A complex number, in mathematics, is a number comprising a real number part and an imaginary number part; it is normally written in the form a + bi, where a and b are real numbers, and i is the square root of minus one.^[1]

Complex numbers are a field in mathematics, with specific notions of addition, subtraction, multiplication and division, satisfying certain axioms. These operations extend the corresponding operations on real numbers, mainly because the product of two imaginary numbers (or the square of one imaginary number) is a negative real number.

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.^[2] The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.^[3] 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.

Complex numbers are used in a number of fields, including: engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

Contents 1 Definitions 1.1 Notation 1.2 Formal development 1.3 Equality 1.4 Operations 1.5 Elementary functions 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 Construction and 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 .

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

where a and b are real numbers, and i is the imaginary unit, which has the property i ² = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part.

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 of 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 + bj or a + jb.

[ Formal development

In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is −1. The definition must therefore be a little less intuitive, building on the knowledge of real numbers. Write C for R², the set of ordered pairs of real numbers, and define operations on complex numbers in C according to

(a, b) + (c, d) = (a + c, b + d)

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

Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. We can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). It is then just a matter of notation to express (a, b) as a + ib.

[ Equality

Two complex numbers are said to be 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. This is obtained by multiplying both the numerator and the denominator with the complex conjugate of the denominator, that is (c − di).

Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. This complex plane is described below.

[ Elementary functions

There are also other elementary functions that can be applied to complex functions. The most important is perhaps the exponential function exp(z), defined in terms of the infinite series

The elementary functions are, loosely, those which can be built using exp and the arithmetic operations given above, as well as taking inverses; in particular, the inverse of the exponential function, the logarithm.

The real-valued logarithm over the positive reals is well-defined (written ln), and the complex logarithm generalises this idea. However, is a multivalued function, unique only up to a multiple of 2πi. so the principal value is often taken by restricting the complex part to the interval (−π,π], giving

where Arg is the principal argument.

The familiar trigonometric functions are composed of these, so are also elementary. For example,

Hyperbolic functions are similarly constructed.

[ The field of complex numbers

A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers form 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. The multiplicative inverse of a + bi is

Examples of other fields are 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.

[ The complex plane

A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001) named after Jean-Robert Argand. The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates. The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.

[ 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. 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.

The complex conjugate of the complex number z = x + yi is defined to be x − yi, written as or . 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 distributes over 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).

[ Geometric interpretation of the operations on complex numbers

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

• 

  

  X = A + B
  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. Thus the addition of two complex numbers is the same as vector addition of two vectors.

• 

  

  X = AB
  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*
  The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other.

These geometric interpretations allow problems of algebra to be translated into geometry. And, conversely, geometric problems can be examined algebraically. For example, the problem of the geometric construction of the 17-gon was by Gauss translated into the analysis of the algebraic equation x¹⁷ = 1. (See Heptadecagon).

[ Polar form

Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. The polar coordinates are r = |z| ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. The representation of a complex number by its polar coordinates is called the polar form of the complex number.

For r = 0 any value of φ describes the same complex number z = 0. To get a unique representation, a conventional choice is to set φ = 0. For r > 0 the argument φ is unique modulo 2π; that is, if any two values of the complex argument differ by an exact integer multiple of 2π, they are considered equivalent. To get a unique representation, a conventional choice is to limit φ to the interval (-π,π], i.e. −π < φ ≤ π. This choice of φ is sometimes called the principal value of arg(z).

[ Conversion from the polar form to the Cartesian form

[ Conversion from the Cartesian form to the polar form

(taking the sign appropriately so that z = e^iφ).

The value of φ can change by any multiple of 2π and still give the same angle. In applied contexts, such as computing, the principal value in the range (−π, +π] of the arg function may be written as atan2. The arg function is sometimes considered as multivalued taking as possible values Arg(z) + 2πk, where k is any integer.

[ 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.

In electronics it is common to use angle notation to represent a phasor with amplitude A and phase θ as:

where θ may be in either radians or degrees. In electronics j is used instead of i because i is used for electric current.

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

Multiplication, division, exponentiation, and root extraction have simple formulas in polar form.

Using sum and difference identities it follows that

and that

Geometrically, multiplication and division of complex numbers is respectively a counterclockwise and clockwise rotation on the argand plane.

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.

Multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching, in particular multiplication by 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.

If c is a complex number and n a positive integer, then any complex number z satisfying zⁿ = c is called an n-th root of c. If c is nonzero, there are exactly n distinct n-th roots of c, which can be found as follows. Write c = re ^iφ 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. 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. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. Every such matrix can be written as

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.

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.

If the matrix elements are themselves complex numbers, the resulting algebra is that of the quaternions. In other words, this matrix representation is one way of expressing the Cayley-Dickson construction of algebras.

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.

While the above is a representation of C in the real matrices (2 x 2), it is not the only one. Any matrix

has the property that its square is the negative of the identity matrix. Then is also isomorphic to the field C.

[ Real vector space

C is a two-dimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. More generally, no field containing a square root of −1 can be ordered.

R-linear maps C → C have the general form

with complex coefficients a and b. Only the first term is C-linear, and only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

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 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. Indeed, the complex numbers are the algebraic closure of the real numbers, as described below.

[ Construction and algebraic characterization

One construction of C is as a field extension of the field R of real numbers, in which a root of x²+1 is added. To construct this extension, begin with the polynomial ring R[x] of the real numbers in the variable x. Because the polynomial x²+1 is irreducible over R, the quotient ring R[x]/(x²+1) will be a field. This extension field will contain two square roots of -1; one of them is selected and denoted i. The set {1, i} will form a basis for the extension field over the reals, which means that each element of the extension field can be written in the form a+ b·i. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers.

Although only roots of x²+1 were explicitly added, the resulting complex field is actually algebraically closed – every polynomial with coefficients in C factors into linear polynomials with coefficients in C. Because each field has only one algebraic closure, up to field isomorphism, the complex numbers can be characterized as the algebraic closure of the real numbers.

The field extension does yield the well-known complex plane, but it only characterizes it algebraically. The field C is characterized up to field isomorphism by the following three properties:

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

One consequence of this characterization is that C contains many proper subfields which are isomorphic to C (the same is true of R, which contains many subfields isomorphic to itself^{[citation needed]}). As described below, topological considerations are needed to distinguish these subfields from the fields C and R themselves.

[ Characterization as a topological field

As just noted, the algebraic characterization of C fails to capture some of its most important topological properties. These properties are key for the study of complex analysis, where the complex numbers are studied as a topological field.

The following properties characterize C as a topological field:^{[citation needed]}

• 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 a field with these properties, one can define a topology by taking the sets

• 

as a base, where x ranges over the field and p ranges over P.

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.

Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting that the nonzero complex numbers are connected, while the nonzero real numbers are not.

[ 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.

[ Applications

Some applications of complex numbers are:

[ Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using thee Laplace transform. The system's poles and zeros Source: this wikipedia article, under GFDL.