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 satisfies the equation i² = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted using the symbol ${\displaystyle \mathbb {C} }$. 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.[note 1][1][2][3][4]

A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i satisfies i² = −1.

Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

${\displaystyle (x+1)^{2}=-9}$

has no real solution, since the square of a real number cannot be negative. Complex numbers, however, provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) taken to satisfy the relation i² = −1, so that solutions to equations like the preceding one can be found. In this case, the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i² = −1:

${\displaystyle ((-1+3i)+1)^{2}=(3i)^{2}=\left(3^{2}\right)\left(i^{2}\right)=9(-1)=-9,}$

${\displaystyle ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}\left(i^{2}\right)=9(-1)=-9.}$

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. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th-century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers—in his attempts to find solutions to cubic equations.[5]

Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.[6] This means that complex numbers can be added, subtracted and multiplied as polynomials in the variable i, under the rule that i² = −1. Furthermore, complex numbers can also be divided by nonzero complex numbers.[3] Overall, the complex number system is a field.

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

The geometric identification of the complex numbers with the complex plane, which is a Euclidean plane (${\displaystyle \mathbb {R} ^{2}}$), makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector—with respect to the canonical standard basis. The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space. 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.

Definition

An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.

Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i² = −1. For example, 2 + 3i is a complex number.[7][3]

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i² + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation i² + 1 = 0 induces the equalities i^4k = 1, i^4k+1 = i, i^4k+2 = −1, and i^4k+3 = −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.

The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.[8][9][3]

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 i² + 1 (see below).[6]

Notation

A real number a can be regarded as a complex number a + 0i, whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi, whose real part is zero. As with polynomials, it is common to write a for a + 0i and bi for 0 + bi. 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.

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. This is often expedient for imaginary parts denoted by expressions, for example, when b is a radical.[10]

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).[1] For example,

${\displaystyle \operatorname {Re} (2+3i)=2\quad }$ and ${\displaystyle \quad \operatorname {Im} (2+3i)=3.}$

The set of all complex numbers is denoted by ${\displaystyle \mathbf {C} }$ (upright bold) or ${\displaystyle \mathbb {C} }$ (blackboard bold).[1]

In some disciplines, particularly in electromagnetism and electrical engineering, j is used instead of i as i is frequently used to represent electric current.[11] In these cases, complex numbers are written as a + bj, or a + jb.

Visualization

Main article: Complex plane

A complex number z, as a point (red) and its position vector (blue)

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. The most immediate space is the Euclidean plane with suitable coordinates, which is then called complex plane or Argand diagram,[12][13][14] named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.

Cartesian complex plane

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

A charted number may be either viewed as the coordinatized point, or as a position vector from the origin to this point. The coordinate values of a complex number z can hence be expressed its Cartesian, rectangular, or algebraic form.

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. 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:

${\displaystyle (a+bi)\cdot i=ai+b(i)^{2}=-b+ai.}$

Polar complex plane

Main article: Polar coordinate system

"Polar form" redirects here. For the higher-dimensional analogue, see Polar decomposition.

Argument φ and modulus r locate a point in the complex plane.

Modulus and argument

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. This leads to the polar form of complex numbers.

The absolute value (or modulus or magnitude) of a complex number z = x + yi is[15]

${\displaystyle r=|z|={\sqrt {x^{2}+y^{2}}}.}$

If z is a real number (that is, if y = 0), then r = |x|. That is, the absolute value of a real number equals its absolute value as a complex number.

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.

The argument of z (in many applications referred to as the "phase" φ)[14] is the angle of the radius Oz with the positive real axis, and is written as ${\displaystyle \arg(z)}$. As with the modulus, the argument can be found from the rectangular form ${\displaystyle x+yi}$[16]—by applying the inverse tangent to the quotient of imaginary-by-real parts. 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

${\displaystyle \varphi =\arg(x+yi)={\begin{cases}2\arctan \left({\dfrac {y}{{\sqrt {x^{2}+y^{2}}}+x}}\right)&{\text{if }}x>0{\text{ or }}y\neq 0,\\\pi &{\text{if }}x<0{\text{ and }}y=0,\\{\text{undefined}}&{\text{if }}x=0{\text{ and }}y=0.\end{cases}}}$

Normally, as given above, the principal value in the interval (−π, π] is chosen. Values in the range [0, 2π) are obtained by adding 2π—if the value is negative. The value of φ is expressed in radians in this article. 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. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the polar angle 0 is common.

The value of φ equals the result of atan2:

${\displaystyle \varphi =\operatorname {atan2} \left(\operatorname {Im} (z),\operatorname {Re} (z)\right).}$

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

${\displaystyle z=r(\cos \varphi +i\sin \varphi ).}$

Using Euler's formula this can be written as

${\displaystyle z=re^{i\varphi }.}$

Using the cis function, this is sometimes abbreviated to

${\displaystyle z=r\operatorname {cis} \varphi .}$

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as[17]

${\displaystyle z=r\angle \varphi .}$

Complex graphs

Main articles: Domain coloring and Riemann surface

A color wheel graph of the expression (z² − 1)(z − 2 − i)²/z² + 2 + 2i

When visualizing complex functions, both a complex input and output are needed. 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. Because of this, other ways of visualizing complex functions have been designed.

In domain coloring the output dimensions are represented by color and brightness, respectively. 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. Dark spots mark moduli near zero, brighter spots are farther away from the origin, the gradation may be discontinuous, but is assumed as monotonous. The colors often vary in steps of π/3 for 0 to 2π from red, yellow, green, cyan, blue, to magenta. These plots are called color wheel graphs. This provides a simple way to visualize the functions without losing information. The picture shows zeros for ±1, (2+i) and poles at ±√−2−2i.

Riemann surfaces are another way to visualize complex functions. 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. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like ${\displaystyle {\sqrt {z}}}$.

History

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). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545,[18] though his understanding was rudimentary.

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

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.[19] 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.[20]

The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term ${\displaystyle {\sqrt {81-144}}=3i{\sqrt {7}}}$ in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive (${\displaystyle {\sqrt {144-81}}=3{\sqrt {7}}}$).[21]

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). It was soon realized (but proved much later)[22] that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form ${\displaystyle x^{3}=px+q}$[note 2] gives the solution to the equation x³ = x as

${\displaystyle {\tfrac {1}{\sqrt {3}}}\left(\left({\sqrt {-1}}\right)^{1/3}+\left({\sqrt {-1}}\right)^{-1/3}\right).}$

At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation z³ = i has solutions −i, ${\displaystyle {\tfrac {\sqrt {3}}{2}}+{\tfrac {1}{2}}i}$ and ${\displaystyle {\tfrac {-{\sqrt {3}}}{2}}+{\tfrac {1}{2}}i}$. Substituting these in turn for ${\displaystyle {{\sqrt {-1}}^{1/3}}}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x³ − x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously,[22] the use of complex numbers is unavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.

The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature[23]

[...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine. ([...] quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.)

A further source of confusion was that the equation ${\displaystyle {\sqrt {-1}}^{2}={\sqrt {-1}}{\sqrt {-1}}=-1}$ seemed to be capriciously inconsistent with the algebraic identity ${\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}}}$, which is valid for non-negative 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 ${\displaystyle {\tfrac {1}{\sqrt {a}}}={\sqrt {\tfrac {1}{a}}}}$) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of √−1 to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

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. 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:

${\displaystyle (\cos \theta +i\sin \theta )^{n}=\cos n\theta +i\sin n\theta .}$

In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:[24]

${\displaystyle \cos \theta +i\sin \theta =e^{i\theta }}$

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

The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799,[25] although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.[26]

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra.[27] 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".[28] It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane,[29][30] largely establishing modern notation and terminology.

If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, √−1 positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. - Gauss[29][30]

In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée,[31][32] Mourey,[33] Warren,[34] Français and his brother, Bellavitis.[35][36]

The English mathematician G.H. 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.[37]

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.

The common terms used in the theory are chiefly due to the founders. Argand called ${\displaystyle \cos \phi +i\sin \phi }$ the direction factor, and ${\displaystyle r={\sqrt {a^{2}+b^{2}}}}$ the modulus;[38] Cauchy (1821) called ${\displaystyle \cos \phi +i\sin \phi }$ the reduced form (l'expression réduite)[39] and apparently introduced the term argument; Gauss used i for ${\displaystyle {\sqrt {-1}}}$,[40] introduced the term complex number for a + bi,[41] and called a² + b² the norm.[42] The expression direction coefficient, often used for ${\displaystyle \cos \phi +i\sin \phi }$, is due to Hankel (1867),[43] and absolute value, for modulus, is due to Weierstrass.

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

Relations and operations

Equality

Two complex numbers are equal if and only if both their real and imaginary parts are equal. That is, complex numbers ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$ are equal if and only if ${\displaystyle \operatorname {Re} (z_{1})=\operatorname {Re} (z_{2})}$ and ${\displaystyle \operatorname {Im} (z_{1})=\operatorname {Im} (z_{2})}$. Nonzero complex numbers written in polar form are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of 2π.

Ordering

Since complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers. In fact, 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. This is because any square in an ordered field is at least 0, but i² = −1.

Conjugate

See also: Complex conjugate

Geometric representation of z and its conjugate ${\displaystyle {\overline {z}}}$ in the complex plane

The complex conjugate of the complex number z = x + yi is given by x − yi. It is denoted by either ${\displaystyle {\overline {z}}}$ or z*.[44] This unary operation on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.

Geometrically, ${\displaystyle {\overline {z}}}$ is the "reflection" of z about the real axis. Conjugating twice gives the original complex number

${\displaystyle {\overline {\overline {z}}}=z,}$

which makes this operation an involution. The reflection leaves both the real part and the magnitude of ${\displaystyle z}$ unchanged, that is

${\displaystyle \operatorname {Re} ({\overline {z}})=\operatorname {Re} (z)\quad }$ and ${\displaystyle \quad |{\overline {z}}|=|z|.}$

The imaginary part and the argument of a complex number ${\displaystyle z}$ change their sign under conjugation

${\displaystyle \operatorname {Im} ({\overline {z}})=-\operatorname {Im} (z)\quad }$ and ${\displaystyle \quad \operatorname {arg} ({\overline {z}})\equiv -\operatorname {arg} (z){\pmod {2\pi }}.}$

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

The product of a complex number ${\displaystyle z=x+yi}$ and its conjugate is known as the absolute square. It is always a positive real number and equals the square of the magnitude of each:

${\displaystyle z\cdot {\overline {z}}=x^{2}+y^{2}=|z|^{2}=|{\overline {z}}|^{2}.}$

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. 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.

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

${\displaystyle \operatorname {Re} (z)={\dfrac {z+{\overline {z}}}{2}},\quad }$ and ${\displaystyle \quad \operatorname {Im} (z)={\dfrac {z-{\overline {z}}}{2i}}.}$

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

Conjugation distributes over the basic complex arithmetic operations:

${\displaystyle {\overline {z\pm w}}={\overline {z}}\pm {\overline {w}},}$

${\displaystyle {\overline {z\cdot w}}={\overline {z}}\cdot {\overline {w}},\quad {\overline {z/w}}={\overline {z}}/{\overline {w}}.}$

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

Addition and subtraction

Addition of two complex numbers can be done geometrically by constructing a parallelogram.

Two complex numbers ${\displaystyle a}$ and ${\displaystyle b}$ are most easily added by separately adding their real and imaginary parts of the summands. That is to say:

${\displaystyle a+b=(x+yi)+(u+vi)=(x+u)+(y+v)i.}$

Similarly, subtraction can be performed as

${\displaystyle a-b=(x+yi)-(u+vi)=(x-u)+(y-v)i.}$

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers ${\displaystyle a}$ and ${\displaystyle b}$, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices ${\displaystyle O}$, and the points of the arrows labeled ${\displaystyle a}$ and ${\displaystyle b}$ (provided that they are not on a line). Equivalently, calling these points ${\displaystyle A,\;B,}$ respectively and the fourth point of the parallelogram ${\displaystyle X,}$ the triangles ${\displaystyle OAB}$ and ${\displaystyle XBA}$ are congruent. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.

Multiplication

Since the real part, the imaginary part, and the indeterminate ${\displaystyle i}$ in a complex number are all considered as numbers in themselves, two complex numbers, given as ${\displaystyle z=x+yi}$ and ${\displaystyle w=u+vi}$ are multiplied under the rules of the distributive property, the commutative properties and the defining property ${\displaystyle i^{2}=-1}$ in the following way

${\displaystyle {\begin{aligned}z\cdot w&=(x+yi)\cdot (u+vi)&\\&=x(u+vi)+yi(u+vi)&&{\text{by the (right) distributive law}}\\&=xu+xvi+yiu+yivi&&{\text{by the (left) distributive law}}\\&=xu+yivi+xvi+yiu&&{\text{by the commutativity of addition}}\\&=xu+yvi^{2}+xvi+yui&&{\text{by the commutativity of multiplication}}\\&=(xu+yvi^{2})+(xvi+yui)&&{\text{by the associativity of addition}}\\&=(xu-yv)+(xvi+yui)&&{\text{by the defining property of }}i\\&=(xu-yv)+(xv+yu)i&&{\text{by the distributive law}}.\end{aligned}}}$

Reciprocal and division

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

${\displaystyle {\frac {1}{z}}={\frac {\overline {z}}{z{\overline {z}}}}={\frac {\overline {z}}{|z|^{2}}}={\frac {\overline {z}}{x^{2}+y^{2}}}={\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i,}$

since non-zero implies that ${\displaystyle x^{2}+y^{2}}$ is greater than zero.

This can be used to express a division of an arbitrary complex number ${\displaystyle w=u+vi}$ by a non-zero complex number ${\displaystyle z}$ as

${\displaystyle {\frac {w}{z}}=w\cdot {\frac {1}{z}}=(u+vi)\cdot \left({\frac {x}{x^{2}+y^{2}}}-{\frac {y}{x^{2}+y^{2}}}i\right)={\frac {1}{x^{2}+y^{2}}}\left((ux+vy)+(vx-uy)i\right).}$

Multiplication and division in polar form

Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by √5, the length of the hypotenuse of the blue triangle.

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z₁ = r₁(cos φ₁ + i sin φ₁) and z₂ = r₂(cos φ₂ + i sin φ₂), because of the trigonometric identities

${\displaystyle \cos(a)\cos(b)-\sin(a)\sin(b)=\cos(a+b)}$

${\displaystyle \cos(a)\sin(b)+\sin(a)\cos(b)=\sin(a+b)}$

we may derive

${\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).}$

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i² = −1. The picture at the right illustrates the multiplication of

${\displaystyle (2+i)(3+i)=5+5i.}$

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

${\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)}$

holds. 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.

Similarly, division is given by

${\displaystyle {\frac {z_{1}}{z_{2}}}={\frac {r_{1}}{r_{2}}}\left(\cos(\varphi _{1}-\varphi _{2})+i\sin(\varphi _{1}-\varphi _{2})\right).}$

Square root

See also: Square roots of negative and complex numbers

The square roots of a + bi (with b ≠ 0) are ${\displaystyle \pm (\gamma +\delta i)}$, where

${\displaystyle \gamma ={\sqrt {\frac {a+{\sqrt {a^{2}+b^{2}}}}{2}}}}$

and

${\displaystyle \delta =\operatorname {sgn}(b){\sqrt {\frac {-a+{\sqrt {a^{2}+b^{2}}}}{2}}},}$

where sgn is the signum function. This can be seen by squaring ${\displaystyle \pm (\gamma +\delta i)}$ to obtain a + bi.[45][46] Here ${\displaystyle {\sqrt {a^{2}+b^{2}}}}$ is called the modulus of a + bi, and the square root sign indicates the square root with non-negative real part, called the principal square root; also ${\displaystyle {\sqrt {a^{2}+b^{2}}}={\sqrt {z{\overline {z}}}},}$ where ${\displaystyle z=a+bi.}$[47]

Exponential function

The exponential function ${\displaystyle \exp(z)}$ can be defined for every complex number z by the power series

${\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}},}$

which has an infinite radius of convergence.

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

${\displaystyle e=\exp(1)=\sum _{n=1}^{\infty }{\frac {1}{n!}}\approx 2.71828.}$

If z is real, one has ${\displaystyle \exp(z)=e^{z}.}$ Analytic continuation allows extending this equality for every complex value of z, and thus to define the complex exponentiation with base e as

${\displaystyle e^{z}=\exp(z).}$

Functional equation

The exponential function satisfies the functional equation ${\displaystyle e^{z+t}=e^{z}e^{t}.}$ This can be proved either by comparing the power series expansion of both members or by applying analytic continuation from the restriction of the equation to real arguments.

Euler's formula

Euler's formula states that, for any real number x,

${\displaystyle e^{ix}=\cos x+i\sin x.}$

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

${\displaystyle e^{x+iy}=e^{x}\cos y+ie^{x}\sin y,}$

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

Exponentiation

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

${\displaystyle x^{z}=e^{z\ln x},}$

where ln denotes the natural logarithm.

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 logarithm

In the real case, the natural logarithm can be defined as the inverse of the exponential function. For extending this to the complex domain, one can start from Euler's formula. It implies that, if a complex number z is written in polar form

${\displaystyle z=r(\cos \varphi +i\sin \varphi ),}$

then its complex logarithm should be

${\displaystyle \ln(z)=\ln(r)+\varphi i.}$

However, because cosine and sine are periodic functions, the addition to ${\displaystyle \varphi }$ of an integer multiple of 2π. does not change z. For example, ${\displaystyle e^{i\pi }=e^{3i\pi }=-1}$, so both ${\displaystyle i\pi }$ and ${\displaystyle 3i\pi }$ are possible values for the natural logarithm of ${\displaystyle -1}$.

Therefore, the complex logarithm must be defined as a multivalued function:

${\displaystyle \ln(z)=\left\{\ln(r)+\varphi i+2\pi ki\mid k\in \mathbb {Z} \right\}.}$

Alternatively, a branch cut can be used to define a true function. If z is not a negative real number, the principal value of the complex logarithm is obtained with ${\displaystyle -\pi <\varphi <\pi .}$ This is an analytic function outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number.

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

${\displaystyle z^{t}=\left\{e^{t\ln r}\,(\cos(\varphi t+2\pi kt)+i\sin(\varphi t+2\pi kt))\}\mid k\in \mathbb {Z} \right\}}$

Integer and fractional exponents

Geometric representation of the 2nd to 6th roots of a complex number z, in polar form re^iφ  where r = |z | and φ = arg z. If z is real, φ = 0 or π. Principal roots are shown in black.

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 ${\displaystyle z^{n}}$ is well defined, and the exponentiation formula simplifies to de Moivre's formula:

${\displaystyle z^{n}=(r(\cos \varphi +i\sin \varphi ))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi ).}$

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

${\displaystyle z^{1/n}={\sqrt[{n}]{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)}$

for 0 ≤ k ≤ n − 1. (Here ${\displaystyle {\sqrt[{n}]{r}}}$ is the usual (positive) nth root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values.

While the nth root of a positive real number r is chosen to be the positive real number c satisfying cⁿ = r, there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root is a n-valued function of z. This implies that, contrary to the case of positive real numbers, one has

${\displaystyle (z^{n})^{1/n}\neq z,}$

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

Properties

Field structure

The set C of complex numbers is a field.[48] Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. 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. 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:

${\displaystyle z_{1}+z_{2}=z_{2}+z_{1},}$

${\displaystyle z_{1}z_{2}=z_{2}z_{1}.}$

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.

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. In fact, in any ordered field, the square of any element is necessarily positive, so i² = −1 precludes the existence of an ordering on C.[49]

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

Solutions of polynomial equations

Given any complex numbers (called coefficients) a₀, ..., a_n, the equation

${\displaystyle a_{n}z^{n}+\dotsb +a_{1}z+a_{0}=0}$

has at least one complex solution z, provided that at least one of the higher coefficients a₁, ..., a_n is nonzero.[50] This is the statement of the fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial x² − 2 does not have a rational root, since √2 is not a rational number) nor the real numbers R (the polynomial x² + a does not have a real root for a > 0, since the square of x is positive for any real number x).

There are various proofs of this theorem, either by 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.

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.

Algebraic characterization

The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Q_p also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).[51] Also, C is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields that are isomorphic to C.

Characterization as a topological field

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

• 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.

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.

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.

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.[52]

Formal construction

Construction as ordered pairs

William Rowan Hamilton introduced the approach to define the set C of complex numbers[53] as the set R² of ordered pairs (a, b) of real numbers, in which the following rules for addition and multiplication are imposed:[48]

${\displaystyle {\begin{aligned}(a,b)+(c,d)&=(a+c,b+d)\\(a,b)\cdot (c,d)&=(ac-bd,bc+ad).\end{aligned}}}$

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

Construction as a quotient field

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. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law

${\displaystyle (x+y)z=xz+yz}$

must hold for any three elements x, y and z of a field. The set R of real numbers does form a field. A polynomial p(X) with real coefficients is an expression of the form

${\displaystyle a_{n}X^{n}+\dotsb +a_{1}X+a_{0},}$

where the a₀, ..., a_n are real numbers. The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.

The set of complex numbers is defined as the quotient ring R[X]/(X ² + 1).[6] This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X ² + 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. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers. The quotient ring is a field, because X² + 1 is irreducible over R, so the ideal it generates is maximal.

The formulas for addition and multiplication in the ring R[X], modulo the relation X² = −1, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field C are isomorphic (as fields).

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.

Matrix representation of complex numbers

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

${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}}$

Here the entries a and b are real numbers. 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:

${\displaystyle {\begin{pmatrix}a&-b\\b&\;\;a\end{pmatrix}}{\begin{pmatrix}c&-d\\d&\;\;c\end{pmatrix}}={\begin{pmatrix}ac-bd&-ad-bc\\bc+ad&\;\;-bd+ac\end{pmatrix}}}$

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

${\displaystyle |z|^{2}={\begin{vmatrix}a&-b\\b&a\end{vmatrix}}=a^{2}+b^{2}.}$

The conjugate ${\displaystyle {\overline {z}}}$ corresponds to the transpose of the matrix.

Though this representation of complex numbers with matrices is the most common, many other representations arise from matrices other than ${\displaystyle {\bigl (}{\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}}{\bigr )}}$ that square to the negative of the identity matrix. See the article on 2 × 2 real matrices for other representations of complex numbers.

Complex analysis

Color wheel graph of sin(1/z). Black parts inside refer to numbers having large absolute values.

Main article: Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Complex exponential and related functions

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. 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. From a more abstract point of view, C, endowed with the metric

${\displaystyle \operatorname {d} (z_{1},z_{2})=|z_{1}-z_{2}|}$

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

${\displaystyle |z_{1}+z_{2}|\leq |z_{1}|+|z_{2}|}$

for any two complex numbers z₁ and z₂.

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^z, is defined as the infinite series

${\displaystyle \exp(z):=1+z+{\frac {z^{2}}{2\cdot 1}}+{\frac {z^{3}}{3\cdot 2\cdot 1}}+\cdots =\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}$

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

Euler's formula states:

${\displaystyle \exp(i\varphi )=\cos(\varphi )+i\sin(\varphi )}$

for any real number φ, in particular

${\displaystyle \exp(i\pi )=-1}$

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

${\displaystyle \exp(z)=w}$

for any complex number w ≠ 0. It can be shown that any such solution z – called complex logarithm of w – satisfies

${\displaystyle \log(w)=\ln |w|+i\arg(w),}$

where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π, π].

Complex exponentiation z^ω is defined as

${\displaystyle z^{\omega }=\exp(\omega \log z),}$

and is multi-valued, except when ${\displaystyle \omega }$ is an integer. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.

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. For example, they do not satisfy

${\displaystyle a^{bc}=\left(a^{b}\right)^{c}.}$

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.

Holomorphic functions

A function f : C → C is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any R-linear map C → C can be written in the form

${\displaystyle f(z)=az+b{\overline {z}}}$

with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand ${\displaystyle b{\overline {z}}}$ is real-differentiable, but does not satisfy the Cauchy–Riemann equations.

Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions f and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions, functions that can locally be written as f(z)/(z − z₀)ⁿ with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.

Applications

Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some of these applications are described below.

Geometry

Shapes

Three non-collinear points ${\displaystyle u,v,w}$ in the plane determine the shape of the triangle ${\displaystyle \{u,v,w\}}$. Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as

${\displaystyle S(u,v,w)={\frac {u-w}{u-v}}.}$

The shape ${\displaystyle S}$ of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle ${\displaystyle \{u,v,w\}}$ is in a similarity class of triangles with the same shape.[54]

Fractal geometry

The Mandelbrot set with the real and imaginary axes labeled.

The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location ${\displaystyle c}$ where iterating the sequence ${\displaystyle f_{c}(z)=z^{2}+c}$ does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where ${\displaystyle c}$ remains constant.

Triangles

Every triangle has a unique Steiner inellipse – an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:[55][56] Denote the triangle's vertices in the complex plane as a = x_A + y_Ai, b = x_B + y_Bi, and c = x_C + y_Ci. Write the cubic equation ${\displaystyle \scriptstyle (x-a)(x-b)(x-c)=0}$, take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

Algebraic number theory

Construction of a regular pentagon using straightedge and compass.

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. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. 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. In this way, algebraic methods can be used to study geometric questions and vice versa. 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.

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.

Analytic number theory

Main article: Analytic number theory

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. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function ζ(s) is related to the distribution of prime numbers.

Improper integrals

In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.

Dynamic equations

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^rt. 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^t.

In applied mathematics

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's zeros and poles are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

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. If a linear, time-invariant (LTI) 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.

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

Signal analysis

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. 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) is the phase.

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

${\displaystyle x(t)=\operatorname {Re} \{X(t)\}}$

and

${\displaystyle X(t)=Ae^{i\omega t}=ae^{i\phi }e^{i\omega t}=ae^{i(\omega t+\phi )}}$

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

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.

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

${\displaystyle {\begin{aligned}\cos((\omega +\alpha )t)+\cos \left((\omega -\alpha )t\right)&=\operatorname {Re} \left(e^{i(\omega +\alpha )t}+e^{i(\omega -\alpha )t}\right)\\&=\operatorname {Re} \left(\left(e^{i\alpha t}+e^{-i\alpha t}\right)\cdot e^{i\omega t}\right)\\&=\operatorname {Re} \left(2\cos(\alpha t)\cdot e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \operatorname {Re} \left(e^{i\omega t}\right)\\&=2\cos(\alpha t)\cdot \cos \left(\omega t\right).\end{aligned}}}$

In physics

Electromagnetism and electrical engineering

Main article: Alternating current

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.

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.

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

${\displaystyle V(t)=V_{0}e^{j\omega t}=V_{0}\left(\cos \omega t+j\sin \omega t\right),}$

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

${\displaystyle v(t)=\mathrm {Re} (V)=\mathrm {Re} \left[V_{0}e^{j\omega t}\right]=V_{0}\cos \omega t.}$

The complex-valued signal ${\displaystyle V(t)}$ is called the analytic representation of the real-valued, measurable signal ${\displaystyle v(t)}$. [57]

Fluid dynamics

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

Quantum mechanics

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

Relativity

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. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Generalizations and related notions

The process of extending the field R of reals to C is known as the Cayley–Dickson construction. 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. In this context the complex numbers have been called the binarions.[58]

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. 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.

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.

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). This means the following: the R-linear map

${\displaystyle \mathbb {C} \rightarrow \mathbb {C} ,z\mapsto wz}$

for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is

${\displaystyle {\begin{pmatrix}\operatorname {Re} (w)&-\operatorname {Im} (w)\\\operatorname {Im} (w)&\;\;\operatorname {Re} (w)\end{pmatrix}}}$,

that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrix

${\displaystyle J={\begin{pmatrix}p&q\\r&-p\end{pmatrix}},\quad p^{2}+qr+1=0}$

has the property that its square is the negative of the identity matrix: J² = −I. Then

${\displaystyle \{z=aI+bJ:a,b\in \mathbf {R} \}}$

is also isomorphic to the field C, and gives an alternative complex structure on R². This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize R, C, H, and O. For example, this notion contains the split-complex numbers, which are elements of the ring R[x]/(x² − 1) (as opposed to R[x]/(x² + 1)). In this ring, the equation a² = 1 has four solutions.

The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q lead to the fields Q_p of p-adic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Q_p, by Ostrowski's theorem. The algebraic closures ${\displaystyle {\overline {\mathbf {Q} _{p}}}}$ of Q_p still carry a norm, but (unlike C) are not complete with respect to it. The completion ${\displaystyle \mathbf {C} _{p}}$ of ${\displaystyle {\overline {\mathbf {Q} _{p}}}}$ turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.

The fields R and Q_p and their finite field extensions, including C, are local fields.

See also

• Algebraic surface
• Circular motion using complex numbers
• Complex-base system
• Complex geometry
• Dual-complex number
• Eisenstein integer
• Euler's identity
• Geometric algebra (which includes the complex plane as the 2-dimensional spinor subspace ${\displaystyle {\mathcal {G}}_{2}^{+}}$)
• Root of unity
• Unit complex number

Notes

1. For an extensive account of the history, from initial skepticism to ultimate acceptance, See (Bourbaki 1998), pages 18-24.
2. In modern notation, Tartaglia's solution is based on expanding the cube of the sum of two cube roots: ${\displaystyle \left({\sqrt[{3}]{u}}+{\sqrt[{3}]{v}}\right)^{3}=3{\sqrt[{3}]{uv}}\left({\sqrt[{3}]{u}}+{\sqrt[{3}]{v}}\right)+u+v}$ With ${\displaystyle x={\sqrt[{3}]{u}}+{\sqrt[{3}]{v}}}$, ${\displaystyle p=3{\sqrt[{3}]{uv}}}$, ${\displaystyle q=u+v}$, u and v can be expressed in terms of p and q as ${\displaystyle u=q/2+{\sqrt {(q/2)^{2}-(p/3)^{3}}}}$ and ${\displaystyle v=q/2-{\sqrt {(q/2)^{2}-(p/3)^{3}}}}$, respectively. Therefore, ${\displaystyle x={\sqrt[{3}]{q/2+{\sqrt {(q/2)^{2}-(p/3)^{3}}}}}+{\sqrt[{3}]{q/2-{\sqrt {(q/2)^{2}-(p/3)^{3}}}}}}$. When ${\displaystyle (q/2)^{2}-(p/3)^{3}}$ is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugate of the first one.

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28. Gauss, Carl Friedrich (1799) "Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse." [New proof of the theorem that any rational integral algebraic function of a single variable can be resolved into real factors of the first or second degree.] Ph.D. thesis, University of Helmstedt, (Germany). (in Latin)
29. Gauss, C.F. (1831). "Anzeige von Theoria residuorum biquadraticorum, commentatio secunda" [Notice on the theory of biquadratic residues, second treatise]. Göttingische gelehrte Anzeigen: 638.
30. Ewald, William B. (1996). "From Kant to Hilbert: A Source Book in the Foundations of Mathematics". Oxford University Press. p. 313. Retrieved 18 March 2020.
31. Adrien Quentin Buée (1745–1845): MacTutor
32. Buée (1806). "Mémoire sur les quantités imaginaires" [Memoir on imaginary quantities]. Philosophical Transactions of the Royal Society of London (in French). 96: 23–88.
33. Mourey, C.V. (1861). La vraies théore des quantités négatives et des quantités prétendues imaginaires [The true theory of negative quantities and of alleged imaginary quantities] (in French). Paris, France: Mallet-Bachelier. 1861 reprint of 1828 original.
34. See:
    • Warren, John (1828). A Treatise on the Geometrical Representation of the Square Roots of Negative Quantities. Cambridge, England: Cambridge University Press.
    • Warren, John (1829). "Consideration of the objections raised against the geometrical representation of the square roots of negative quantities". Philosophical Transactions of the Royal Society of London. 119: 241–254.
    • Warren, John (1829). "On the geometrical representation of the powers of quantities, whose indices involve the square roots of negative numbers". Philosophical Transactions of the Royal Society of London. 119: 339–359.
35. Français, J.F. (1813). "Nouveaux principes de géométrie de position, et interprétation géométrique des symboles imaginaires" [New principles of the geometry of position, and geometric interpretation of complex [number] symbols]. Annales des mathématiques pures et appliquées (in French). 4: 61–71.
36. Caparrini, Sandro (2000), "On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers", in Kim Williams (ed.), Two Cultures, Birkhäuser, p. 139, ISBN 978-3-7643-7186-9 Extract of page 139 Archived 2 December 2016 at the Wayback Machine
37. Hardy, G.H.; Wright, E.M. (2000) [1938], An Introduction to the Theory of Numbers, OUP Oxford, p. 189 (fourth edition), ISBN 978-0-19-921986-5
38. Argand (1814). "Reflexions sur la nouvelle théorie des imaginaires, suives d'une application à la demonstration d'un theorème d'analise" [Reflections on the new theory of complex numbers, followed by an application to the proof of a theorem of analysis]. Annales de mathématiques pures et appliquées (in French). 5: 197–209. On p. 204, Argand defines the modulus of a complex number but he doesn't name it: "Dans ce qui suit, les accens, indifféremment placés, seront employés pour indiquer la grandeur absolue des quantités qu'ils affectent; ainsi, si ${\displaystyle a=m+n{\sqrt {-1}}}$, ${\displaystyle m}$ et ${\displaystyle n}$ étant réels, on devra entendre que ${\displaystyle a_{'}}$ ou ${\displaystyle a'={\sqrt {m^{2}+n^{2}}}}$." (In what follows, accent marks, wherever they're placed, will be used to indicate the absolute size of the quantities to which they're assigned; thus if ${\displaystyle a=m+n{\sqrt {-1}}}$, ${\displaystyle m}$ and ${\displaystyle n}$ being real, one should understand that ${\displaystyle a_{'}}$ or ${\displaystyle a'={\sqrt {m^{2}+n^{2}}}}$.)
    On p. 208, Argand defines and names the module and the direction factor of a complex number: " ... ${\displaystyle a={\sqrt {m^{2}+n^{2}}}}$ pourrait être appelé le module de ${\displaystyle a+b{\sqrt {-1}}}$, et représenterait la grandeur absolue de la ligne ${\displaystyle a+b{\sqrt {-1}}}$, tandis que l'autre facteur, dont le module est l'unité, en représenterait la direction." ( ... ${\displaystyle a={\sqrt {m^{2}+n^{2}}}}$ could be called the module of ${\displaystyle a+b{\sqrt {-1}}}$ and would represent the absolute size of the line ${\displaystyle a+b{\sqrt {-1}}}$ [Note: Argand represented complex numbers as vectors.], whereas the other factor [namely, ${\displaystyle {\tfrac {a}{\sqrt {a^{2}+b^{2}}}}+{\tfrac {b}{\sqrt {a^{2}+b^{2}}}}{\sqrt {-1}}}$], whose module is unity [1], would represent its direction.)
    See also:
    • Jeff Miller (1999) Earliest Known Uses of Some of the Words of Mathematics (M), Internet Archive Wayback Machine
39. Cauchy, Augustin Louis (1821). Cours d'analyse de l'École royale polytechnique (in French). vol. 1. Paris, France: L'Imprimerie Royale. p. 183.
40. Gauss, C.F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148. From p. 96: "Quemadmodum scilicet arithmetica sublimior in quaestionibus hactenus pertractatis inter solos numeros integros reales versatur, ita theoremata circa residua biquadratica tunc tantum in summa simplicitate ac genuina venustate resplendent, quando campus arithmeticae ad quantitates imaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae a + bi, denotantibus i, pro more quantitatem imaginariam √ -1, atque a, b indefinite omnes numeros reales integros inter -${\displaystyle \infty }$ et +${\displaystyle \infty }$." (Of course just as the higher arithmetic has been investigated so far in problems only among real integer numbers, so theorems regarding biquadratic residues then shine in greatest simplicity and genuine beauty, when the field of arithmetic is extended to imaginary quantities, so that, without restrictions on it, numbers of the form a + bi — i denoting by convention the imaginary quantity √ -1, and the variables a, b [denoting] all real integer numbers between -${\displaystyle \infty }$ and +${\displaystyle \infty }$ — constitute an object.)
41. Gauss, C.F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148. From p. 96: "Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeantur." (We will call such numbers [namely, numbers of the form a + bi ] "complex integer numbers", so that real [numbers] are regarded not as the opposite of complex [numbers] but [as] a type [of number that] is, so to speak, contained within them.)
42. Gauss, C.F. (1831). "Theoria residuorum biquadraticorum. Commentatio secunda" [Theory of biquadratic residues. Second memoir.]. Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores (in Latin). 7: 89–148. From p. 98: "Productum numeri complexi per numerum ipsi conjunctum utriusque normam vocamus. Pro norma itaque numeri realis, ipsius quadratum habendum est." (We call a "norm" the product of a complex number [e.g,. a + ib ] with its conjugate [a - ib ]. Therefore the square of a real number should be regarded as its norm.)
43. Hankel, Hermann (1867). Vorlesungen über die complexen Zahlen und ihre Functionen [Lectures About the Complex Numbers and Their Functions] (in German). vol. 1. Leipzig, [Germany]: Leopold Voss. p. 71. From p. 71: "Wir werden den Factor (cos φ + i sin φ) haüfig den Richtungscoefficienten nennen." (We will often call the factor (cos φ + i sin φ) the "coefficient of direction".)
44. For the former notation, See (Apostol 1981), pages 15–16.
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Works cited

• Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0-07-000657-7
• Apostol, Tom (1981). Mathematical analysis. Addison-Wesley.
• Solomentsev, E.D. (2001) [1994], "Complex number", Encyclopedia of Mathematics, EMS Press

Further reading

• Penrose, Roger (2005), The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred A. Knopf, ISBN 978-0-679-45443-4
• Derbyshire, John (2006), Unknown Quantity: A Real and Imaginary History of Algebra, Joseph Henry Press, ISBN 978-0-309-09657-7
• Needham, Tristan (1997), Visual Complex Analysis, Clarendon Press, ISBN 978-0-19-853447-1

Mathematical

• Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0-07-000657-7
• Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 978-0-387-90328-6
• Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons, ISBN 978-0-470-21152-6
• Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 978-0-486-65812-4
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.5 Complex Arithmetic", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8
• Solomentsev, E.D. (2001) [1994], "Complex number", Encyclopedia of Mathematics, EMS Press

Historical

• Bourbaki, Nicolas (1998), "Foundations of mathematics § logic: set theory", Elements of the history of mathematics, Springer
• Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9
• Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
• Nahin, Paul J. (1998), An Imaginary Tale: The Story of ${\displaystyle \scriptstyle {\sqrt {-1}}}$, Princeton University Press, ISBN 978-0-691-02795-1

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

• Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991), Numbers (hardcover ed.), Springer, ISBN 978-0-387-97497-2

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