NationMaster - Encyclopedia: Complex argument

In mathematics, a complex number is an expression of the form Image File history File links Wikibooks-logo-en. ... Euclid, detail from The School of Athens by Raphael. ...

$a + bi ,$

where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with 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. When the imaginary part b is 0, the complex number is just the real number a. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. ... In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) 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.

Complex numbers can be added, subtracted, multiplied, and divided in a similar way to real numbers; but they have additional elegant properties. For example, every polynomial algebraic equation has a complex number as a solution, not just some, as in the real numbers. In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ...

In some fields (in particular, electrical engineering, where i is a symbol for current), complex numbers are written as a + bj. This article treats electronics engineering as a subfield of electrical engineering, though this is not typical use in some areas. ... Electric current is the flow of electric charge. ...

Definitions

Notation and operations

The set of all complex numbers is usually denoted by C, or in blackboard bold by $mathbb{C}$ . The real numbers, R, may be regarded as "lying in" C by considering every real number as a complex: $a = a + 0 i$ . In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... An example of blackboard bold letters. ...

Complex numbers are added, subtracted, and multiplied 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. ...

(a + bi) + (c + di) = (a+c) + (b+d)i

(a + bi) − (c + di) = (a−c) + (b−d)i

(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i

Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed. 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 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, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. An adjective is a part of speech which modifies a noun, usually describing it or making its meaning more specific. ... In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and these days... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ...

The complex number field

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations: An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...

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

$( a , b ) cdot ( c , d ) = ( ac - bd , bc + ad ). ,$

So defined, the complex numbers form a field, the complex number field, denoted by C. 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. ...

Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1).

In C, we have:

additive identity ("zero"): (0, 0)
multiplicative identity ("one"): (1, 0)
additive inverse of (a,b): (−a, −b)
multiplicative inverse (reciprocal) of non-zero (a, b): $left({aover a^2+b^2},{-bover a^2+b^2}right).$

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below. In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1. ... 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 relative to a field is any element of a given field containing such that is a solution of a polynomial equation of the form: anxn + anâˆ’1xnâˆ’1 + Â·Â·Â· + a1x + a0 = 0 where n is a positive integer called the degree of the polynomial, every coefficient... 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

-from the creater, wshun 02:20, 24 Dec 2004 (UTC) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand). In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... 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. ...

The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the circular coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... Eulers formula, named after Leonhard Euler (pronounced oiler), is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

$z = x + iy = r (cos phi + isin phi ) = r e^{i phi}. ,$

Additionally the notation r cis φ is sometimes used.

Note that the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent. The word modulo is the Latin ablative of modulus. ... The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

By simple trigonometric identities, we see that In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

$r_1 e^{iphi_1} cdot r_2 e^{iphi_2} = r_1 r_2 e^{i(phi_1 + phi_2)} ,$

and that

$frac{r_1 e^{iphi_1}} {r_2 e^{iphi_2}} = frac{r_1}{r_2} e^{i (phi_1 - phi_2)}. ,$

Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching. Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...

Multiplication with i corresponds to a counter clockwise rotation by 90 degrees ( $π / 2$ radians). The geometric content of the equation i² = −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. A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized Â°, is a measurement of plane angle, representing 1ï¼360 of a full rotation. ... The radian (symbol: rad, or a superscript c ( half circle)) is the SI unit of plane angle. ...

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = r e^iφ is defined as |z| = r. Algebraically, if z = a + ib, then $| z | = sqrt{a^2+b^2}.$

One can check readily that the absolute value has three important properties:

$| z | = 0 ,$ iff $z = 0 ,$

$| z + w | leq | z | + | w | ,$

$| z w | = | z | ; | w | ,$

for all complex numbers z and w. It then follows, for example, that $| 1 | = 1$ and $| z / w | = | z | / | w |$ . By defining the distance function d(z, w) = |z − w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers. â†” â‡” â‰¡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Common alternative phrases to iff or if and only if include Q is necessary and sufficient for P and P... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... 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 larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...

The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as $bar{z}$ or $z^*,$ . As seen in the figure, $bar{z}$ is the "reflection" of z about the real axis. The following can be checked: In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

$overline{z+w} = bar{z} + bar{w}$

$overline{zw} = bar{z}bar{w}$

$overline{(z/w)} = bar{z}/bar{w}$

$bar{bar{z}}=z$

$bar{z}=z$ if and only if z is real

$|z|=|bar{z}|$

$|z|^2 = zbar{z}$

$z^{-1} = bar{z}|z|^{-2}$ 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. It has been suggested that this article or section be merged with Logical biconditional. ...

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); note, however, that conjugation is not differentiable (see holomorphic). 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. ...

Complex number division

Given a complex number (a + bi) which is to be divided by another complex number (c + di) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easy to derive. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. This causes the denominator to simplify into a real number:

${a + bi over c + di} = {(a + bi) (c - di) over (c + di) (c - di)} = {(ac + bd) + (bc - ad) i over c^2 + d^2}$

$= left({ac + bd over c^2 + d^2}right) + ileft( {bc - ad over c^2 + d^2} right).$

Matrix representation of complex numbers

While usually not useful, alternative representations of complex fields can give some insight into their nature. One particularly elegant representation interprets every complex number as 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 numbers or, more generally, of elements of a ring-like algebraic structure. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ...

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

with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as

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

which suggests that we should identify the real number 1 with the 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 −1.

The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a 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 described by the transpose of the matrix corresponding to z. In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or... 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. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

If the matrix elements are themselves complex numbers, then the resulting algebra is that of the quaternions. In this way, the matrix representation can be seen as a 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. ...

Geometric interpretation of the operations on complex numbers

The point X is the sum of A and B.

Choose a point in the plane which will be the origin, $0$ . Given two points A and B in the plane, their sum is the point X in the plane such that the triangles with vertices 0, A, B and X, B, A are similar. Image File history File links Complex_numbers_addition. ... Image File history File links Complex_numbers_addition. ... A triangle is one of the basic shapes of geometry: a two-dimensional figure with three vertices and three sides which are straight line segments. ... Several equivalence relations in mathematics are called similarity. ...

The point X is the product of A and B.

Choose in addition a point in the plane different from zero, which will be the unity, 1. Given two points A and B in the plane, their product is the point X in the plane such that the triangles with vertices 0, 1, A, and 0, B, X are similar. Image File history File links Complex_numbers_multiplication. ... Image File history File links Complex_numbers_multiplication. ...

The point X is the complex conjugate of A.

Given a point A in the plane, its complex conjugate is a point X in the plane such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other. Image File history File links Complex_numbers_conjugation. ... Image File history File links Complex_numbers_conjugation. ... In geometry, the mirror image of an object or two-dimensional figure is the virtual image formed by reflection in a plane mirror; it is of the same size as the original object, yet different, unless the object or figure has reflection symmetry (also known in the terminology of modern...

Some properties

Real vector space

C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, an ordered field is a field (F,+,*) together with a total order ≤ on F that is compatible with the algebraic operations in the following sense: if a ≤ b then a + c ≤ b + c if 0 ≤ a and 0 ≤ b then 0 ≤ a b It follows from these axioms...

R-linear maps C → C 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; also 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 and 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 most striking result 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 shows that the complex numbers are an algebraically closed field. In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... 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 of degree has exactly roots (zeros), counted with multiplicity. ... 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 (root) in F (i. ...

Indeed, the complex number field is the algebraic closure of the real number field, and Cauchy constructed complex numbers in this way. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X² + 1: 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 (root) in F (i. ... 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 in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... 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, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring becomes 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 a special kind of ideal which is in some sense maximal, that is not contained in any other non-trivial ideal of the ring. ...

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...

its characteristic is 0
its transcendence degree over the prime field is the cardinality of the continuum
it is algebraically closed

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: Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... 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 are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... 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 the nonzero complex numbers are connected whereas the nonzero real numbers are not. Lev Semenovich Pontryagin (Russian: Ð›ÐµÐ² Ð¡ÐµÐ¼Ñ‘Ð½Ð¾Ð²Ð¸Ñ‡ ÐŸÐ¾Ð½Ñ‚Ñ€ÑÐ³Ð¸Ð½) (3 September 1908 â€“ 3 May 1988) was a Soviet Russian mathematician. ... 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. Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... 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 that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ...

Applications

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 the application to living systems, see perceptual control theory. ... 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 mathematics, the Laplace transform is a powerful technique for analyzing linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems, to name just a few. ... In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0. ... In control theory, a root locus is the locus of the poles and zeros of a systems S-function as the system gain is varied from 0 to infinity. ... 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. ...

in the right half plane, it will be unstable,
all in the left half plane, it will be stable,
on the imaginary axis, it will have marginal stability.

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. ... The word stability has a number of technical meanings, all related to the common meaning of the word. ... 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 as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency. Signal analysis is the extraction of information from a signal. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ... 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... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ...

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 the real part of complex valued functions of the form Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

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

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ...

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 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. This article treats electronics engineering as a subfield of electrical engineering, though this is not typical use in some areas. ... International danger high voltage symbol. ... In electricity, current refers to electric current, which is the flow of electric charge. ... Resistor symbols (US and Japan) Resistor symbols (Europe) A pack of resistors A resistor is a two-terminal electrical or electronic component that resists an electric current by producing a voltage drop between its terminals in accordance with Ohms law. ... A capacitor is a device that stores energy in the electric field created between a pair of conductors on which equal magnitude but opposite sign electric charges have been placed. ... An inductor is a passive electrical device employed in electrical circuits for its property of inductance. ... In electrical engineering, Impedance is a measure of opposition to a sinusoidal electric current. ... Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. ... It has been suggested that this article or section be merged into image processing. ... In mathematics, wavelets, wavelet analysis, and the wavelet transform refers to the representation of a signal in terms of a finite length or fast decaying oscillating waveform (known as the mother wavelet). ... In computer science, data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than a more obvious representation would use, through use of specific encoding schemes. ... A digital system is one that uses numbers, especially binary numbers, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (an analog system) or non-numeric symbols such as letters or icons. ... A schematic representation of hearing. ... Look up Video in Wiktionary, the free dictionary Video is the technology of capturing, recording, processing, transmitting, and reconstructing moving pictures, typically using celluloid film, electronic signals, or digital media. ...

Improper integrals

In applied fields, the use of complex analysis is 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 also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C. A simple introduction to this subject is provided in Basics of quantum mechanics. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ...

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... It has been suggested that Einsteins theory of gravitation be merged into this article or section. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ...

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) = e^rt. 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 Lf = g, where the differential operator L is a linear operator. ...

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in 2d. Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... In fluid dynamics, potential flow, also know as irrotational flow (of incompressible fluids) is steady flow defined by the equations (zero rotation = no viscosity) (zero divergence = volume conservation) Equivalently, where: v is the vector fluid velocity Φ is the fluid flow potential, scalar × is curl · is divergence. ...

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. ... A rendering of the Mandelbrot set In mathematics, the Mandelbrot set is defined as the connectedness locus of the family of complex quadratic polynomials. ... Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. ...

History

The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Hellenized Egyptian mathematician and inventor Heron of Alexandria in the 1st century CE, when he considered the volume of an impossible frustum of a pyramid ^{[citation needed]}, though negative numbers were not conceived in the Hellenistic world. In India, the mathematician and astronomer Madhava of Sangamagrama in the 14th century gave approximations of transcendental numbers (non-algebraic complex numbers) by means of continued fractions. In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or... A negative number is a number that is less than zero, such as −3. ... The term Hellenistic (established by the German historian Johann Gustav Droysen) in the history of the ancient world is used to refer to the shift from a culture dominated by ethnic Greeks, however scattered geographically, to a culture dominated by Greek-speakers of whatever ethnicity, and from the political dominance... 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. ... Heros aeolipile Hero (or Heron) of Alexandria (c. ... The 1st century was that century which lasted from 1 to 100. ... The Common Era (CE), sometimes known as the Christian Era or Current Era, is the period of measured time beginning with the year 1 until the present. ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... Geometric shape created by connecting a polygonal base to an apex For other versions including architectural Pyramids, 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 peoples that were conquered by Alexander the Great. ... The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BC) and Vedic civilization (1500-500 BC) to modern India (21st century CE). ... Indian science has a very ancient history going back to the Vedas. ... Madhava (à¤®à¤¾à¤§à¤µ) of Sangamagrama (1350-1425) was a major mathematician from Kerala, South India. ... This 14th-century statue from south India depicts the gods Shiva (on the left) and Uma (on the right). ... In mathematics, a transcendental number is any real number that is not algebraic, that is, not the solution of a non-zero polynomial equation with integer (or, equivalently, rational) coefficients. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ...

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 $x 3 - x = 0$ : (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. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ... In mathematics and elsewhere, the adjective quartic means fourth order, such as the function . ... In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... Niccolo Fontana Tartaglia. ... Gerolamo Cardano or Jerome Cardan or Girolamo Cardan (September 24, 1501 - September 21, 1576) was a celebrated Italian Renaissance mathematician, physician, astrologer, and gambler. ...

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

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

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 (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 $frac{1}{sqrt{a}}=sqrt{frac{1}{a}}$ ) 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. For other things named Descartes, see Descartes (disambiguation). ... 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. ... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number. ... 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 Abraham de Moivre (May 26, 1667 in Vitry-le-FranÃ§ois, Champagne, France â€“ November 27, 1754 in London, England) was a French mathematician famous for de Moivres formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. ... Leonhard was the first to use the term function to describe an expression involving various arguments; i. ... De Moivres formula states that for any real number x and any integer n, 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 (pronounced oiler), is a mathematical formula in the subfield of complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...

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

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). ... (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields(is also considered as the last complete mathematician meaning he contributed to every existing field of his time), including number theory, analysis, differential geometry, geodesy, magnetism, 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. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Jean-Robert Argand was an accountant and bookkeeper in Paris who was only an amateur mathematician. ... Augustin Louis Cauchy 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 FinnÃ¸y. ...

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

The expression direction coefficient, often used for $cosφ + i sinφ$ , 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. ... Augustus De Morgan (June 27, 1806 â€“ March 18, 1871) was an Indian-born British mathematician and logician. ... August Ferdinand MÃ¶bius (November 17, 1790, Schulpforta, Saxony, Germany - September 26, 1868, Leipzig) was a German mathematician and theoretical astronomer. ... Peter Gustav Lejeune Dirichlet. ...

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 $a + b i$ , where a and b are integral, or rational (and i is one of the two roots of $x 2 + 1 = 0$ ). His student, Ferdinand Eisenstein, studied the type $a + b ω$ , where $ω$ is a complex root of $x 3 - 1 = 0$ . Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity $x k - 1 = 0$ 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 similar (but not identical) properties to those familiar from the integers. ... 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 topology and related branches of mathematics, a closed set is a set whose complement is open. ... (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields(is also considered as the last complete mathematician meaning he contributed to every existing field of his time), including number theory, analysis, differential geometry, geodesy, magnetism, 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 â€“ June 22, 1925) was a German mathematician. ... 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, Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane. Karl WeierstraÃŸ 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. // Biography 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. ... Henri PoincarÃ©, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912), generally known as Henri PoincarÃ©, 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 (1851 - 1913) was a Scottish-Canadian logician, physicist, and mathematician. ...

The formally correct definition using pairs of real numbers was given in the 19th century. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...

External links

John and Betty's Journey Through Complex Numbers
Complex Number from MathWorld
SOS Math - Complex Variables
Windows calculator that supports complex numbers
Algebraic Structure of Complex Numbers from cut-the-knot

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Contents

Definitions

Notation and operations

The complex number field

The complex plane

Absolute value, conjugation and distance

Complex number division

Matrix representation of complex numbers

Geometric interpretation of the operations on complex numbers

Some properties

Real vector space

Solutions of polynomial equations

Algebraic characterization

Characterization as a topological field

Complex analysis

Applications

Control theory

Signal analysis

Improper integrals

Quantum mechanics

Relativity

Applied mathematics

Fluid dynamics

Fractals

History

See also

Further reading

External links

Contents

Definitions GA_googleFillSlot("encyclopedia_square");

Notation and operations

The complex number field

The complex plane

Absolute value, conjugation and distance

Complex number division

Matrix representation of complex numbers

Geometric interpretation of the operations on complex numbers

Some properties

Real vector space

Solutions of polynomial equations

Algebraic characterization

Characterization as a topological field

Complex analysis

Applications

Control theory

Signal analysis

Improper integrals

Quantum mechanics

Relativity

Applied mathematics

Fluid dynamics

Fractals

History

See also

Further reading

External links

Definitions