In
mathematics, a complex number is an element of a
number system
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers ca ...
that extends the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s with a specific element denoted , called the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and ...
and satisfying the
equation
In mathematics, an equation is a formula that expresses the equality (mathematics), equality of two Expression (mathematics), expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may h ...
; every complex number can be expressed in the form
, where and are real numbers. Because no real number satisfies the above equation, was called an
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
by
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Math ...
. For the complex number
, is called the , and is called the . The set of complex numbers is denoted by either of the symbols
or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the
mathematical sciences
The mathematical sciences are a group of areas of study that includes, in addition to mathematics, those academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.
Statist ...
as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.
Complex numbers allow solutions to all
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
s, even those that have no solutions in real numbers. More precisely, the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynom ...
asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation
has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions
and
.
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule
combined with the
associative,
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, and
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic ...
s. Every nonzero complex number has a
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
. This makes the complex numbers a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
that has the real numbers as a subfield. The complex numbers also form a
real vector space
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
of dimension two, with as a
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the ...
.
This standard basis makes the complex numbers a
Cartesian plane
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, called the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
which is identified to the horizontal axis of the complex plane. The complex numbers of
absolute value one form the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. The addition of a complex number is a
translation
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
in the complex plane, and the multiplication by a complex number is a
similarity centered at the origin. The
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an Imaginary number, imaginary part equal in magnitude but opposite in Sign (mathematics), sign. That is, (if a and b are real, then) the complex ...
is the
reflection symmetry with respect to the real axis. The complex absolute value is a
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
.
In summary, the complex numbers form a rich structure that is simultaneously an
algebraically closed field, a
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
over the reals, and a
Euclidean vector space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension two.
Definition
A complex number is a number of the form , where and are
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, and is an indeterminate satisfying . For example, is a complex number.
This way, a complex number is defined as a
polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
with real coefficients in the single indeterminate , for which the relation is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation induces the equalities and which hold for all integers ; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in , again of the form with real coefficients
The real number is called the ''real part'' of the complex number ; the real number is called its ''imaginary part''. To emphasize, the imaginary part does not include a factor ; that is, the imaginary part is , not .
Formally, the complex numbers are defined as the
quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
of the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
in the indeterminate , by the
ideal generated by the polynomial (see
below).
Notation
A real number can be regarded as a complex number , whose imaginary part is 0. A purely
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
is a complex number , whose real part is zero. As with polynomials, it is common to write for and for . Moreover, when the imaginary part is negative, that is, , it is common to write instead of ; for example, for , can be written instead of .
Since the multiplication of the indeterminate and a real is commutative in polynomials with real coefficients, the polynomial may be written as This is often expedient for imaginary parts denoted by expressions, for example, when is a radical.
The real part of a complex number is denoted by ,
, or
; the imaginary part of a complex number is denoted by ,
, or
For example,
The
set of all complex numbers is denoted by
(
blackboard bold) or (upright bold).
In some disciplines, particularly in
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
and
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, is used instead of as is frequently used to represent
electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
. In these cases, complex numbers are written as , or .
Visualization
A complex number can thus be identified with an
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
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,''
named after
Jean-Robert Argand Jean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram and is know ...
. 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 viewed either as the
coordinatized point or as a
position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
from the origin to this point. The coordinate values of a complex number can hence be expressed in 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
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
, 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 corresponds to rotating the position vector
counterclockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite s ...
by a quarter
turn (
90°) about the origin—a fact which can be expressed algebraically as follows:
Polar complex plane
Modulus and argument
An alternative option for coordinates in the complex plane is the
polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the ...
that uses the distance of the point from the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
(), and the angle subtended between the
positive real axis
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
and the line segment in a counterclockwise sense. This leads to the polar form
:
of a complex number, where is the
absolute value of , and
is the
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
of .
The ''absolute value'' (or ''modulus'' or ''magnitude'') of a complex number is
If is a real number (that is, if ), then . That is, the absolute value of a real number equals its absolute value as a complex number.
By
Pythagoras' theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, the absolute value of a complex number is the distance to the origin of the point representing the complex number in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
.
The ''argument'' of (in many applications referred to as the "phase" )
is the angle of the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form —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 -function, and avoids a more subtle case-by-case analysis
Normally, as given above, the
principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
in the interval is chosen. If the arg value is negative, values in the range or can be obtained by adding . The value of is expressed in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s in this article. It can increase by any integer multiple of and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through . 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 equals the result of
atan2
In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive :
Together, and 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''
Using
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
this can be written as
Using the function, this is sometimes abbreviated to
In
angle notation, often used in
electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification a ...
to represent a
phasor
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
with amplitude and phase , it is written as
Complex graphs
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
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', ...
, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.
In
domain coloring
In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the complex plane to different colors and brightness, d ...
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 for to 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 and poles at
History
The solution in
radicals (without
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
) of a general
cubic equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
, when all three of its roots are real numbers, contains the square roots of
negative numbers
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed ma ...
, a situation that cannot be rectified by factoring aided by the
rational root test
In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or theorem) states a constraint on rational solutions of a polynomial equation
:a_nx^n+a_x^+\cdots+a_0 = 0
with integer coefficients a_i\in\m ...
, if the cubic is
irreducible; this is the so-called ''
casus irreducibilis'' ("irreducible case"). This conundrum led Italian mathematician
Gerolamo Cardano
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
to conceive of complex numbers in around 1545 in his ''Ars Magna'', though his understanding was rudimentary; moreover he later dismissed complex numbers as "subtle as they are useless". Cardano did use imaginary numbers, but described using them as “mental torture.” This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably
Scipione del Ferro
Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation.
Life
Scipione del Ferro was born in Bologna, in northern Italy, to Floriano and Filip ...
, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Since they ignored the answers with the imaginary numbers, Cardano found them useless.
Work on the problem of general polynomials ultimately led to the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynom ...
, which shows that with complex numbers, a solution exists to every
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
of degree one or higher. Complex numbers thus form an
algebraically closed field, where any polynomial equation has a
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
.
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
Rafael Bombelli (baptised on 20 January 1526; died 1572) was an Italian mathematician. Born in Bologna, he is the author of a treatise on algebra and is a central figure in the understanding of imaginary numbers.
He was the one who finally mana ...
. A more abstract formalism for the complex numbers was further developed by the Irish mathematician
William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ...
, who extended this abstraction to the theory of
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
.
The earliest fleeting reference to
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
s of
negative number
In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s can perhaps be said to occur in the work of the
Greek mathematician Hero of Alexandria
Hero of Alexandria (; grc-gre, Ἥρων ὁ Ἀλεξανδρεύς, ''Heron ho Alexandreus'', also known as Heron of Alexandria ; 60 AD) was a Greek mathematician and engineer who was active in his native city of Alexandria, Roman Egypt. He ...
in the 1st century
AD, where in his ''
Stereometrica'' he considered, apparently in error, the volume of an impossible
frustum
In geometry, a (from the Latin for "morsel"; plural: ''frusta'' or ''frustums'') is the portion of a solid (normally a pyramid or a cone) that lies between two parallel planes cutting this solid. In the case of a pyramid, the base faces are ...
of a
pyramid to arrive at the term
in his calculations, which today would simplify to
. Negative quantities were not conceived of in
Hellenistic mathematics
Greek mathematics refers to mathematics texts and ideas stemming from the Archaic through the Hellenistic and Roman periods, mostly extant from the 7th century BC to the 4th century AD, around the shores of the Eastern Mediterranean. Greek mathe ...
and Hero merely replaced it by its positive
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
Cubic may refer to:
Science and mathematics
* Cube (algebra), "cubic" measurement
* Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
** Cubic crystal system, a crystal system w ...
and
quartic polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s were discovered by Italian mathematicians (see
Niccolò Fontana Tartaglia
Niccolò Fontana Tartaglia (; 1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Repu ...
,
Gerolamo Cardano
Gerolamo Cardano (; also Girolamo or Geronimo; french: link=no, Jérôme Cardan; la, Hieronymus Cardanus; 24 September 1501– 21 September 1576) was an Italian polymath, whose interests and proficiencies ranged through those of mathematician, ...
). It was soon realized (but proved much later)
[ that these formulas, even if one were interested only 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 gives the solution to the equation as
At first glance this looks like nonsense. However, formal calculations with complex numbers show that the equation has three solutions: Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of . 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, the use of complex numbers is unavoidable. Rafael Bombelli was the first to address explicitly 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
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Math ...
in 1637, who was at pains to stress their unreal nature
A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for non-negative real numbers and , and which was also used in complex number calculations with one of , positive and the other negative. The incorrect use of this identity (and the related identity ) in the case when both and are negative even bedeviled Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ...
. This difficulty eventually led to the convention of using the special symbol in place of 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
''Elements of Algebra'' is an elementary mathematics textbook written by mathematician Leonhard Euler around 1765 in German. It was first published in Russian as "''Universal Arithmetic''" (''Универсальная арифметика''), tw ...
'', 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
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.
He move ...
noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula
In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that
:\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx,
where is the imaginary unit (). ...
:
In 1748, Euler went further and obtained Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
of complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
:
by formally manipulating complex power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
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 Danish
Danish may refer to:
* Something of, from, or related to the country of Denmark
People
* A national or citizen of Denmark, also called a "Dane," see Demographics of Denmark
* Culture of Denmark
* Danish people or Danes, people with a Danish ance ...
– Norwegian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
...
Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's ''A Treatise of Algebra''.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy
The Royal Danish Academy of Fine Arts ( da, Det Kongelige Danske Kunstakademi - Billedkunst Skolerne) has provided education in the arts for more than 250 years, playing its part in the development of the art of Denmark.
History
The Royal Dan ...
but went largely unnoticed. In 1806 Jean-Robert Argand Jean-Robert Argand (, , ; July 18, 1768 – August 13, 1822) was an amateur mathematician. In 1806, while managing a bookstore in Paris, he published the idea of geometrical interpretation of complex numbers known as the Argand diagram and is know ...
independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynom ...
. Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
had earlier published an essentially topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". 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:
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, 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.
In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren
A warren is a network of wild rodent or lagomorph, typically rabbit burrows. Domestic warrens are artificial, enclosed establishment of animal husbandry dedicated to the raising of rabbits for meat and fur. The term evolved from the medieval Angl ...
, Français and his brother, Bellavitis.
The English mathematician G.H. Hardy
Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of po ...
remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Norwegian Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
Augustin-Louis Cauchy and Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
together brought the fundamental ideas of complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
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 the ''direction factor'', and the ''modulus''; Cauchy (1821) called the ''reduced form'' (l'expression réduite) and apparently introduced the term ''argument''; Gauss used for , introduced the term ''complex number'' for , and called the ''norm''. The expression ''direction coefficient'', often used for , is due to Hankel (1867), and ''absolute value,'' for ''modulus,'' is due to Weierstrass.
Later classical writers on the general theory include Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
, Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christ ...
, Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...
, Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
, Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.
Life
Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, ...
, Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
and many others. Important work (including a systematization) in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger
Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.
Biography
He was born at Ybbs on the Danube and studied at the Unive ...
in 1927.
Relations and operations
Equality
Complex numbers have a similar definition of equality to real numbers; two complex numbers and are equal if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicond ...
both their real and imaginary parts are equal, that is, if and . Nonzero complex numbers written in polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
are equal if and only if they have the same magnitude and their arguments differ by an integer multiple of .
Ordering
Unlike the real numbers, there is no natural ordering of the complex numbers. In particular, there is no linear ordering
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X:
# a \leq a ( reflexive ...
on the complex numbers that is compatible with addition and multiplication. Hence, the complex numbers do not have the structure of an ordered field. One explanation for this is that every non-trivial sum of squares in an ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
is nonzero, and is a non-trivial sum of squares. Thus, complex numbers are naturally thought of as existing on a two-dimensional plane.
Conjugate
The ''complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
'' of the complex number is given by . It is denoted by either or . This unary operation
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation o ...
on complex numbers cannot be expressed by applying only their basic operations addition, subtraction, multiplication and division.
Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number
which makes this operation an involution
Involution may refer to:
* Involute, a construction in the differential geometry of curves
* '' Agricultural Involution: The Processes of Ecological Change in Indonesia'', a 1963 study of intensification of production through increased labour inpu ...
. The reflection leaves both the real part and the magnitude of unchanged, that is
and
The imaginary part and the argument of a complex number change their sign under conjugation
For details on argument and magnitude, see the section on Polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
.
The product of a complex number and its conjugate is known as the '' absolute square''. It is always a non-negative real number and equals the square of the magnitude of each:
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 can be extracted using the conjugation:
Moreover, a complex number is real if and only if it equals its own conjugate.
Conjugation distributes over the basic complex arithmetic operations:
Conjugation is also employed in inversive geometry
Inversive activities are processes which self internalise the action concerned. For example, a person who has an Inversive personality internalises his emotions from any exterior source. An inversive heat source would be a heat source where all th ...
, 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 In electrical engineering, the maximum power transfer theorem states that, to obtain ''maximum'' external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its ou ...
is looked for.
Addition and subtraction
Two complex numbers and are most easily added by separately adding their real and imaginary parts. That is to say:
Similarly, subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
can be performed as
Multiplication of a complex number and a real number can be done similarly by multiplying separately and the real and imaginary parts of :
In particular, subtraction can be done by negating the subtrahend
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
(that is multiplying it with ) and adding the result to the minuend
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
:
Using the visualization of complex numbers in the complex plane, addition has the following geometric interpretation: the sum of two complex numbers and , interpreted as points in the complex plane, is the point obtained by building a parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equ ...
from the three vertices , and the points of the arrows labeled and (provided that they are not on a line). Equivalently, calling these points , , respectively and the fourth point of the parallelogram the triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
s and are congruent.
Multiplication and square
The rules of the distributive property
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
, the commutative properties (of addition and multiplication), and the defining property apply to complex numbers. It follows that
In particular,
Reciprocal and division
Using the conjugation, the reciprocal of a nonzero complex number can always be broken down to
since ''non-zero'' implies that is greater than zero.
This can be used to express a division of an arbitrary complex number by a non-zero complex number as
Multiplication and division in polar form
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers and , because of the trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvi ...
we may derive
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 corresponds to a quarter- turn counter-clockwise, which gives back . The picture at the right illustrates the multiplication of
Since the real and imaginary part of are equal, the argument of that number is 45 degrees, or (in radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
). 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
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas
In mathematics, Machin-like formulae are a popular technique for computing to a large number of digits. They are generalizations of John Machin's formula from 1706:
:\frac = 4 \arctan \frac - \arctan \frac
which he used to compute to 100 d ...
– are used for high-precision approximations of .
Similarly, division is given by
Square root
The square roots of (with ) are , where
and
where is the signum function. This can be seen by squaring to obtain . Here is called the modulus of , and the square root sign indicates the square root with non-negative real part, called the principal square root; also where .
Exponential function
The exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
can be defined for every complex number by the power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
which has an infinite radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
.
The value at of the exponential function is Euler's number
The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an express ...
If is real, one has
Analytic continuation allows extending this equality for every complex value of , and thus to define the complex exponentiation with base as
Functional equation
The exponential function satisfies the functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
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
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
states that, for any real number ,
The functional equation implies thus that, if and are real, one has
which is the decomposition of the exponential function into its real and imaginary parts.
Complex logarithm
In the real case, the natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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 is written in polar form
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
with then with
as complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to ...
one has a proper inverse:
However, because cosine and sine are periodic functions, the addition of an integer multiple of to does not change . For example, , so both and are possible values for the natural logarithm of .
Therefore, if the complex logarithm is not to be defined as a multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
one has to use a branch cut and to restrict the codomain, resulting in the bijective
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
function
If is not a non-positive real number (a positive or a non-real number), the resulting principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of the complex logarithm is obtained with . It is an analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
outside the negative real numbers, but it cannot be prolongated to a function that is continuous at any negative real number , where the principal value is .
Exponentiation
If is real and complex, the exponentiation is defined as
where denotes the natural logarithm.
It seems natural to extend this formula to complex values of , but there are some difficulties resulting from the fact that the complex logarithm is not really a function, but a multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
.
It follows that if is as above, and if is another complex number, then the ''exponentiation'' is the multivalued function
Integer and fractional exponents
If, in the preceding formula, is an integer, then the sine and the cosine are independent of . Thus, if the exponent is an integer, then is well defined, and the exponentiation formula simplifies to de Moivre's formula
In mathematics, de Moivre's formula (also known as de Moivre's theorem and de Moivre's identity) states that for any real number and integer it holds that
:\big(\cos x + i \sin x\big)^n = \cos nx + i \sin nx,
where is the imaginary unit (). ...
:
The th roots of a complex number are given by
for . (Here is the usual (positive) th root of the positive real number .) Because sine and cosine are periodic, other integer values of do not give other values.
While the th root of a positive real number is chosen to be the ''positive'' real number satisfying , there is no natural way of distinguishing one particular complex th root of a complex number. Therefore, the th root is a -valued function of . This implies that, contrary to the case of positive real numbers, one has
since the left-hand side consists of values, and the right-hand side is a single value.
Properties
Field structure
The set of complex numbers is a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
. 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 , its additive inverse 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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name ...
of addition and multiplication for any two complex numbers and :
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, is not an ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
, that is to say, it is not possible to define a relation that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so precludes the existence of an ordering on
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 analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
, complex matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, complex polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
, and complex Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
.
Solutions of polynomial equations
Given any complex numbers (called coefficients) , the equation
has at least one complex solution ''z'', provided that at least one of the higher coefficients is nonzero. This is the statement of the fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynom ...
, of Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and Jean le Rond d'Alembert. Because of this fact, is called an algebraically closed field. This property does not hold for the rational number, field of rational numbers (the polynomial does not have a rational root, since square root of 2, √2 is not a rational number) nor the real numbers (the polynomial does not have a real root for , since the square of is positive for any real number ).
There are various proofs of this theorem, by either analytic methods such as Liouville's theorem (complex analysis), Liouville's theorem, or topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
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 For example, any non-empty complex square matrix has at least one (complex) eigenvalue.
Algebraic characterization
The field has the following three properties:
* First, it has characteristic (algebra), characteristic 0. This means that for any number of summands (all of which equal one).
* Second, its transcendence degree over , the prime field of 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 For example, the algebraic closure of the field of the p-adic number, -adic number also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields). Also, 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 contains many proper subfields that are isomorphic to .
Characterization as a topological field
The preceding characterization of describes only the algebraic aspects of That is to say, the properties of neighborhood (topology), nearness and continuity (topology), continuity, which matter in areas such as Mathematical analysis, analysis and topology, are not dealt with. The following description of as a topological ring, topological field (that is, a field that is equipped with a topological space, topology, which allows the notion of convergence) does take into account the topological properties. contains a subset (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
* is closed under addition, multiplication and taking inverses.
* If and are distinct elements of , then either or is in .
* If is any nonempty subset of , then for some in
Moreover, has a nontrivial involution (mathematics), involutive automorphism (namely the complex conjugation), such that is in for any nonzero in
Any field with these properties can be endowed with a topology by taking the sets as a base (topology), base, where ranges over the field and ranges over . With this topology is isomorphic as a ''topological'' field to
The only connected space, connected locally compact topological ring, topological fields are and This gives another characterization of as a topological field, since can be distinguished from because the nonzero complex numbers are connected space, connected, while the nonzero real numbers are not.
Formal construction
Construction as ordered pairs
William Rowan Hamilton
Sir William Rowan Hamilton LL.D, DCL, MRIA, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the Andrews Professor of Astronomy at Trinity College Dublin, and Royal Astronomer of Irel ...
introduced the approach to define the set of complex numbers as the set of of real numbers, in which the following rules for addition and multiplication are imposed:
It is then just a matter of notation to express as .
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 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
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic ...
must hold for any three elements , and of a field. The set of real numbers does form a field. A polynomial with real coefficients is an expression of the form
where the are real numbers. The usual addition and multiplication of polynomials endows the set of all such polynomials with a ring (mathematics), ring structure. This ring is called the polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variab ...
over the real numbers.
The set of complex numbers is defined as the quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. ...
This extension field contains two square roots of , namely (the cosets of) and , respectively. (The cosets of) and form a basis of 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 of real numbers. The quotient ring is a field, because is Irreducible polynomial, irreducible over so the ideal it generates is Maximal ideal, maximal.
The formulas for addition and multiplication in the ring modulo the relation , correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field are isomorphism, isomorphic (as fields).
Accepting that is algebraically closed, since it is an algebraic extension of in this approach, is therefore the algebraic closure of
Matrix representation of complex numbers
Complex numbers can also be represented by matrix (mathematics), matrices that have the form:
Here the entries and are real numbers. As the sum and product of two such matrices is again of this form, these matrices form a subring of the ring matrices.
A simple computation shows that the map:
is a ring isomorphism from the field of complex numbers to the ring of these matrices. This isomorphism associates the square of the absolute value of a complex number with the determinant of the corresponding matrix, and the conjugate of a complex number with the transpose of the matrix.
The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrix, rotation matrices by using this correspondence between complex numbers and such matrices. The action of the matrix on a vector corresponds to the multiplication of by . In particular, if the determinant is , there is a real number such that the matrix has the form:
In this case, the action of the matrix on vectors and the multiplication by the complex number are both the rotation (mathematics), rotation of the angle .
Complex analysis
The study of functions of a complex variable is known as complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
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 graph of a function of two variables, 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 convergent sequence, 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, , endowed with the metric (mathematics), metric
is a complete metric space, which notably includes the triangle inequality
for any two complex numbers and .
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the ''exponential function
The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
'' , also written , is defined as the infinite series
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 (function), 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
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
'' states:
for any real number , in particular
, which is Euler's identity.
Unlike in the situation of real numbers, there is an infinite set, infinitude of complex solutions of the equation
for any complex number . It can be shown that any such solution – called complex logarithm
In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related:
* A complex logarithm of a nonzero complex number z, defined to ...
of – satisfies
where arg is the arg (mathematics), argument defined #Polar form, above, and ln the (real) natural logarithm
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. As arg is a multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
, unique only up to a multiple of , log is also multivalued. The principal value
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positi ...
of log is often taken by restricting the imaginary part to the interval (mathematics), interval .
Complex exponentiation is defined as
and is multi-valued, except when is an integer. For , for some natural number , this recovers the non-uniqueness of th 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 Exponentiation#Failure of power and logarithm identities, failure of power and logarithm identities. For example, they do not satisfy
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'': → is called Holomorphic function, holomorphic if it satisfies the Cauchy–Riemann equations. For example, any Linear transformation#Definition and first consequences, -linear map → can be written in the form
with complex coefficients and . This map is holomorphic if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicond ...
. The second summand 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 and that agree on an arbitrarily small open subset of necessarily agree everywhere. Meromorphic functions, functions that can locally be written as with a holomorphic function , still share some of the features of holomorphic functions. Other functions have essential singularity, essential singularities, such as at .
Applications
Complex numbers have applications in many scientific areas, including signal processing, control theory, electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
, fluid dynamics, quantum mechanics, cartography, and Vibration#Vibration analysis, vibration analysis. Some of these applications are described below.
Geometry
Shapes
Three collinearity, non-collinear points in the plane determine the Shape#Similarity classes, shape of the triangle . Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as
The shape 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 is in a shape#Similarity classes, similarity class of triangles with the same shape.
Fractal geometry
The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location where iterating the sequence does not diverge (stability theory), diverge when Iteration, iterated infinitely. Similarly, Julia sets have the same rules, except where 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 Focus (geometry), foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem: Denote the triangle's vertices in the complex plane as , , and . Write the cubic equation
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
, 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
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in . ''Argumentum a fortiori, 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 , the algebraic closure of , which also contains all algebraic numbers, 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 (mathematics), field theory to the number field containing root of unity, roots of unity, it can be shown that it is not possible to construct a regular nonagon compass and straightedge constructions, using only compass and straightedge – a purely geometric problem.
Another example is the Gaussian integers; that is, numbers of the form , where and are integers, which can be used to classify Fermat's theorem on sums of two squares, sums of squares.
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 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 of the Linear differential equation#Homogeneous equations with constant coefficients, 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 . Likewise, in difference equations, the complex roots 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 .
Linear algebra
Eigendecomposition of a matrix, Eigendecomposition is a useful tool for computing matrix powers and matrix exponentials. However, it often requires the use of complex numbers, even if the matrix is real (for example, a rotation matrix).
Complex numbers often generalize concepts originally conceived in the real numbers. For example, the conjugate transpose generalizes the transpose, Hermitian matrix, hermitian matrices generalize Symmetric matrix, symmetric matrices, and Unitary matrix, unitary matrices generalize Orthogonal matrix, orthogonal matrices.
In applied mathematics
Control theory
In control theory, systems are often transformed from the time domain to the complex 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 BIBO stability, 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 of the corresponding is the amplitude and the Argument (complex analysis), argument is the phase (waves), 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
and
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 use digital versions of Fourier analysis (and wavelet analysis) to transmit, Data compression, compress, restore, and otherwise process Digital data, digital Sound, audio signals, still images, and video signals.
Another example, relevant to the two side bands of amplitude modulation of AM radio, is:
In physics
Electromagnetism and electrical engineering
In electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, the Fourier transform is used to analyze varying voltages and Electric current, 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 Electrical impedance, impedance. This approach is called phasor calculus.
In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
, or, more particularly, , which is generally in use to denote instantaneous electric current.
Since the voltage in an AC electric circuit, circuit is oscillating, it can be represented as
To obtain the measurable quantity, the real part is taken:
The complex-valued signal is called the analytic signal, analytic representation of the real-valued, measurable signal .
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 relativity, 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 Wick rotation, 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 of reals to is known as the Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions and octonions which (as a real vector space) are of dimension 4 and 8, respectively.
In this context the complex numbers have been called the binarions.[ ]
Just as by applying the construction to reals the property of ordered field, ordering is lost, properties familiar from real and complex numbers vanish with each extension. The quaternions lose commutativity, that is, for some quaternions , and the multiplication of octonions, additionally to not being commutative, fails to be associative: for some octonions .
Reals, complex numbers, quaternions and octonions are all normed division algebras over . By Hurwitz's theorem (normed division algebras), 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 thought of as an -Algebra (ring theory), algebra (an -vector space with a multiplication), with respect to the basis . This means the following: the -linear map
for some fixed complex number can be represented by a matrix (once a basis has been chosen). With respect to the basis , this matrix is
that is, the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of in the 2 × 2 real matrices, it is not the only one. Any matrix
has the property that its square is the negative of the identity matrix: . Then
is also isomorphic to the field and gives an alternative complex structure on This is generalized by the notion of a linear complex structure.
Hypercomplex numbers also generalize and For example, this notion contains the split-complex numbers, which are elements of the ring (as opposed to for complex numbers). In this ring, the equation has four solutions.
The field is the completion of the field of rational numbers, with respect to the usual absolute value metric (mathematics), metric. Other choices of metric (mathematics), metrics on lead to the fields of p-adic number, -adic numbers (for any prime number ), which are thereby analogous to . There are no other nontrivial ways of completing than and by Ostrowski's theorem. The algebraic closures of still carry a norm, but (unlike ) are not complete with respect to it. The completion of turns out to be algebraically closed. By analogy, the field is called -adic complex numbers.
The fields and their finite field extensions, including are called local fields.
See also
* Algebraic surface
* Circular motion#Using complex numbers, Circular motion using complex numbers
* Complex-base system
* Complex geometry
* Dual-complex number
* Eisenstein integer
* Euler's identity
* Geometric algebra#Unit pseudoscalars, Geometric algebra (which includes the complex plane as the 2-dimensional Spinor#Two dimensions, spinor subspace )
* Unit complex number
Notes
References
Works cited
*
*
*
*
*
Further reading
*
*
*
Mathematical
*
*
*
*
*
*
Historical
*
*
*
* — A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
* — An advanced perspective on the historical development of the concept of number.
{{DEFAULTSORT:Complex Number
Composition algebras
Complex numbers,