See also: Complex plane

In mathematics, the complex numbers are the extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:^[1]

Every complex number can be written in the form x + iy, where x and y are real numbers called the real part and the imaginary part of the complex number, respectively. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, the real numbers may be described informally in several different ways Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e. g. , negative real numbers can be obtained by squaring complex (imaginary) numbers.

Complex numbers were first discovered by the Italian mathematician Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations ^[2]. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. In Algebra, casus irreducibilis ( Latin for "the irreducible case" is one of the cases that may arise in attempting to solve a Cubic equation This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

The rules for addition, subtraction, multiplication, and division of complex numbers were first developed by the Italian mathematician Rafael Bombelli. Rafael Bombelli (1526–1572 was an Italian Mathematician. Born in Bologna, he is the author of a treatise on Algebra and is a central figure A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician

Complex numbers are used in many different fields including applications in engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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 In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

Contents 1 Definitions 1.1 Notation 1.2 Equality 1.3 Operations 2 The field of complex numbers 2.1 The complex plane 2.2 Absolute value, conjugation and distance 2.3 Geometric interpretation of the operations on complex numbers 3 Polar form 3.1 Conversion from the polar form to the Cartesian form 3.2 Conversion from the Cartesian form to the polar form 3.3 Notation of the polar form 3.4 Multiplication, division, exponentiation, and root extraction in the polar form 4 Some properties 4.1 Matrix representation of complex numbers 4.2 Real vector space 4.3 Solutions of polynomial equations 4.4 Algebraic characterization 4.5 Characterization as a topological field 5 Complex analysis 6 Applications 6.1 Control theory 6.2 Signal analysis 6.3 Improper integrals 6.4 Quantum mechanics 6.5 Relativity 6.6 Applied mathematics 6.7 Fluid dynamics 6.8 Fractals 7 History 8 See also 9 Notes 10 References 10.1 Mathematical references 10.2 Historical references 11 Further reading 12 External links

Definitions

Notation

The set of all complex numbers is usually denoted by C, or in blackboard bold by . Blackboard bold is a Typeface style often used for certain symbols in Mathematics and Physics texts in which certain lines of the symbol (usually vertical

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

where a and b are real numbers, and i is the imaginary unit, which has the property i ² = −1. In Mathematics, the real numbers may be described informally in several different ways Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation The real number a is called the real part of the complex number, and the real number b is the imaginary part. In Mathematics, the real part of a Complex number z is the first element of the Ordered pair of Real numbers representing z In Mathematics, the imaginary part of a Complex number z is the second element of the ordered pair of Real numbers representing z

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

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

In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + jb. Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

Equality

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. ↔ In other words, if the two complex numbers are written as a + bi and c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d.

Operations

Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i ² = −1:

• Addition:
• Subtraction:
• Multiplication:
• Division:

where c and d are not both zero. In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

It is also possible to represent complex numbers as ordered pairs of real numbers, so that the complex number a + ib corresponds to (a, b). In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry In this representation, the algebraic operations have the following formulas:

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

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

Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property This complex plane is described below. In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis

The field of complex numbers

A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, The complex numbers are a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess:

• An additive identity ("zero"), 0 + 0i.
• A multiplicative identity ("one"), 1+ 0i.
• An additive inverse of every complex number. The additive inverse of a+bi is -a-bi.
• A multiplicative inverse (reciprocal) of every nonzero complex number. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which The multiplicative inverse of a+bi is

Other fields include the real numbers and the rational numbers. When each real number a is identified with the complex number a+ 0i, the field of real numbers R becomes a subfield of C.

The complex numbers C can also be characterized as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below. In Mathematics, the closure of a set S consists of all points which are intuitively "close to S " In Mathematics, an algebraic number is a Complex number that is a root of a non-zero Polynomial in one variable with rational (or In Mathematics, particularly Abstract algebra, an algebraic closure of a field K is an Algebraic extension of K that is

The complex plane

A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001) named after Jean-Robert Argand. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis Jean-Robert Argand ( July 18, 1768 - August 13, 1822) was a non-professional Mathematician. The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = re^iφ is defined as | z | = r. Algebraically, if z = x + yi, then

The absolute value has three important properties:

where if and only if

(triangle inequality)

for all complex numbers z and w. ↔ In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater These imply that | 1 | = 1 and | z / w | = | z | / | w | . By defining the distance function d(z,w) = | z − w | , we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity. In Mathematics, a metric space is a set where a notion of Distance (called a metric) 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" In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output

The complex conjugate of the complex number z = x + yi is defined to be x − yi, written as or . In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. As seen in the figure, is the "reflection" of z about the real axis, and so both and are real numbers. Many identities relate complex numbers and their conjugates:

  if and only if z is real

  if and only if z is purely imaginary

  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 complex-differentiable (see holomorphic function). Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

Geometric interpretation of the operations on complex numbers

The operations of addition, multiplication, and complex conjugation in the complex plane admit natural geometrical interpretations.

• The sum of two points A and B of the complex plane is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Geometry, two sets of points are called congruent if one can be transformed into the other by an Isometry, i

• 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. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other

• The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other. "Mirror Image" is an episode of the Television series The Twilight Zone.

These geometric interpretations allow problems of geometry to be translated into algebra. And, conversely, geometric problems can be examined algebraically. For example, the problem of the geometric construction of the 17-gon is thus translated into the analysis of the algebraic equation x¹⁷ = 1.

Polar form

Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by The polar coordinates are r =  |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. For r = 0 any value of φ describes the same number. To get a unique representation, a conventional choice is to set arg(0) = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i. e. −π < φ ≤ π. The representation of a complex number by its polar coordinates is called the polar form of the complex number.

Conversion from the polar form to the Cartesian form

Conversion from the Cartesian form to the polar form

(See arg function and atan2. In Mathematics the arg function is a logical function that extracts the angular component of a Complex number or function In Computing, atan2 is a two-argument function that makes it easy to find the angle round the origin of a point )

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

Notation of the polar form

The notation of the polar form as

is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ. Using Euler's formula it can also be written as

which is called exponential form. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

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

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

Using sum and difference identities it follows that

and that

Exponentiation with integer exponents; according to De Moivre's formula,

from which it follows that

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any

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

Multiplication by i corresponds to a counter-clockwise rotation by 90 degrees (π/2 radians). This article describes the unit of angle For other meanings see Degree. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57 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.

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

where represents the usual (positive) n-th root of the positive real number r. If c = 0, then the only n-th root of c is 0 itself, which as n-th root of 0 is considered to have multiplicity n.

Some properties

Matrix representation of complex numbers

While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, the real numbers may be described informally in several different ways Every such matrix has the form

where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. In Mathematics, commutativity is the ability to change the order of something without changing the end result Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective Every such matrix can be written as

which suggests that we should identify the real number 1 with the identity 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 the 2×2 matrix that represents −1.

The square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n

If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

Real vector space

C is a two-dimensional real vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. In Mathematics and Set theory, a total order, linear order, simple order, or (non-strict ordering is a Binary relation In Mathematics, an ordered field is a field together with a Total ordering of its elements that agrees in a certain sense with the field operations More generally, no field containing a square root of −1 can be ordered.

R-linear maps C → C have the general form

with complex coefficients a and b. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, the Cauchy-Riemann differential equations in Complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two

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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients

Indeed, the complex number field C is the algebraic closure of the real number field, and Cauchy constructed the field of complex numbers in this way. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients It can also be characterized as the quotient ring of the polynomial ring R[X] over the ideal generated by the polynomial X² + 1:

This is indeed a field because X² + 1 is irreducible over the real numbers, hence generating a maximal ideal, in R[X]. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. 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 set 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 The image of X in this quotient ring is the imaginary unit i.

Algebraic characterization

The field C is (up to field isomorphism) characterized by the following three facts:

• 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. 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 Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In the jargon of Mathematics, the statement that "Property P characterizes object X " means not simply that X has property P, but that In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients 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, a Galois group is a group associated with a certain type of Field extension. In Mathematics, the Infinite Cardinal numbers are represented by the Hebrew letter \aleph ( aleph) indexed with a subscript that runs

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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The following properties characterize C as a topological field:

• C is a field. In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are
• 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

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. In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are In Mathematics, a Dedekind cut, named after Richard Dedekind, in a Totally ordered set S is a partition of it into two non-empty In Mathematics, the real numbers may be described informally in several different ways 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 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. Lev Semenovich Pontryagin ( Russian Лев Семёнович Понтрягин ( 3 September 1908 &ndash 3 May 1988) was a In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology and related branches of Mathematics, a Topological space is called locally compact if roughly speaking each small portion of the space looks In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of

Complex analysis

For more details on this topic, see Complex analysis. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex

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. Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex 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 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). Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real 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 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, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex A three-dimensional graph is the graph of a function of two Variables f ( x, y)

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. Control theory is an interdisciplinary branch of Engineering and Mathematics, that deals with the behavior of Dynamical systems The desired output Time domain is a term used to describe the analysis of mathematical functions or physical signals with respect to Time. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency In Mathematics, the Laplace transform is one of the best known and most widely used Integral transforms It is commonly used to produce an easily soluble algebraic The system's poles and zeros are then analyzed in the complex plane. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Complex analysis, a zero of a Holomorphic function f is a Complex number a such that f ( a) = 0 The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. In Control theory, the root locus is the locus of the poles and zeros of a Transfer function as the System gain K is varied A Nyquist plot is used in automatic control and Signal processing for assessing the stability of a system with Feedback. A Nichols plot is a graph used in Signal processing in which the Logarithm of the magnitude is plotted against the phase of a Frequency response on

In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i. In Complex analysis, a pole of a Meromorphic function is a certain type of singularity that behaves like the singularity at z = 0 In Complex analysis, a zero of a Holomorphic function f is a Complex number a such that f ( a) = 0 e. have real part greater than or less than zero. If a system has poles that are

• 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. Instability in systems is generally characterized by some of the Outputs or internal states growing without Bounds. Bibo redirects here For the Egyptian football player nicknamed Bibo see Mahmoud El-Khateeb. In the theory of Dynamical systems, and Control theory, a continuous linear Time-invariant system is marginally stable If and only if

If a system has zeros in the right half plane, it is a nonminimum phase system. In Control theory and Signal processing, a linear time-invariant system is said to be minimum-phase if the system and its inverse are causal

Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg(z) the phase. Frequency is a measure of the number of occurrences of a repeating event per unit Time. Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0

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

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series Do not confuse with Angular velocity In Physics (specifically Mechanics and Electrical engineering) angular frequency

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Electrical tension (or voltage after its SI unit, the Volt) is the difference of electrical potential between two points of an electrical Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. 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. |- align = "center"| |width = "25"| | |- align = "center"| || Potentiometer |- align = "center"| | | |- align = "center"| Resistor| | A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors An inductor is a passive electrical component designed to provide Inductance in a circuit Electrical impedance, or simply impedance, describes a measure of opposition to a sinusoidal Alternating current (AC (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i. ) This approach is called phasor calculus. In Physics and Engineering, a phase vector ("phasor" is a representation of a Sine wave whose amplitude ( A) phase ( θ) 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. Digital signal processing ( DSP) is concerned with the representation of the signals by a sequence of numbers or symbols and the processing of these signals Digital image processing is the use of computer Algorithms to perform Image processing on Digital images As a subfield of Digital signal processing A wavelet is a mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution A digital system uses discrete (discontinuous values usually but not always Symbolized Numerically (hence called "digital" to represent information for Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies Video is the technology of electronically capturing, Recording, processing storing transmitting and reconstructing a sequence of Still images

Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. In Calculus, an improper integral is the limit of a Definite integral as an endpoint of the interval of integration approaches either a specified Several methods exist to do this; see methods of contour integration. In complex analysis contour integration is a method of evaluating certain Integrals along paths in the complex plane

Quantum mechanics

The complex number field is relevant in the mathematical formulation of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of Quantum mechanics. This article assumes some familiarity with Analytic geometry and the concept of a limit. The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Matrix mechanics is a formulation of Quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS (This is no longer standard in classical relativity, but is used in an essential way in quantum field theory. In Physics, Wick rotation, named after Gian-Carlo Wick, is a method of finding a solution to a problem in Minkowski space from a solution to a related problem In quantum field theory (QFT the forces between particles are mediated by other particles ) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity. In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually

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. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Linear algebra, one associates a Polynomial to every Square matrix, its characteristic polynomial. In Mathematics, a linear differential equation is a Differential equation of the form Ly = f \ where the Differential

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in two dimensions. Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion

Fractals

Certain fractals are plotted in the complex plane e. A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" g. Mandelbrot set and Julia set. In Mathematics, the Mandelbrot set, named after Benoît In Complex dynamics, the Julia set J(f\ of a Holomorphic function f\ informally consists of those points whose long-time behavior under

History

The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid,^[3] though negative numbers were not conceived in the Hellenistic world. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose A negative number is a Number that is less than zero, such as −2 Greece (Ελλάδα transliterated: Elláda, historically, Ellás,) officially the Hellenic Republic (Ελληνική Δημοκρατία 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 Hero (or Heron) of Alexandria ( Ήρων ο Αλεξανδρεύς) (c The 1st century was the Century that lasted from 1 to 100 according the Julian calendar. Elements special cases and related concepts Each plane section is a base of the frustum A pyramid is a Building where the upper surfaces are triangular and converge on one point This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period.

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). This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In Mathematics, a quartic equation is one which can be expressed as a Quartic function equalling zero In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Niccolò Fontana Tartaglia (1499/1500 Brescia, Italy &ndash December 13, 1557, Venice, Italy was a Mathematician It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0:

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 in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x³ – x = 0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). Geometric interpretation Geometrically imaginary numbers are found on the vertical axis of the complex number plane 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.

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system "Moivre" redirects here for the French commune see Moivre Marne. To de Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:

and to Euler (1748) Euler's formula of complex analysis:

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. De Moivre's formula, named after Abraham de Moivre, states that for any Complex number (and in particular for any Real number) x and any This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex Caspar Wessel ( June 8, 1745 - March 25, 1818) was a Danish-Norwegian Mathematician. Year 1799 ( MDCCXCIX) was a Common year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus. John Wallis ( November 23, 1616 - October 28, 1703) was an English mathematician who is given partial credit for the

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. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician 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. Jean-Robert Argand ( July 18, 1768 - August 13, 1822) was a non-professional Mathematician. 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. Niels Henrik Abel (August 5 1802 &ndash April 6 1829 was a noted Norwegian Mathematician who proved the impossibility of solving the Quintic equation

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). Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) was a German Mathematician. Leopold Kronecker ( December 7, 1823 – December 29, 1891) was a German Mathematician and Logician who argued Arturo Dell'Acqua Bellavitis (b 1947 is the current director of the Milan Triennale Foundation and Exposition Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. 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. August Ferdinand Möbius ( November 17, 1790 &ndash September 26, 1868, (ˈmøbiʊs was a German Mathematician and Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician , as in the case of real numbers.

A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set 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). Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German His student, Ferdinand Eisenstein, studied the type a + bω, where ω is a complex root of x³ − 1 = 0. Ferdinand Gotthold Max Eisenstein ( 16 April, 1823 – 11 October, 1852) was a German Mathematician. 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. In Number theory, a cyclotomic field is a Number field obtained by adjoining a complex Root of unity to Q, the field of Rational numbers In Mathematics, the n th roots of unity, or de Moivre numbers are all the Complex numbers that yield 1 when raised to a given power This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. Ernst Eduard Kummer ( 29 January 1810 - 14 May 1893) 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 Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Bonaventure Berloty, Henri Poincaré, Eduard Study, and Alexander MacFarlane. Karl Theodor Wilhelm Weierstrass ( Weierstraß) ( October 31, 1815 &ndash February 19, 1897) was a German mathematician See also Hermann Schwarz (philosopher (1864&ndash1951 A different Hermann Schwarz was a founder of Rohde & Schwarz, a German manufacturer Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Otto Ludwig Hölder ( December 22, 1859 - August 29, 1937) was a German Mathematician born in Stuttgart. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Eduard Study ( March 23, 1862 &ndash January 6, 1930) was a German Mathematician known for work on Invariant theory Alexander Macfarlane ( April 21 1851 – August 28, 1913) was a Scottish - Canadian Logician Physicist

See also

Notes

References

Mathematical references

• Ahlfors, Lars (1979), Complex analysis (3rd ed. Lars Valerian Ahlfors ( April 18, 1907 – October 11, 1996) was a Finnish Mathematician, remembered for his work in the ), McGraw-Hill, ISBN 978-0070006577 
• Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0-387-90328-3 
• Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0 
• Solomentsev, E. Dan Pedoe (1910&ndash1998 was an English-born mathematician and Geometer with a career spanning more than sixty years D. (2001), “Complex number”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

Historical references

• Nahin, Paul J. The Encyclopaedia of Mathematics is a large reference work in Mathematics. (1998), An Imaginary Tale: The Story of (hardcover ed. ), Princeton University Press, ISBN 0-691-02795-1 

A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

• H. -D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel, R. Remmert (1991), Numbers (hardcover ed. ), Springer, ISBN 0-387-97497-0 

An advanced perspective on the historical development of the concept of number.

Further reading

• The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
• Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
• Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-198-53447-7 (hardcover, 1997). Tristan Needham is the author of the highly originalbook Visual Complex Analysis in which he uses a geometric approach to develop Complex analysis (which he says was History of complex numbers and complex analysis with compelling and useful visual interpretations.

External links

• Euler's work on Complex Roots of Polynomials at Convergence
• John and Betty's Journey Through Complex Numbers
• Eric W. Weisstein, Complex Number at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• "Argand Diagram" by Eric W. Weisstein, The Wolfram Demonstrations Project, 2007, and Argand Diagram on Weisstein's MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• SOS Math - Complex Variables
• Algebraic Structure of Complex Numbers from cut-the-knot
• Complex Numbers Module by John H. Mathews
• IMO Compendium Training Materials contains a text on applications of complex numbers to euclidean geometry
• myElectrical.com Complex Number Ccalculator
• solvemymath.com Complex Numbers Calculator
• Interactive Visual Representation of Complex Numbers

Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in Mathematics.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic