysis: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ...

$cos theta + isin theta = e ^{itheta }. ,$

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus. Caspar Wessel (June 8, 1745 - March 25, 1818) was a Norwegian-Danish mathematician. ... 1799 was a common year starting on Tuesday (see link for calendar). ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually GauÃŸ, Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... John Wallis John Wallis (November 22, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the development of modern calculus. ...

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that $pmsqrt{-1}$ should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known. Graphical representation of quaternion units product as 90Â°-rotation in 4D-space, ij = k, ji = -k, ij = -ji This page describes the mathematical entity. ... Jean-Robert Argand was an accountant and bookkeeper in Paris who was only an amateur mathematician. ... Augustin Louis Cauchy (August 21, 1789 â€“ May 23, 1857) was a French mathematician. ... Niels Henrik Abel (August 5, 1802â€“April 6, 1829), Norwegian mathematician, was born in Nedstrand, near FinnÃ¸y where his father acted as rector. ...

The common terms used in the theory are chiefly due to the founders. Argand called cosφ + isinφ the direction factor, and $r = sqrt{a^2+b^2}$ the modulus; Cauchy (1828) called cosφ + isinφ the reduced form (l'expression réduite); Gauss used i for $sqrt{-1}$, introduced the term complex number for a + bi, and called a2 + b2 the norm.

The expression direction coefficient, often used for cosφ + isinφ, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers. Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Arturo DellAcqua Bellavitis Arturo DellAcqua Bellavitis (b. ... The tone or style of this article or section may not be appropriate for Wikipedia. ... August Ferdinand MÃ¶bius. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 â€“ May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ...

A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form a + bi, where a and b are integral, or rational (and i is one of the two roots of x2 + 1 = 0). His student, Ferdinand Eisenstein, studied the type a + bω, where ω is a complex root of x3 − 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity xk − 1 = 0 for higher values of k. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set. ... Johann Carl Friedrich Gauss (pronounced ,  ; in German usually GauÃŸ, Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. ... Ferdinand Gotthold Max Eisenstein (April 16, 1823 - October 11, 1852) was a German mathematician. ... In mathematics, the nth roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... In mathematics, the n-th roots of unity or de Moivre numbers, named after Abraham de Moivre (1667 - 1754), are complex numbers located on the unit circle. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... In mathematics an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Kummer, and lead to Dedekinds definition of ideals for rings. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Galois at the age of fifteen from the pencil of a classmate. ...

$F(x) = 0.$

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Bonaventure Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane. Karl Theodor Wilhelm Weierstrass (WeierstraÃŸ) (October 31, 1815 â€“ February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ... Karl Hermann Amandus Schwarz (25 January 1843 â€“ 30 November 1921) was a German mathematician, known for his work in complex analysis. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 â€“ February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Otto Ludwig HÃ¶lder (December 22, 1859 - August 29, 1937) was a mathematician born in Stuttgart, Germany. ... Jules Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Eduard Study (23 March 1862 - 6 Jan 1930) was a 19th-century German mathematician known for work on invariant theory of ternary forms (1889). ... Alexander Macfarlane (Blairgowrie, Scotland, April 21, 1851 â€“ Chatham, Ontario, August 28, 1913) was a Scottish-Canadian logician, physicist, and mathematician. ...

In mathematics, a square root (âˆš) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In physics, circular motion is rotation along a circle: a circular path or a circular orbit. ... In mathematics, complex geometry is the application of complex numbers to plane geometry. ... de Moivres formula, named after Abraham de Moivre, states that for any complex number (and, in particular, for any real number) x and any integer n it holds that The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry. ... For other uses, see List of topics named after Leonhard Euler. ... The term hypercomplex number has been used in mathematics for the elements of algebras that extend or go beyond complex number arithmetic. ... In mathematics, a local field is a special type of field which has the additional property that it is a complete metric space with respect to a discrete valuation. ... Initial image of a Mandelbrot set zoom sequence with continuously coloured environment The Mandelbrot set is a set of points in the complex plane, the boundary of which forms a fractal. ... Graphical representation of quaternion units product as 90Â°-rotation in 4D-space, ij = k, ji = -k, ij = -ji This page describes the mathematical entity. ... The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection â€” details are given below). ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...

## Notes

1. ^ K. D. Joshi, Foundations of Discrete Mathematics, 1989, Wiley, p. 398 ISBN 0470211520
2. ^ Burton, David (1995). "7", The History of Mathematics, 3rd, New York: McGraw-Hill, 294. ISBN 0-07-009465-9.
3. ^ A brief history of complex numbers

## References

### Mathematical references

• Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0070006577
• Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0-387-90328-3
• Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0
• Solomentsev, E.D. (2001), “Complex number”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

Lars Valerian Ahlfors (April 18, 1907 â€“ October 11, 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis. ... Dan Pedoe (1910 to 1998) was an English-born mathematician and geometer with a career spanning more than sixty years. ... The Encyclopaedia of Mathematics is a large reference work in mathematics. ...

### Historical references

• Nahin, Paul J. (1998), An Imaginary Tale: The Story of $sqrt{-1}$ (hardcover ed.), Princeton University Press, ISBN 0-691-02795-1
A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
• H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert (1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0
An advanced perspective on the historical development of the concept of number.

• The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
• Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
• Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-198-53447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.

Sir Roger Penrose, OM, FRS (born 8 August 1931) is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College. ... Tristan Needham is the author of the highly original book Visual Complex Analysis (Oxford, 1997) in which he uses a geometric approach to develop complex analysis (which he says was inspired by Isaac Newtons original geometric approach to ordinary calculus). ...

Results from FactBites:

 Complex Numbers - MSN Encarta (834 words) The complex numbers are the field C of numbers of the form x+iy, where x and y are real numbers and i is the imaginary unit equal to the square root of -1, sqrt(-1). Complex analysis, which combines complex numbers with ideas from calculus, has been widely applied to subjects as different as the theory of numbers and the design of airplane wings. The study of complex functions was continued by French mathematician Augustin Louis Cauchy, who in 1825 generalized the real definite integral of calculus to functions of a complex variable.
 Complex number (3325 words) Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z. The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard.
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