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In mathematics, a complex number is a number of the form Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... A number is an abstract entity that represents a count or measurement. ...

where a and b are real numbers, and i is 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 may be described informally in several different ways. ... 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. If z = a + bi, the real part (a) is denoted Re(z), and the imaginary part (b) is denoted Im(z).


Complex numbers can be added, subtracted, multiplied, and divided like real numbers, but they have additional elegant properties. For example, real numbers alone do not provide a solution for every polynomial algebraic equation with real coefficients, while complex numbers do (the fundamental theorem of algebra). In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a 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 some fields (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j. It follows that complex numbers are written as a + bj. Electrical Engineers design power systems… … and complex electronic circuits. ... Electric current is by definition the flow of electric charge. ... 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 . ...

Contents

Definitions

Equality

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. That is, a + bi = c + di 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...


Notation and operations

The set of all complex numbers is usually denoted by C, or in blackboard bold by . The real numbers, R, may be regarded as "lying in" C by considering every real number as a complex: a = a + 0i. 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 i2 = −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. ...

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, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In mathematics, a Lie algebra is an algebraic structure whose main use is 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: In mathematics, 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 (the first and second elements are also known as left and right projections). ...

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. You may be looking for an Injective function, in which (f(a)=f(b)) -> a=b, or a Bijection function, which is both injective and surjective (ie. ... 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):

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. The reciprocal function: y = 1/x. ... 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 in the complex plane.
Geometric representation of z and its conjugate in the complex plane.

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). Image File history File links Imaginarynumber. ... 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). ... 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. ...


The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the polar coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z. These numbers are connected by The polar coordinate system is a two-dimensional coordinate system in which points are given by an angle and a distance from the pole, called the origin in the Cartesian coordinate system. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

Together with Euler's formula we have 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. ...

The notation cis φ is sometimes used for cos φ + i sin φ.


The complex argument of 0 is not defined by the equations above. There are two possible approaches for this case. The first is to consider arg(0) an undefined form, just like 0/0. The other is to choose some fixed value and to define arg(0) to have that value. For this approach, a conventional choice is to set arg(0) = 0.


Note that for a non-zero complex number 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 (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... The integers are commonly denoted by the above symbol. ...


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. ...

and that

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. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ...


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 °, is a measurement of plane angle, representing 1/360 of a full rotation. ... The radian is a unit of plane angle. ...


Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = r eiφ is defined as |z| = r. Algebraically, if z = a + ib, then


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

if and only if
(triangle inequality)

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) = |zw| 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. It has been suggested that this article or section be merged with Logical biconditional. ... 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. ... 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 their index increases indefinitely. ... 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 = a + ib is defined to be a - ib, written as or . As seen in the figure, 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. ...

  if and only if z is real
  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. ) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function is not 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. ...


Complex fractions

We can divide a complex number (a + bi) by another complex number (c + di) whose magnitude is non-zero in two ways. 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:

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, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, the real numbers may be described informally in several different ways. ...

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

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

and the imaginary unit i with

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. In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×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. ...

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, 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

Consider a plane. One point is the origin, 0. Another point is the unity, 1. Two intersecting planes in three-dimensional space In mathematics, a plane is a fundamental two-dimensional object. ...

X = A + B
X = A + B

Image File history File links Complex_numbers_addition. ... Image File history File links Complex_numbers_addition. ...

Addition

The sum of two points A and B is the point X = A + B such that the triangles with vertices 0, A, B and X, B, A are congruent. A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... An example of congruence. ...

X = AB
X = AB

Image File history File links Complex_numbers_multiplication. ... Image File history File links Complex_numbers_multiplication. ...

Multiplication

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.

X = A*
X = A*

Image File history File links Complex_numbers_conjugation. ... Image File history File links Complex_numbers_conjugation. ...

Conjugation

The complex conjugate of a point A is a point X = A* such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other. This article is about the Twilight Zone episode. ...


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. In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... 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...


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. ...

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

corresponds to rotations combined with scaling, while the function

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 more 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 variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... 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 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 variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... 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. ...

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 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: 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. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... 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

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... In mathematics, the real numbers may be described informally in several different ways. ... 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. ... Connected and disconnected subspaces of R². 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 R². 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. ... To meet Wikipedias quality standards, this article may require cleanup. ... 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. ...


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. In engineering and mathematics, control theory deals with the behavior of dynamical systems. ... 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, 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. ... 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. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... 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... Sine waves of various frequencies; the bottom 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 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. ...

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 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… … and complex electronic circuits. ... International safety symbol Caution, risk of electric shock (ISO 3864), colloquially known as 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, IEC) 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. ... Capacitors: SMD ceramic at top left; SMD tantalum at bottom left; through-hole tantalum at top right; through-hole electrolytic at bottom right. ... 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. ... Digital image processing is the use of computer algorithms to perform image processing on digital images. ... 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 and information theory, data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than an unencoded representation would use through use of specific encoding schemes. ... A digital system is one that uses discrete numbers, especially binary numbers, or non-numeric symbols such as letters or icons, for input, processing, transmission, storage, or display, rather than a continuous spectrum of values (an analog system). ... It has been suggested that this article be split into articles entitled Sound, Sound pressure and Sound pressure level, accessible from a disambiguation page. ... Video is the technology of electronically capturing, recording, processing, storing, transmitting, and reconstructing a sequence of still images which represent scenes in motion. ...


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. The more limited original formulations of Schrödinger and Heisenberg are also in terms of complex numbers. Fig. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... Erwin Rudolf Josef Alexander Schrödinger (August 12, 1887 – January 4, 1961) was an Austrian physicist who achieved fame for his contributions to quantum mechanics, especially the Schrödinger equation, for which he received the Nobel Prize in 1933. ... Werner Karl Heisenberg (December 5, 1901 – 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. ...


Relativity

In special and general relativity, some formulae for the metric on spacetime become simpler if one takes the time variable to be imaginary. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. ...


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 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. ... Initial image of a Mandelbrot set zoom sequence with continuously colored environment. ... 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 Greek mathematician and inventor Heron of Alexandria in the 1st century CE, when he considered the volume of an impossible frustum of a pyramid [1], though negative numbers were not conceived in the Hellenistic world. In mathematics, a square root of a number x is a number 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. ... Heros aeolipile Hero (or Heron) of Alexandria (c. ... The 1st century was that century which lasted from 1 to 100 according the Gregorian calendar. ... The Common Era (CE), is the period of measured time beginning with the year 1 on the Gregorian calendar. ... A frustum is the portion of a solid â€“ normally a cone or pyramid â€“ which lies between two parallel planes cutting the solid. ... A pyramid is any three-dimensional structure where the upper surfaces are triangular and converge on one point. ... The term Hellenistic (derived from HéllÄ“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. ...


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 x3x = 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 in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... 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. ...

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions −i, and . Substituting these in turn for into the cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3x = 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 seemed to be capriciously inconsistent with the algebraic identity , 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 ) 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 to guard against this mistake. René Descartes (March 31, 1596 – February 11, 1650), also known as Cartesius, was a noted French philosopher, mathematician, and scientist. ... 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 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. ...

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. ... 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. ...

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 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, 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 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 – May 23, 1857) was a French mathematician. ... Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...


The common terms used in the theory are chiefly due to the founders. Argand called cosφ + isinφ the direction factor, and the modulus; Cauchy (1828) called cosφ + isinφ the reduced form (l'expression réduite); Gauss used i for , 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. ... Augustus De Morgan (June 27, 1806 – March 18, 1871) was an Indian-born British mathematician and logician. ... August Ferdinand Möbius (pronounced , November 17, 1790 - September 26, 1868) 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 + 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 topology and related branches of mathematics, a closed set is a set whose complement is open. ... (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, 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, Düsseldorf, Germany – June 22, 1925, Göttingen) 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. ...

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 Weierstraß Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – 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 – 30 November 1921) was a German mathematician, known for his work in complex analysis. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Otto Ludwig Hölder (December 22, 1859 - August 29, 1937) was a mathematician born in Stuttgart, Germany. ... Jules Henri Poincaré (April 29, 1854 – 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 (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. ...


See also

In mathematics, a square root of a number x is a number 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 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. ... For other meanings, see List of topics named after Leonhard Euler In mathematical analysis, Eulers identity, named after Leonhard Euler, is the equation where is Eulers number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one... In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, octonions, biquaternions and sedenions. ... Euler redirects here. ... 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 colored environment. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... A rendering of the Riemann Sphere. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... 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 . ...

Further reading

  • An Imaginary Tale: The Story of , by Paul J. Nahin; Princeton University Press; ISBN 0-691-02795-1 (hardcover, 1998). A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
  • Numbers, by H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert; Springer; ISBN 0-387-97497-0 (hardcover, 1991). 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 mabern algebra.

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. ...

References

  1. ^ http://people.bath.ac.uk/aab20/complexnumbers.html

External links


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