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

A complex number is a number which can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where i² = −1.^[1] In this expression, a is called the real part and b the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point (a, b). A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with only real numbers.

Complex numbers are used in many scientific fields, including engineering, electromagnetism, quantum physics, applied mathematics, and chaos theory. Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious", during his attempts to find solutions to cubic equations in the 16th century.^[2]

[ Overview

has no real solution, since the square of a real number is either 0 or positive. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where , so that solutions to equations like the preceding one can be found. In this case the solutions are −1 ± 3i. In fact not only quadratic equations, but all polynomial equations in a single variable can be solved using complex numbers.

[ Definition

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

A complex number is a number that can be expressed in the form

where a and b are real numbers and i is the imaginary unit, satisfying i² = −1. For example, −3.5 + 2i is a complex number. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write a − bi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i.

The set of all complex numbers is denoted by or .

The real number a of the complex number z = a + bi is called the real part of z, and the real number b is often called the imaginary part. By this convention the imaginary part is a real number – not including the imaginary unit: hence b, not bi, is the imaginary part.^[3][4] The real part is denoted by Re(z) or ℜ(z), and the imaginary part b is denoted by Im(z) or ℑ(z). For example,

Some authors write a+ib instead of a+bi (scalar multiplication between b and i is commutative). In some disciplines, in particular electromagnetism and electrical engineering, j is used instead of i, since i is frequently used for electric current. In these cases complex numbers are written as a + bj or a + jb.

A real number a can usually be regarded as a complex number with an imaginary part of zero, that is to say, a + 0i. However the sets are defined differently and have slightly different operations defined, for instance comparison operations are not defined for complex numbers. A pure imaginary number is a complex number whose real part is zero, that is to say, of the form 0 + bi.

[ Complex plane

Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; is the rectangular expression of the point.

A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-Robert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.

The defining characteristic of a position vector is that it has magnitude and direction. These are emphasised in a complex number's polar form and it turns out notably that the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds to rotating a complex number counterclockwise through 90° about the origin: .

[ History in brief

Main section: History

The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.

Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.^[5] 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.

[ Elementary operations

[ Conjugation

Geometric representation of and its conjugate in the complex plane

The complex conjugate of the complex number z = x + yi is defined to be x − yi. It is denoted or . Geometrically, is the "reflection" of z about the real axis. In particular, conjugating twice gives the original complex number: .

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

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

Conjugation distributes over the standard arithmetic operations:

The reciprocal of a nonzero complex number z = x + yi is given by

This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. Inversive geometry, a branch of geometry studying more general reflections than ones about a line, can also be expressed in terms of complex numbers.

[ Addition and subtraction

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

Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:

Similarly, subtraction is defined by

Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.

[ Multiplication and division

(distributive law)

(commutative law of addition—the order of the summands can be changed)

(commutative law of multiplication—the order of the multiplicands can be changed)

(fundamental property of the imaginary unit).

The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division:

Division can be defined in this way because of the following observation:

As shown earlier, is the complex conjugate of the denominator . The real part c and the imaginary part d of the denominator must not both be zero for division to be defined.

[ Square root

The square roots of a + bi (with b ≠ 0) are , where

and

where sgn is the signum function. This can be seen by squaring to obtain a + bi.^[6][7] Here is called the modulus of a + bi, and the square root with non-negative real part is called the principal square root.

[ Polar form

Figure 2: The argument φ and modulus r locate a point on an Argand diagram; or are polar expressions of the point.

[ Absolute value and argument

Another way of encoding points in the complex plane other than using the x- and y-coordinates is to use the distance of a point P to O, the point whose coordinates are (0, 0) (the origin), and the angle of the line through P and O. This idea leads to the polar form of complex numbers.

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

If z is a real number (i.e., y = 0), then r = |x|. In general, by Pythagoras' theorem, r is the distance of the point P representing the complex number z to the origin.

The argument or phase of z is the angle of the radius OP with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form :^[8]

The value of φ must always be expressed in radians. It can change by any multiple of 2π and still give the same angle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principal value in the interval is chosen. Values in the range are obtained by adding if the value is negative. The polar angle for the complex number 0 is undefined, but arbitrary choice of the angle 0 is common.

The value of φ equals the result of atan2: .

Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form

Using Euler's formula this can be written as

Using the cis function, this is sometimes abbreviated to

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ it is written as^[9]

[ Multiplication, division and exponentiation in polar form

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

The relevance of representing complex numbers in polar form stems from the fact that the formulas for multiplication, division and exponentiation are simpler than the ones using Cartesian coordinates. Given two complex numbers z₁ = r₁(cos φ₁ + isin φ₁) and z₂ =r₂(cos φ₂ + isin φ₂) the formula for multiplication is

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, which gives back i ² = −1. The picture at the right illustrates the multiplication of

Since the real and imaginary part of 5+5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangle 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 π.

Similarly, division is given by

This also implies de Moivre's formula for exponentiation of complex numbers with integer exponents:

The n-th roots of z are given by

for any integer k satisfying 0 ≤ k ≤ n − 1. Here is the usual (positive) nth root of the positive real number r. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cⁿ = x there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z is considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely defined number. Formulas such as

(which holds for positive real numbers), do in general not hold for complex numbers.

[ Properties

[ Field structure

The set C 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 z, its negative −z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z₁ and z₂:

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

Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z₁ < z₂ that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i² = −1 precludes the existence of an ordering on C.

When the underlying field for a mathematical topic or construct is the field of complex numbers, the thing's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

[ Solutions of polynomial equations

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

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

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

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

[ Algebraic characterization

The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ... + 1 ≠ 0 for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Q_p also satisfies these three properties, so these two fields are isomorphic. Also, C is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields which are isomorphic to C.

[ Characterization as a topological field

The preceding characterization of C describes the algebraic aspects of C, only. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:

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

Moreover, C has a nontrivial involutive automorphism (namely the complex conjugation), such that xx^∗ is in P for any nonzero x in C.

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

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

[ Formal construction

[ Formal development

Above, complex numbers have been defined by introducing i, the imaginary unit, as a symbol. More rigorously, the set C of complex numbers can be defined as the set R² of ordered pairs (a, b) of real numbers. In this notation, the above formulas for addition and multiplication read

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

Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with an addition, subtraction, multiplication and division operations which behave as is familiar from, say, rational numbers. For example, the distributive law

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

where the a₀, ..., a_n are real numbers. The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. This ring is called polynomial ring. The quotient ring R[X]/(X²+1) can be shown to be a field. This extension field contains two square roots of −1, namely (the cosets of) X and −X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X²+1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a,b) of real numbers. Moreover, the above formulas for addition etc. correspond to the ones yielded by this abstract algebraic approach – the two definitions of the field C are said to be isomorphic (as fields). Together with the above-mentioned fact that C is algebraically closed, this also shows that C is an algebraic closure of R.

[ Matrix representation of complex numbers

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

Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices. The geometric description of the multiplication of complex numbers can also be phrased in terms of rotation matrices by using this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:

The conjugate corresponds to the transpose of the matrix.

Though this representation of complex numbers with matrices is the most common, many other representations arise from matrices other than that square to the negative of the identity matrix. See thee article on 2 × 2 real matrices for other representations of complex numbers.

[ Complex analysis

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