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In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:

- $i^2=-1.,$

Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively.

Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers.

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

The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.

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

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

- $a\; +\; bi\; ,$

For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + ib, the real part a is denoted Re(z) or ℜ(z), and the imaginary part b is denoted Im(z) or ℑ(z).

The real numbers, R, may be regarded as a subset of C by considering every real number a complex number with an imaginary part of zero; that is, the real number a is identified with the complex number . Complex numbers with a real part of zero are called imaginary numbers; instead of writing , that imaginary number is usually denoted as just bi. If b equals 1, instead of using or 1i, the number is denoted as i.

In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj.

- * Addition: $,(a\; +\; bi)\; +\; (c\; +\; di)\; =\; (a\; +\; c)\; +\; (b\; +\; d)i$

- * Subtraction: $,(a\; +\; bi)\; -\; (c\; +\; di)\; =\; (a\; -\; c)\; +\; (b\; -\; d)i$

- * Multiplication: $,(a\; +\; bi)\; (c\; +\; di)\; =\; ac\; +\; bci\; +\; adi\; +\; bd\; i^2\; =\; (ac\; -\; bd)\; +\; (bc\; +\; ad)i$

- * Division: $,frac\{(a\; +\; bi)\}\{(c\; +\; di)\}\; =\; left(\{ac\; +\; bd\; over\; c^2\; +\; d^2\}right)\; +\; left(\{bc\; -\; ad\; over\; c^2\; +\; d^2\}\; right)i,,$

It is also possible to represent complex numbers as ordered pairs of real numbers, so that the complex number a + ib corresponds to (a, b). In this representation, the algebraic operations have the following formulas:

- (a, b) + (c, d) = (a + c, b + d)

- (a, b)(c, d) = (ac − bd, bc + ad)

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

A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers are a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess:

- An additive identity ("zero"), 0 + 0i.
- A multiplicative identity ("one"), 1 + 0i.
- An additive inverse of every complex number. The additive inverse of a + bi is −a − bi.
- A multiplicative inverse (reciprocal) of every nonzero complex number. The multiplicative inverse of a + bi is $\{aover\; a^2+b^2\}+\; left(\{-bover\; a^2+b^2\}right)i.$

Other fields include the real numbers and the rational numbers. When each real number a is identified with the complex number a + 0i, the field of real numbers R becomes a subfield of C.

The complex numbers C can also be characterized as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

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

The absolute value has three important properties:

- $|\; z\; |\; geq\; 0,\; ,$ where $|\; z\; |\; =\; 0\; ,$ if and only if $z\; =\; 0\; ,$

- $|\; z\; +\; w\; |\; leq\; |\; z\; |\; +\; |\; w\; |\; ,$ (triangle inequality)

- $|\; z\; cdot\; w\; |\; =\; |\; z\; |\; cdot\; |\; w\; |\; ,$

for all complex numbers z and w. These imply that $|1|=1$ and $|z/w|=|z|/|w|$. By defining the distance function $d(z,w)=|z-w|$, we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

The complex conjugate of the complex number $z=x+yi$ is defined to be $x-yi$, written as $bar\{z\}$ or $z^*,$. As seen in the figure, $bar\{z\}$ is the "reflection" of z about the real axis, and so both $z+bar\{z\}$ and $zcdotbar\{z\}$ are real numbers. Many identities relate complex numbers and their conjugates:

- $overline\{z+w\}\; =\; bar\{z\}\; +\; bar\{w\}$

- $overline\{zcdot\; w\}\; =\; bar\{z\}cdotbar\{w\}$

- $overline\{(z/w)\}\; =\; bar\{z\}/bar\{w\}$

- $bar\{bar\{z\}\}=z$

- $bar\{z\}=z$ if and only if z is real

- $bar\{z\}=-z$ if and only if z is purely imaginary

- $operatorname\{Re\},(z)\; =\; tfrac\{1\}\{2\}(z+bar\{z\})$

- $operatorname\{Im\},(z)\; =\; tfrac\{1\}\{2i\}(z-bar\{z\})$

- $|z|=|bar\{z\}|$

- $|z|^2\; =\; zcdotbar\{z\}$

- $z^\{-1\}\; =\; frac\{bar\{z\}\}\{|z|^\{2\}\}$ if z is non-zero.

The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates.

That conjugation commutes with all the algebraic operations (and many functions; e.g. $sinbar\; z=overline\{sin\; z\}$) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function $f(z)\; =\; bar\{z\}$ is not complex-differentiable (see holomorphic function).

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

- The sum of two points A and B of the complex plane is the point X = A + B such that the triangles with vertices 0, A, B, and X, B, A, are congruent. Thus the addition of two complex numbers is the same as vector addition of two vectors.
- The product of two points A and B is the point X = AB such that the triangles with vertices 0, 1, A, and 0, B, X, are similar.
- The complex conjugate of a point A is the point X = A* such that the triangles with vertices 0, 1, A, and 0, 1, X, are mirror images of each other.

These geometric interpretations allow problems of geometry to be translated into algebra. And, conversely, geometric problems can be examined algebraically. For example, the problem of the geometric construction of the 17-gon is thus translated into the analysis of the algebraic equation x^{17} = 1.

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

- $x\; =\; r\; cos\; varphi$

- $y\; =\; r\; sin\; varphi$

- $r\; =\; |z|\; =\; sqrt\{x^2+y^2\}$

- $varphi\; =\; arg(z)\; =\; operatorname\{atan2\}(y,x)$

The value of φ can change by any multiple of 2π and still give the same angle. The function atan2 gives the principal value in the range (−π, +π]. If a non-negative value of φ in the range [0, 2π) is desired, add 2π to any negative value.

The arg function is sometimes considered as multivalued taking as possible values atan2(y, x) + 2πk, where k is any integer.

- $z\; =\; r,(cos\; varphi\; +\; isin\; varphi\; ),$

- $z\; =\; r,mathrm\{e\}^\{i\; varphi\},$

Using sum and difference identities it follows that

- $r\_1,e^\{ivarphi\_1\}\; cdot\; r\_2,e^\{ivarphi\_2\}$

and that

- $frac\{r\_1,e^\{ivarphi\_1\}\}\{r\_2,e^\{ivarphi\_2\}\}$

Exponentiation with integer exponents; according to De Moivre's formula,

- $(cosvarphi\; +\; isinvarphi)^n\; =\; cos(nvarphi)\; +\; isin(nvarphi),,$

from which it follows that

- $(r(cosvarphi\; +\; isinvarphi))^n\; =\; (r,e^\{ivarphi\})^n\; =\; r^n,e^\{invarphi\}\; =\; r^n,(cos\; nvarphi\; +\; mathrm\{i\}\; sin\; n\; varphi).,$

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

Multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching, in particular multiplication by i corresponds to a counter-clockwise rotation by 90 degrees (π/2 radians). The geometric content of the equation i^{ 2} = −1 is that a sequence of two 90 degree rotations results in a 180 degree (π radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

If c is a complex number and n a positive integer, then any complex number z satisfying z^{n} = c is called an n-th root of c. If c is nonzero, there are exactly n distinct n-th roots of c, which can be found as follows. Write $c=re^\{ivarphi\}$ with real numbers r > 0 and φ, then the set of n-th roots of c is

- $\{\; sqrt[n]r,e^\{i(frac\{varphi+2kpi\}\{n\})\}\; mid\; kin\{0,1,ldots,n-1\}\; ,\; \},$

- $$

a & -b

b & ;; aend{bmatrix}

where a and b are real numbers. The sum and product of two such matrices is again of this form, and the product operation on matrices of this form is commutative. Every non-zero matrix of this form is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field, isomorphic to the field of complex numbers. Every such matrix can be written as

- $$

a & -b

b & ;; aend{bmatrix} = a begin{bmatrix}

1 & ;; 0

0 & ;; 1end{bmatrix} + b begin{bmatrix}

0 & -1

1 & ;; 0end{bmatrix} which suggests that we should identify the real number 1 with the identity matrix

- $$

1 & ;; 0

0 & ;; 1end{bmatrix}, and the imaginary unit i with

- $$

0 & -1

1 & ;; 0end{bmatrix},

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to the 2×2 matrix that represents −1.

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

- $|z|^2\; =$

a & -b

b & aend{vmatrix}

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

It should also be noted that the two eigenvalues of the 2x2 matrix representing a complex number are the complex number itself and its conjugate.

R-linear maps C → C have the general form

- $f(z)=az+boverline\{z\}$

The function

- $f(z)=az,$

- $f(z)=boverline\{z\}$

One construction of C is as a field extension of the field R of real numbers, in which a root of x^{2}+1 are added. To construct this extension, begin with the ring R[x] of polynomials of the real numbers in the variable x. Because the polynomial x^{2}+1 is irreducible over R, the quotient ring R[x]/(x^{2}+1) will be a field. This extension field will contain two square roots of -1; one of them is selected and denoted i. The set {1, i} will form a basis for the extension field over the reals, which means that each element of the extension field can be written in the form a+ b·i. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers.

Although only roots of x^{2}+1 were explicitly added, the resulting complex field is actually algebraically closed – every polynomial with coefficients in C factors into linear polynomials with coefficients in C. Because each field has only one algebraic closure, up to field isomorphism, the complex numbers can be characterized as the algebraic closure of the real numbers.

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

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

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

The following properties characterize C as a topological field:

- C is a field.
- C contains a subset P of nonzero elements satisfying:
- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x-y or y-x is in P
- If S is any nonempty subset of P, then S+P=x+P for some x in P.
- C has a nontrivial involutive automorphism x→x*, fixing P and such that xx* is in P for any nonzero x in C.

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

- $B(x,p)\; =\; \{y\; |\; p\; -\; (y-x)(y-x)^*in\; P\}$

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

To see that these properties characterize C as a topological field, one notes that P ∪ {0} ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization.

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

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

In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are

- in the right half plane, it will be unstable,
- all in the left half plane, it will be stable,
- on the imaginary axis, it will have marginal stability.

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

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

- $f\; (t\; )\; =\; z\; e^\{iomega\; t\}\; ,$

In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This approach is called phasor calculus. This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and Wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

The complex number field is relevant in the mathematical formulation 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.

Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0:

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

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z^{3} = i has solutions –i, $\{scriptstylefrac\{sqrt\{3\}\}\{2\}\}+\{scriptstylefrac\{1\}\{2\}\}i$ and $\{scriptstylefrac\{-sqrt\{3\}\}\{2\}\}+\{scriptstylefrac\{1\}\{2\}\}i$. Substituting these in turn for $\{scriptstylesqrt\{-1\}^\{1/3\}\}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x^{3} – x = 0.

This was doubly unsettling since not even negative numbers were considered to be on firm ground at the time. The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation $sqrt\{-1\}^2=sqrt\{-1\}sqrt\{-1\}=-1$ seemed to be capriciously inconsistent with the algebraic identity $sqrt\{a\}sqrt\{b\}=sqrt\{ab\}$, which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity $scriptstyle\; 1/sqrt\{a\}=sqrt\{1/a\}$) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of $sqrt\{-1\}$ to guard against this mistake.

The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To de Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula:

- $(cos\; theta\; +\; isin\; theta)^\{n\}\; =\; cos\; n\; theta\; +\; isin\; n\; theta\; ,$

and to Euler (1748) Euler's formula of complex analysis:

- $cos\; theta\; +\; isin\; theta\; =\; e\; ^\{itheta\; \}.\; ,$

The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion. The idea of the graphic representation of complex numbers had appeared, however, as early as 1685, in Wallis's De Algebra tractatus.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that $pmsqrt\{-1\}$ should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. Mention should also be made of an excellent little treatise by Mourey (1828), in which the foundations for the theory of directional numbers are scientifically laid. The general acceptance of the theory is not a little due to the labors of Augustin Louis Cauchy and Niels Henrik Abel, and especially the latter, who was the first to boldly use complex numbers with a success that is well known.

The common terms used in the theory are chiefly due to the founders. Argand called $cos\; phi\; +\; isin\; phi$ the direction factor, and $r\; =\; sqrt\{a^2+b^2\}$ the modulus; Cauchy (1828) called $cos\; phi\; +\; isin\; phi$ the reduced form (l'expression réduite); Gauss used i for $sqrt\{-1\}$, introduced the term complex number for $a+bi$, and called $a^2+b^2$ the norm.

The expression direction coefficient, often used for $cos\; phi\; +\; i\; sin\; phi$, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.

Following Cauchy and Gauss have come a number of contributors of high rank, of whom the following may be especially mentioned: Kummer (1844), Leopold Kronecker (1845), Scheffler (1845, 1851, 1880), Bellavitis (1835, 1852), Peacock (1845), and De Morgan (1849). Möbius must also be mentioned for his numerous memoirs on the geometric applications of complex numbers, and Dirichlet for the expansion of the theory to include primes, congruences, reciprocity, etc., as in the case of real numbers.

A complex ring or field is a set of complex numbers which is closed under addition, subtraction, and multiplication. Gauss studied complex numbers of the form $a\; +\; bi$, where a and b are integral, or rational (and i is one of the two roots of $x^2\; +\; 1\; =\; 0$). His student, Ferdinand Eisenstein, studied the type $a\; +\; bomega$, where $omega$ is a complex root of $x^3\; -\; 1\; =\; 0$. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity $x^k\; -\; 1\; =\; 0$ for higher values of $k$. This generalization is largely due to Kummer, who also invented ideal numbers, which were expressed as geometrical entities by Felix Klein in 1893. The general theory of fields was created by Évariste Galois, who studied the fields generated by the roots of any polynomial equation in one variable.

The late writers (from 1884) on the general theory include Weierstrass, Schwarz, Richard Dedekind, Otto Hölder, Henri Poincaré, Eduard Study, and Alexander MacFarlane.

- Circular motion using complex numbers
- Complex base systems
- Complex geometry
- De Moivre's formula
- Domain coloring
- Euler's identity
- Hypercomplex number
- Local field
- Mandelbrot set
- Mathematical diagram
- Quaternion
- Riemann sphere (extended complex plane)
- Split-complex number
- Square root of complex numbers
- Imaginary number/Imaginary unit

- {{citation|title=An Imaginary Tale: The Story of $sqrt\{-1\}$|first=Paul J.|last=Nahin|publisher=Princeton University Press|isbn=0-691-02795-1|year=1998|edition=hardcover}}
- :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.

- The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.
- Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
- Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-198-53447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.

- Euler's work on Complex Roots of Polynomials at Convergence. MAA Mathematical Sciences Digital Library.
- John and Betty's Journey Through Complex Numbers
- MathWorld articles Complex number and Argand Diagram, and demonstration "Argand Diagram"
- Complex Numbers Module by John H. Mathews
- Dimensions: a math film. Chapter 5 presents an introduction to complex arithmetic and stereographic projection. Chapter 6 discusses transformations of the complex plane, Julia sets, and the Mandelbrot set.

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