# Complex number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i represents the imaginary unit, satisfying the equation i = −1. Complex number_sentence_0

Because no real number satisfies this equation, i is called an imaginary number. Complex number_sentence_1

For the complex number a + bi, a is called the real part, and b is called the imaginary part. Complex number_sentence_2

The set of complex numbers is denoted by either of the symbols ℂ or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world. Complex number_sentence_3

Complex numbers allow solutions to certain equations that have no solutions in real numbers. Complex number_sentence_4

For example, the equation Complex number_sentence_5

has no real solution, since the square of a real number cannot be negative. Complex number_sentence_6

Complex numbers, however, provide a solution to this problem. Complex number_sentence_7

The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) taken to satisfy the relation i = −1, so that solutions to equations like the preceding one can be found. Complex number_sentence_8

In this case, the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i = −1: Complex number_sentence_9

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. Complex number_sentence_10

In contrast, some polynomial equations with real coefficients have no solution in real numbers. Complex number_sentence_11

The 16th-century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers—in his attempts to find solutions to cubic equations. Complex number_sentence_12

Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i. Complex number_sentence_13

This means that complex numbers can be added, subtracted and multiplied as polynomials in the variable i, under the rule that i = −1. Complex number_sentence_14

Furthermore, complex numbers can also be divided by nonzero complex numbers. Complex number_sentence_15

Overall, the complex number system is a field. Complex number_sentence_16

Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane, by using the horizontal axis for the real part, and the vertical axis for the imaginary part. Complex number_sentence_17

The complex number a + bi can be identified with the point (a, b) in the complex plane. Complex number_sentence_18

A complex number whose real part is zero is said to be purely imaginary, and the points for these numbers lie on the vertical axis of the complex plane. Complex number_sentence_19

Similarly, a complex number whose imaginary part is zero can be viewed as a real number, whose point lies on the horizontal axis of the complex plane. Complex number_sentence_20

Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude), and a particular angle known as the argument of the complex number. Complex number_sentence_21

The geometric identification of the complex numbers with the complex plane, which is a Euclidean plane (ℝ), makes their structure as a real 2-dimensional vector space evident. Complex number_sentence_22

Real and imaginary parts of a complex number may be taken as components of a vector—with respect to the canonical standard basis. Complex number_sentence_23

The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. Complex number_sentence_24

However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space. Complex number_sentence_25

For example, the multiplication of two complex numbers always yields again a complex number, and should not be mistaken for the usual "products" involving vectors, like the scalar multiplication, the scalar product or other (sesqui)linear forms, available in many vector spaces; and the broadly exploited vector product exists only in an orientation-dependent form in three dimensions. Complex number_sentence_26

## Definition Complex number_section_0

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i = −1. Complex number_sentence_27

For example, 2 + 3i is a complex number. Complex number_sentence_28

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i + 1 = 0 is imposed. Complex number_sentence_29

Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. Complex number_sentence_30

The relation i + 1 = 0 induces the equalities i = 1, i = i, i = −1, and i = −i, which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form a + bi with real coefficients a, b. Complex number_sentence_31

The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. Complex number_sentence_32

To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi. Complex number_sentence_33

Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial i + 1 (see below). Complex number_sentence_34

## Notation Complex number_section_1

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. Complex number_sentence_35

A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. Complex number_sentence_36

As with polynomials, it is common to write a for a + 0i and bi for 0 + bi. Complex number_sentence_37

Moreover, when the imaginary part is negative, that is, b = −|b| < 0, it is common to write a − |b|i instead of a + (−|b|)i; for example, for b = −4, 3 − 4i can be written instead of 3 + (−4)i. Complex number_sentence_38

Since the multiplication of the indeterminate i and a real is commutative in polynomials with real coefficients, the polynomial a + bi may be written as a + ib. Complex number_sentence_39

This is often expedient for imaginary parts denoted by expressions, for example, when b is a radical. Complex number_sentence_40

The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). Complex number_sentence_41

For example, Complex number_sentence_42

The set of all complex numbers is denoted by C (upright bold) or ℂ (blackboard bold). Complex number_sentence_43

In some disciplines, particularly in electromagnetism and electrical engineering, j is used instead of i as i is frequently used to represent electric current. Complex number_sentence_44

In these cases, complex numbers are written as a + bj, or a + jb. Complex number_sentence_45

## Visualization Complex number_section_2

Main article: Complex plane Complex number_sentence_46

A complex number z can thus be identified with an ordered pair (Re(z), Im(z)) of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. Complex number_sentence_47

The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram, named after Jean-Robert Argand. Complex number_sentence_48

Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere. Complex number_sentence_49

### Cartesian complex plane Complex number_section_3

The definition of the complex numbers involving two arbitrary real values immediately suggests the use of Cartesian coordinates in the complex plane. Complex number_sentence_50

The horizontal (real) axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical (imaginary) axis, with increasing values upwards. Complex number_sentence_51

A charted number may be viewed either as the coordinatized point or as a position vector from the origin to this point. Complex number_sentence_52

The coordinate values of a complex number z can hence be expressed in its Cartesian, rectangular, or algebraic form. Complex number_sentence_53

Notably, the operations of addition and multiplication take on a very natural geometric character, when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Complex number_sentence_54

Viewed in this way, the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin—a fact which can be expressed algebraically as follows: Complex number_sentence_55

### Polar complex plane Complex number_section_4

Main article: Polar coordinate system Complex number_sentence_56

"Polar form" redirects here. Complex number_sentence_57

For the higher-dimensional analogue, see Polar decomposition. Complex number_sentence_58

#### Modulus and argument Complex number_section_5

An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (O), and the angle subtended between the positive real axis and the line segment Oz in a counterclockwise sense. Complex number_sentence_59

This leads to the polar form of complex numbers. Complex number_sentence_60

The absolute value (or modulus or magnitude) of a complex number z = x + yi is Complex number_sentence_61

If z is a real number (that is, if y = 0), then r = |x|. Complex number_sentence_62

That is, the absolute value of a real number equals its absolute value as a complex number. Complex number_sentence_63

By Pythagoras' theorem, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the complex plane. Complex number_sentence_64

The argument of z (in many applications referred to as the "phase" φ) is the angle of the radius Oz with the positive real axis, and is written as arg z. Complex number_sentence_65

As with the modulus, the argument can be found from the rectangular form x + yi—by applying the inverse tangent to the quotient of imaginary-by-real parts. Complex number_sentence_66

By using a half-angle identity, a single branch of the arctan suffices to cover the range of the arg-function, (−π, π], and avoids a more subtle case-by-case analysis Complex number_sentence_67

Normally, as given above, the principal value in the interval (−π, π] is chosen. Complex number_sentence_68

Values in the range [0, 2π) are obtained by adding 2π—if the value is negative. Complex number_sentence_69

It can increase by any integer multiple of 2π and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through z. Complex number_sentence_71

Hence, the arg function is sometimes considered as multivalued. Complex number_sentence_72

The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle 0 is common. Complex number_sentence_73

The value of φ equals the result of atan2: Complex number_sentence_74

Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Complex number_sentence_75

Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form Complex number_sentence_76

Using Euler's formula this can be written as Complex number_sentence_77

Using the cis function, this is sometimes abbreviated to Complex number_sentence_78

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as Complex number_sentence_79

### Complex graphs Complex number_section_6

Main articles: Domain coloring and Riemann surface Complex number_sentence_80

When visualizing complex functions, both a complex input and output are needed. Complex number_sentence_81

Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Complex number_sentence_82

Because of this, other ways of visualizing complex functions have been designed. Complex number_sentence_83

In domain coloring the output dimensions are represented by color and brightness, respectively. Complex number_sentence_84

Each point in the complex plane as domain is ornated, typically with color representing the argument of the complex number, and brightness representing the magnitude. Complex number_sentence_85

Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. Complex number_sentence_86

The colors often vary in steps of π/3 for 0 to 2π from red, yellow, green, cyan, blue, to magenta. Complex number_sentence_87

These plots are called color wheel graphs. Complex number_sentence_88

This provides a simple way to visualize the functions without losing information. Complex number_sentence_89

The picture shows zeros for ±1, (2+i) and poles at ±√−2−2i. Complex number_sentence_90

Riemann surfaces are another way to visualize complex functions. Complex number_sentence_91

Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. Complex number_sentence_92

However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Complex number_sentence_93

Unlike domain coloring, Riemann surfaces can represent multivalued functions like √z. Complex number_sentence_94

## History Complex number_section_7

The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). Complex number_sentence_95

This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary. Complex number_sentence_96

Work on the problem of general polynomials 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. Complex number_sentence_97

Complex numbers thus form an algebraically closed field, where any polynomial equation has a root. Complex number_sentence_98

Many mathematicians contributed to the development of complex numbers. Complex number_sentence_99

The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. Complex number_sentence_100

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 number_sentence_101

The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano). Complex number_sentence_102

It was soon realized (but proved much later) that these formulas, even if one was interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. Complex number_sentence_103

As an example, Tartaglia's formula for a cubic equation of the form x = px + q gives the solution to the equation x = x as Complex number_sentence_104

The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature Complex number_sentence_105

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. Complex number_sentence_106

For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula: Complex number_sentence_107

In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis: Complex number_sentence_108

by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities. Complex number_sentence_109

The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra. Complex number_sentence_110

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. Complex number_sentence_111

In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Complex number_sentence_112

Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". Complex number_sentence_113

It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. Complex number_sentence_114

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis. Complex number_sentence_115

The English mathematician G.H. Complex number_sentence_116 Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise. Complex number_sentence_117

Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case. Complex number_sentence_118

Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Complex number_sentence_119

## Relations and operations Complex number_section_8

### Ordering Complex number_section_10

Unlike the real numbers, there is no natural ordering of the complex numbers. Complex number_sentence_120

In particular, there is no linear ordering on the complex numbers that is compatible with addition and multiplication – the complex numbers cannot have the structure of an ordered field. Complex number_sentence_121

This is e.g. because every non-trivial sum of squares in an ordered field is ≠ 0, and i + 1 = 0 is a non-trivial sum of squares. Complex number_sentence_122

Thus, complex numbers are naturally thought of as existing on a two-dimensional plane. Complex number_sentence_123

### Conjugate Complex number_section_11

The complex conjugate of the complex number z = x + yi is given by x − yi. Complex number_sentence_125

It is denoted by either z or z*. Complex number_sentence_126

This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division. Complex number_sentence_127

Geometrically, z is the "reflection" of z about the real axis. Complex number_sentence_128

Conjugating twice gives the original complex number Complex number_sentence_129

which makes this operation an involution. Complex number_sentence_130

The reflection leaves both the real part and the magnitude of z unchanged, that is Complex number_sentence_131

The imaginary part and the argument of a complex number z change their sign under conjugation Complex number_sentence_132

For details on argument and magnitude, see the section on Polar form. Complex number_sentence_133

The product of a complex number z = x + yi and its conjugate is known as the absolute square. Complex number_sentence_134

It is always a positive real number and equals the square of the magnitude of each: Complex number_sentence_135

This property can be used to convert a fraction with a complex denominator to an equivalent fraction with a real denominator by expanding both numerator and denominator of the fraction by the conjugate of the given denominator. Complex number_sentence_136

This process is sometimes called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number), because it resembles the method to remove roots from simple expressions in a denominator. Complex number_sentence_137

The real and imaginary parts of a complex number z can be extracted using the conjugation: Complex number_sentence_138

Moreover, a complex number is real if and only if it equals its own conjugate. Complex number_sentence_139

Conjugation distributes over the basic complex arithmetic operations: Complex number_sentence_140

Conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line. Complex number_sentence_141

In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is looked for. Complex number_sentence_142

### Addition and subtraction Complex number_section_12

Two complex numbers a and b are most easily added by separately adding their real and imaginary parts of the summands. Complex number_sentence_143

That is to say: Complex number_sentence_144

Similarly, subtraction can be performed as Complex number_sentence_145

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers a and b, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices O, and the points of the arrows labeled a and b (provided that they are not on a line). Complex number_sentence_146

Equivalently, calling these points A, B, respectively and the fourth point of the parallelogram X the triangles OAB and XBA are congruent. Complex number_sentence_147

A visualization of the subtraction can be achieved by considering addition of the negative subtrahend. Complex number_sentence_148

### Multiplication Complex number_section_13

Since the real part, the imaginary part, and the indeterminate i in a complex number are all considered as numbers in themselves, two complex numbers, given as z = x + yi and w = u + vi are multiplied under the rules of the distributive property, the commutative properties and the defining property i = -1 in the following way Complex number_sentence_149

### Reciprocal and division Complex number_section_14

Using the conjugation, the reciprocal of a nonzero complex number z = x + yi can always be broken down to Complex number_sentence_150

since non-zero implies that {{math>|x + y}} is greater than zero. Complex number_sentence_151

This can be used to express a division of an arbitrary complex number w = u + vi by a non-zero complex number z as Complex number_sentence_152

### Multiplication and division in polar form Complex number_section_15

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Complex number_sentence_153

Given two complex numbers z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), because of the trigonometric identities Complex number_sentence_154

we may derive Complex number_sentence_155

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. Complex number_sentence_156

For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i = −1. Complex number_sentence_157

The picture at the right illustrates the multiplication of Complex number_sentence_158

Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). Complex number_sentence_159

On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Complex number_sentence_160

Thus, the formula Complex number_sentence_161

holds. Complex number_sentence_162

As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π. Complex number_sentence_163

Similarly, division is given by Complex number_sentence_164

### Square root Complex number_section_16

and Complex number_sentence_166

### Exponential function Complex number_section_17

which has an infinite radius of convergence. Complex number_sentence_167

The value at 1 of the exponential function is Euler's number Complex number_sentence_168

#### Euler's formula Complex number_section_19

Euler's formula states that, for any real number y, Complex number_sentence_169

The functional equation implies thus that, if x and y are real, one has Complex number_sentence_170

which is the decomposition of the exponential function into its real and imaginary parts. Complex number_sentence_171

### Complex logarithm Complex number_section_20

as complex logarithm one has a proper inverse: Complex number_sentence_172

Therefore, if the complex logarithm is not to be defined as a multivalued function Complex number_sentence_173

one has to use a branch cut and to restrict the codomain, resulting in the bijective function Complex number_sentence_174

### Exponentiation Complex number_section_21

If x > 0 is real and z complex, the exponentiation is defined as Complex number_sentence_175

where ln denotes the natural logarithm. Complex number_sentence_176

It seems natural to extend this formula to complex values of x, but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function. Complex number_sentence_177

It follows that if z is as above, and if t is another complex number, then the exponentiation is the multivalued function Complex number_sentence_178

#### Integer and fractional exponents Complex number_section_22

If, in the preceding formula, t is an integer, then the sine and the cosine are independent of k. Thus, if the exponent n is an integer, then z is well defined, and the exponentiation formula simplifies to de Moivre's formula: Complex number_sentence_179

The n nth roots of a complex number z are given by Complex number_sentence_180

While the nth root of a positive real number r is chosen to be the positive real number c satisfying c = r, there is no natural way of distinguishing one particular complex nth root of a complex number. Complex number_sentence_181

Therefore, the nth root is a n-valued function of z. Complex number_sentence_182

This implies that, contrary to the case of positive real numbers, one has Complex number_sentence_183

since the left-hand side consists of n values, and the right-hand side is a single value. Complex number_sentence_184

## Properties Complex number_section_23

### Field structure Complex number_section_24

The set C of complex numbers is a field. Complex number_sentence_185

Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Complex number_sentence_186

Second, for any complex number z, its additive inverse −z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Complex number_sentence_187

Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z1 and z2: Complex number_sentence_188

These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field. Complex number_sentence_189

Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. Complex number_sentence_190

In fact, in any ordered field, the square of any element is necessarily positive, so i = −1 precludes the existence of an ordering on C. Complex number_sentence_191

When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. Complex number_sentence_192

For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra. Complex number_sentence_193

### Solutions of polynomial equations Complex number_section_25

Given any complex numbers (called coefficients) a0, ..., an, the equation Complex number_sentence_194

has at least one complex solution z, provided that at least one of the higher coefficients a1, ..., an is nonzero. Complex number_sentence_195

This is the statement of the fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert. Complex number_sentence_196

Because of this fact, C is called an algebraically closed field. Complex number_sentence_197

This property does not hold for the field of rational numbers Q (the polynomial x − 2 does not have a rational root, since √2 is not a rational number) nor the real numbers R (the polynomial x + a does not have a real root for a > 0, since the square of x is positive for any real number x). Complex number_sentence_198

There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root. Complex number_sentence_199

Because of this fact, theorems that hold for any algebraically closed field apply to C. For example, any non-empty complex square matrix has at least one (complex) eigenvalue. Complex number_sentence_200

### Algebraic characterization Complex number_section_26

The field C has the following three properties: first, it has characteristic 0. Complex number_sentence_201

This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one). Complex number_sentence_202

Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Complex number_sentence_203

Third, it is algebraically closed (see above). Complex number_sentence_204

It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Qp also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Complex number_sentence_205

Also, C is isomorphic to the field of complex Puiseux series. Complex number_sentence_206

However, specifying an isomorphism requires the axiom of choice. Complex number_sentence_207

Another consequence of this algebraic characterization is that C contains many proper subfields that are isomorphic to C. Complex number_sentence_208

### Characterization as a topological field Complex number_section_27

The preceding characterization of C describes only the algebraic aspects of C. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. Complex number_sentence_209

The following description of C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. Complex number_sentence_210

C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions: Complex number_sentence_211

Complex number_unordered_list_0

• P is closed under addition, multiplication and taking inverses.Complex number_item_0_0
• If x and y are distinct elements of P, then either x − y or y − x is in P.Complex number_item_0_1
• If S is any nonempty subset of P, then S + P = x + P for some x in C.Complex number_item_0_2

Moreover, C has a nontrivial involutive automorphism x ↦ x* (namely the complex conjugation), such that x x* is in P for any nonzero x in C. Complex number_sentence_212

Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p − (y − x)(y − x)* ∈ P }  as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to C. Complex number_sentence_213

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 because the nonzero complex numbers are connected, while the nonzero real numbers are not. Complex number_sentence_214

## Formal construction Complex number_section_28

### Construction as ordered pairs Complex number_section_29

William Rowan Hamilton introduced the approach to define the set C of complex numbers as the set R of ordered pairs (a, b) of real numbers, in which the following rules for addition and multiplication are imposed: Complex number_sentence_215

It is then just a matter of notation to express (a, b) as a + bi. Complex number_sentence_216

### Construction as a quotient field Complex number_section_30

Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. Complex number_sentence_217

This characterization relies on the notion of fields and polynomials. Complex number_sentence_218

A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. Complex number_sentence_219

For example, the distributive law Complex number_sentence_220

must hold for any three elements x, y and z of a field. Complex number_sentence_221

The set R of real numbers does form a field. Complex number_sentence_222

A polynomial p(X) with real coefficients is an expression of the form Complex number_sentence_223

where the a0, ..., an are real numbers. Complex number_sentence_224

The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. Complex number_sentence_225

This ring is called the polynomial ring over the real numbers. Complex number_sentence_226

The set of complex numbers is defined as the quotient ring R[X]/(X + 1). Complex number_sentence_227

This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. Complex number_sentence_228

(The cosets of) 1 and X form a basis of R[X]/(X + 1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Complex number_sentence_229

Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers. Complex number_sentence_230

The quotient ring is a field, because X + 1 is irreducible over R, so the ideal it generates is maximal. Complex number_sentence_231

The formulas for addition and multiplication in the ring R[X], modulo the relation X = −1, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. Complex number_sentence_232

So the two definitions of the field C are isomorphic (as fields). Complex number_sentence_233

Accepting that C is algebraically closed, since it is an algebraic extension of R in this approach, C is therefore the algebraic closure of R. Complex number_sentence_234

### Matrix representation of complex numbers Complex number_section_31

Complex numbers a + bi can also be represented by 2 × 2 matrices that have the following form: Complex number_sentence_235

Here the entries a and b are real numbers. Complex number_sentence_236

The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices, the product being: Complex number_sentence_237

The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. Complex number_sentence_238

Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix: Complex number_sentence_239

The conjugate z corresponds to the transpose of the matrix. Complex number_sentence_240

## Complex analysis Complex number_section_32

Main article: Complex analysis Complex number_sentence_241

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 number_sentence_242

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). Complex number_sentence_243

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 number_sentence_244

### Complex exponential and related functions Complex number_section_33

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. Complex number_sentence_245

A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. Complex number_sentence_246

This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. Complex number_sentence_247

From a more abstract point of view, C, endowed with the metric Complex number_sentence_248

is a complete metric space, which notably includes the triangle inequality Complex number_sentence_249

for any two complex numbers z1 and z2. Complex number_sentence_250

Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp z, also written e, is defined as the infinite series Complex number_sentence_251

The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. Complex number_sentence_252

For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Complex number_sentence_253

Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation. Complex number_sentence_254

Euler's formula states: Complex number_sentence_255

for any real number φ, in particular Complex number_sentence_256

Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation Complex number_sentence_257

for any complex number w ≠ 0. Complex number_sentence_258

It can be shown that any such solution z – called complex logarithm of w – satisfies Complex number_sentence_259

where arg is the argument defined above, and ln the (real) natural logarithm. Complex number_sentence_260

As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. Complex number_sentence_261

The principal value of log is often taken by restricting the imaginary part to the interval (−π, π]. Complex number_sentence_262

Complex exponentiation z is defined as Complex number_sentence_263

and is multi-valued, except when ω is an integer. Complex number_sentence_264

For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above. Complex number_sentence_265

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. Complex number_sentence_266

For example, they do not satisfy Complex number_sentence_267

Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right. Complex number_sentence_268

### Holomorphic functions Complex number_section_34

A function f : C → C is called holomorphic if it satisfies the Cauchy–Riemann equations. Complex number_sentence_269

For example, any R-linear map C → C can be written in the form Complex number_sentence_270

Complex analysis shows some features not apparent in real analysis. Complex number_sentence_271

For example, any two holomorphic functions f and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Complex number_sentence_272

Meromorphic functions, functions that can locally be written as f(z)/(z − z0) with a holomorphic function f, still share some of the features of holomorphic functions. Complex number_sentence_273

Other functions have essential singularities, such as sin(1/z) at z = 0. Complex number_sentence_274

## Applications Complex number_section_35

Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Complex number_sentence_275

Some of these applications are described below. Complex number_sentence_276

### Algebraic number theory Complex number_section_40

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C. A fortiori, the same is true if the equation has rational coefficients. Complex number_sentence_277

The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Complex number_sentence_278

Compared to Q, the algebraic closure of Q, which also contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. Complex number_sentence_279

In this way, algebraic methods can be used to study geometric questions and vice versa. Complex number_sentence_280

With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem. Complex number_sentence_281

Another example are Gaussian integers, that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums of squares. Complex number_sentence_282

### Analytic number theory Complex number_section_41

Main article: Analytic number theory Complex number_sentence_283

Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. Complex number_sentence_284

This is done by encoding number-theoretic information in complex-valued functions. Complex number_sentence_285

For example, the Riemann zeta function ζ(s) is related to the distribution of prime numbers. Complex number_sentence_286

### Improper integrals Complex number_section_42

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Complex number_sentence_287

Several methods exist to do this; see methods of contour integration. Complex number_sentence_288

### Dynamic equations Complex number_section_43

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form f(t) = e. Likewise, in difference equations, the complex roots r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form f(t) = r. Complex number_sentence_289

### In applied mathematics Complex number_section_44

#### Control theory Complex number_section_45

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. Complex number_sentence_290

The system's zeros and poles are then analyzed in the complex plane. Complex number_sentence_291

The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. Complex number_sentence_292

In the root locus method, it is important whether zeros and poles are in the left or right half planes, that is, have real part greater than or less than zero. Complex number_sentence_293

If a linear, time-invariant (LTI) system has poles that are Complex number_sentence_294

Complex number_unordered_list_1

• in the right half plane, it will be unstable,Complex number_item_1_3
• all in the left half plane, it will be stable,Complex number_item_1_4
• on the imaginary axis, it will have marginal stability.Complex number_item_1_5

If a system has zeros in the right half plane, it is a nonminimum phase system. Complex number_sentence_295

#### Signal analysis Complex number_section_46

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. Complex number_sentence_296

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. Complex number_sentence_297

For a sine wave of a given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg z is the phase. Complex number_sentence_298

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 Complex number_sentence_299

and Complex number_sentence_300

where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above. Complex number_sentence_301

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. Complex number_sentence_302

Another example, relevant to the two side bands of amplitude modulation of AM radio, is: Complex number_sentence_303

### In physics Complex number_section_47

#### Electromagnetism and electrical engineering Complex number_section_48

Main article: Alternating current Complex number_sentence_304

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. Complex number_sentence_305

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. Complex number_sentence_306

This approach is called phasor calculus. Complex number_sentence_307

In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I, which is generally in use to denote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current. Complex number_sentence_308

Since the voltage in an AC circuit is oscillating, it can be represented as Complex number_sentence_309

To obtain the measurable quantity, the real part is taken: Complex number_sentence_310

#### Fluid dynamics Complex number_section_49

In fluid dynamics, complex functions are used to describe potential flow in two dimensions. Complex number_sentence_311

#### Quantum mechanics Complex number_section_50

The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. Complex number_sentence_312

The original foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics – make use of complex numbers. Complex number_sentence_313

#### Relativity Complex number_section_51

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. Complex number_sentence_314

(This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex number_sentence_315

Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity. Complex number_sentence_316

## Generalizations and related notions Complex number_section_52

The process of extending the field R of reals to C is known as the Cayley–Dickson construction. Complex number_sentence_317

It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively. Complex number_sentence_318

In this context the complex numbers have been called the binarions. Complex number_sentence_319

Just as by applying the construction to reals the property of ordering is lost, properties familiar from real and complex numbers vanish with each extension. Complex number_sentence_320

The quaternions lose commutativity, that is, x·y ≠ y·x for some quaternions x, y, and the multiplication of octonions, additionally to not being commutative, fails to be associative: (x·y)·z ≠ x·(y·z) for some octonions x, y, z. Complex number_sentence_321

Reals, complex numbers, quaternions and octonions are all normed division algebras over R. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley–Dickson construction, fail to have this structure. Complex number_sentence_322

The Cayley–Dickson construction is closely related to the regular representation of C, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis (1, i). Complex number_sentence_323

This means the following: the R-linear map Complex number_sentence_324

for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). Complex number_sentence_325

With respect to the basis (1, i), this matrix is Complex number_sentence_326

that is, the one mentioned in the section on matrix representation of complex numbers above. Complex number_sentence_327

While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Complex number_sentence_328

Any matrix Complex number_sentence_329

has the property that its square is the negative of the identity matrix: J = −I. Complex number_sentence_330

Then Complex number_sentence_331

is also isomorphic to the field C, and gives an alternative complex structure on R. This is generalized by the notion of a linear complex structure. Complex number_sentence_332

Hypercomplex numbers also generalize R, C, H, and O. Complex number_sentence_333

For example, this notion contains the split-complex numbers, which are elements of the ring R[x]/(x − 1) (as opposed to R[x]/(x + 1)). Complex number_sentence_334

In this ring, the equation a = 1 has four solutions. Complex number_sentence_335

The fields R and Qp and their finite field extensions, including C, are local fields. Complex number_sentence_336