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 can ...
that extends the
real numbers with a specific element denoted , called the
imaginary unit and satisfying the
equation ; 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 by
René Descartes. 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 (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
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 polynomia ...
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. This makes the complex numbers a
field 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.
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. 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 which is identified to the horizontal axis of the complex plane. The complex numbers of
absolute value one form the
unit circle. The addition of a complex number is a
translation in the complex plane, and the multiplication by a complex number is a
similarity centered at the origin. The
complex conjugation is the
reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D the ...
with respect to the real axis. The complex absolute value is a
Euclidean norm.
In summary, the complex numbers form a rich structure that is simultaneously an
algebraically closed field, a
commutative algebra 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 s ...
of dimension two.
Definition
A complex number is a number of the form , where and are
real numbers, and is an indeterminate satisfying . For example, is a complex number.
This way, a complex number is defined as a
polynomial 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 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 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
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pro ...
) 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, is used instead of as is frequently used to represent
electric current. In these cases, complex numbers are written as , or .
Visualization
A complex number can thus be identified with an
ordered pair 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 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 a ...
, 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 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 that uses the distance of the point from the
origin (), 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 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 ...
, 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 (in many applications referred to as the "phase" )
is the angle of the
radius 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 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 tha ...
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:
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 this can be written as
Using the function, this is sometimes abbreviated to
In
angle notation
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sine wave, sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and Phase (waves), initial phase (''θ'') are time-inva ...
, often used in
electronics to represent a
phasor 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, 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) 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, if the cubic is
irreducible; this is the so-called ''
casus irreducibilis
In algebra, ''casus irreducibilis'' (Latin for "the irreducible case") is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots th ...
'' ("irreducible case"). This conundrum led Italian mathematician
Gerolamo Cardano 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, 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 polynomia ...
, 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 (mathematics), field, often the field of the rational numbers. For many authors, the term '' ...
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
Rafael Bombelli (baptised
Baptism (from grc-x-koine, βάπτισμα, váptisma) is a form of ritual purification—a characteristic of many religions throughout time and geography. In Christianity, it is a Christian sacrament of initia ...
. 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.
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 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 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 solution
A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, divisi ...
s for the roots of
cubic and
quartic polynomials 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 Republi ...
,
Gerolamo Cardano). 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 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. 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 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 of complex analysis:
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 Danish–Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
* Norwegians, both a nation and an ethnic group native to Norway
* Demographics of Norway
*The Norwegian language, including ...
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
On ...
Caspar Wessel
Caspar Wessel (8 June 1745, Vestby – 25 March 1818, Copenhagen) was a Danish– Norwegian mathematician and cartographer. In 1799, Wessel was the first person to describe the geometrical interpretation of complex numbers as points in the comp ...
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 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 polynomia ...
. 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, Français and his brother, Bellavitis.
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 Norwegian
Norwegian, Norwayan, or Norsk may refer to:
*Something of, from, or related to Norway, a country in northwestern Europe
* Norwegians, both a nation and an ethnic group native to Norway
* Demographics of Norway
*The Norwegian language, including ...
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
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
were necessarily using them routinely before Gauss published his 1831 treatise.
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 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, Otto Hölder, 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 grou ...
, Henri Poincaré, 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 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 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 fiel ...
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 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, 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 ...
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 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 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 triangles and are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
.
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 arithmet ...
, 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 involvin ...
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 tha ...
). 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 – 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 can be defined for every complex number by the power series
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 expressi ...
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 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 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 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
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
, 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 of the complex logarithm is obtained with . It 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 , 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. 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 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 fiel ...
, 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 matrix, complex polynomial, and complex Lie algebra.
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 polynomia ...
, 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
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
. Because of this fact, is called an algebraically closed field. This property does not hold for the field of rational numbers (the polynomial does not have a rational root, since √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, 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
In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
, or a proof combining Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
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 0. This means that for any number of summands (all of which equal one).
* Second, its transcendence degree
In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...
over , the prime field
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive ide ...
of is the cardinality of the continuum.
* Third, it is algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
(see above).
It can be shown that any field having these properties is isomorphic (as a field) to For example, the algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ( ...
of the field of the -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
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series
: \begin
x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\
&=x^+ 2x^ + x^ + 2x^ + x^ + ...
. 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 nearness and continuity, which matter in areas such as analysis and topology
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 ...
, are not dealt with. The following description of as a topological field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is w ...
(that is, a field that is equipped with a topology
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 ...
, 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 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, where ranges over the field and ranges over . With this topology is isomorphic as a ''topological'' field to
The only connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
locally compact 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
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
, while the nonzero real numbers are not.
Formal construction
Construction as ordered pairs
William Rowan Hamilton 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