Complex Number

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## Definition

## Notation

## Visualisation

### Cartesian complex plane

### Polar complex plane

#### Modulus and argument

### Complex graphs

## History

## Relations and operations

### Equality

### Ordering

### Conjugate

### Addition and subtraction

### Multiplication

### Reciprocal and division

### Multiplication and division in polar form

### Square root

### Exponentiation

#### Euler's formula

#### Natural logarithm

#### Integer and fractional exponents

## Properties

### Field structure

### Solutions of polynomial equations

### Algebraic characterization

### Characterization as a topological field

## Formal construction

### Construction as ordered pairs

### Construction as a quotient field

### Matrix representation of complex numbers

## Complex analysis

### Holomorphic functions

## Applications

### Geometry

#### Shapes

#### Fractal geometry

#### Triangles

### Algebraic number theory

### Analytic number theory

### Improper integrals

### Dynamic equations

### In applied mathematics

#### Control theory

#### Signal analysis

### In physics

#### Electromagnetism and electrical engineering

#### Fluid dynamics

#### Quantum mechanics

#### Relativity

## See also

## Notes

## References

#### Works cited

## Further reading

### Mathematical

### Historical

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Complex Number

A **complex number** is a number that can be expressed in the form *a* + *bi*, where *a* and *b* are real numbers, and *i* is a solution of the equation *x*^{2} = -1. Because no real number satisfies this equation, *i* is called an imaginary number. For the complex number *a* + *bi*, *a* is called the **real part**, and *b* is called the **imaginary part**. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers, and are fundamental in many aspects of the scientific description of the natural world.^{[note 1]}^{[1]}

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

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate *i* (sometimes called the imaginary unit) that is taken to satisfy the relation *i*^{2} = -1, so that solutions to equations like the preceding one can be found. In this case the solutions are -1 + 3*i* and -1 - 3*i*, as can be verified using the fact that *i*^{2} = -1:

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.^{[2]}

Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number *i*.^{[3]} This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable *i*, with the rule *i*^{2} = -1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.

Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number *a* + *bi* can be identified with the point (*a*, *b*) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.

The geometric identification of the complex numbers with the complex plane, which is a Euclidean plane (), makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis. The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space; e.g., the multiplication of two complex numbers always yields again a complex number, and should not be mistaken for the usual "products" involving vectors, like the *scalar multiplication*, the *scalar product* or other (sesqui)linear forms, available in many vector spaces; and the broadly exploited *vector product* exists only in an orientation-dependent form in three dimensions.

Based on the concept of real numbers, a complex number is a number of the form *a* + *bi*, where a and b are real numbers and *i* is an indeterminate satisfying *i*^{2} = -1. For example, 2 + 3*i* is a complex number.^{[4]}

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate *i*, for which the relation *i*^{2} + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation *i*^{2} + 1 = 0 induces the equalities *i*^{4k} = 1 , *i*^{4k+1} = *i* , *i*^{4k+2} = -1 , and *i*^{4k+3} = -i , which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form *a* + *bi* with real coefficients a, b.

The real number a is called the *real part* of the complex number *a* + *bi*; the real number b is called its *imaginary part*. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not *bi*.^{[5]}^{[6]}

Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate *i*, by the ideal generated by the polynomial *i*^{2} + 1 (see below).^{[7]}

A real number a can be regarded as a complex number *a* + 0*i* whose imaginary part is 0. A purely imaginary number *bi* is a complex number 0 + *bi* whose real part is zero. As with polynomials, it is common to write a for *a* + 0*i* and *bi* for 0 + *bi*. Moreover, when the imaginary part is negative, i.e., *b* = -*|b|* < 0, it is common to write *a* - *|b|i* instead of *a* + (-*|b|*)*i*; for example, for *b* = -4, 3 - 4*i* can be written instead of 3 + (-4)*i*.

Since in polynomials with real coefficients the multiplication of the indeterminate *i* and a real is commutative, the polynomial *a* + *bi* may be written as *a* + *ib*. This is often expedient for imaginary parts denoted by expressions, e.g., when b is a radical.^{[8]}

The real part of a complex number z is denoted by Re(*z*) or R(*z*); the imaginary part of a complex number z is denoted by Im(*z*) or I(*z*). For example,

- and

The set of all complex numbers is denoted by (upright bold) or (blackboard bold).

In some disciplines, in particular electromagnetism and electrical engineering, *j* is used instead of *i* since i is frequently used to represent electric current.^{[9]} In these cases complex numbers are written as *a* + *bj* or *a* + *jb*.

A complex number z can thus be identified with an ordered pair (Re(*z*), Im(*z*)) of real numbers, which in turn may be interpreted as coordinates of a point in a two-dimensional space. The most immediate space is the Euclidean plane with suitable coordinates, which is then called **complex plane** or **Argand diagram,**^{[10]}^{[11]} named after Jean-Robert Argand. Another prominent space on which the coordinates may be projected is the two-dimensional surface of a sphere, which is then called Riemann sphere.

The definition of the complex numbers involving two arbitrary real values immediately suggest 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, increasing values upwards.

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

Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition, while multiplication (see below) corresponds to multiplying their magnitudes and adding the angles they make with the real axis. Viewed in this way the multiplication of a complex number by *i* corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin

An alternative option for coordinates in the complex plane is the polar coordinate system that uses the distance of the point z from the origin (*O*), and the angle subtended between the positive real axis and the line segment *Oz* in a counterclockwise sense. This leads to the polar form of complex numbers.

The *absolute value* (or *modulus* or *magnitude*) of a complex number *z* = *x* + *yi* is^{[12]}

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

By Pythagoras' theorem, the absolute value of complex number is the distance to the origin of the point representing the complex number in the complex plane.

The *argument* of z (in many applications referred to as the "phase" ?) is the angle of the radius *Oz* with the positive real axis, and is written as . As with the modulus, the argument can be found from the rectangular form ^{[13]} by applying the inverse tangent to the quotient of imaginary-by-real parts. By using a half-angle identity a single branch of the arctan suffices to cover the range of the arg-function, (-?, ?], and avoids a more subtle case-by-case analysis

Normally, as given above, the principal value in the interval (-?, ?] is chosen. Values in the range [0, 2?) are obtained by adding 2? if the value is negative. The value of ? is expressed in radians in this article. It can increase by any integer multiple of 2? and still give the same angle, viewed as subtended by the rays of the positive real axis and from the origin through *z*. Hence, the arg function is sometimes considered as multivalued. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common.

The value of ? equals the result of atan2:

Together, r 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 cis function, this is sometimes abbreviated to

In angle notation, often used in electronics to represent a phasor with amplitude r and phase ?, it is written as^{[14]}

When visualizing complex functions, both a complex input and output are needed. Because each complex number is represented in two dimensions, visually graphing a complex function would require the perception of a four dimensional space, which is possible only in projections. Because of this, other ways of visualizing complex functions have been designed.

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

Riemann surfaces are another way to visualize complex functions.^{[further explanation needed]} Riemann surfaces can be thought of as deformations of the complex plane; while the horizontal axes represent the real and imaginary inputs, the single vertical axis only represents either the real or imaginary output. However, Riemann surfaces are built in such a way that rotating them 180 degrees shows the imaginary output, and vice versa. Unlike domain coloring, Riemann surfaces can represent multivalued functions like .

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,^{[15]} though his understanding was rudimentary.

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

Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli.^{[16]} 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 considers, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced it by its positive ().^{[17]}

The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form ^{[note 2]} gives the solution to the equation *x*^{3} = *x* as

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

The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature^{[18]}

[...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.

([...] quelquefois seulement imaginaires c'est-à-dire que l'on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu'il n'y a quelquefois aucune quantité qui corresponde à celle qu'on imagine.)

A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity ) 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 to guard against this mistake.^{[]} Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.

In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:

In 1748 Leonhard 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 Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's *De Algebra tractatus*.

Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of -1". 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. 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.^{[19]}

The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.^{[20]}

"If this subject has hitherto been considered from the wrong viewpoint and thus enveloped in mystery and surrounded by darkness, it is largely an unsuitable terminology which should be blamed. Had +1, -1 and ?-1, instead of being called positive, negative and imaginary (or worse still, impossible) unity, been given the names say,of direct, inverse and lateral unity, there would hardly have been any scope for such obscurity." - Gauss

^{[21]}

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 (1828) called the *reduced form* (l'expression réduite) and apparently introduced the term *argument*; Gauss used *i* for , introduced the term *complex number* for *a* + *bi*, and called *a*^{2} + *b*^{2} 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, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.

Two complex numbers are equal if and only if both their real and imaginary parts are equal. That is, complex numbers and are equal if and only if and . If the complex numbers are written in polar form, they are equal if and only if they have the same argument and the same magnitude.

Since complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers. In fact, there is no linear ordering on the complex numbers that is compatible with addition and multiplication - the complex numbers cannot have the structure of an ordered field. This is because any square in an ordered field is at least 0, but *i*^{2} = -1.

The *complex conjugate* of the complex number *z* = *x* + *yi* is given by *x* - *yi*. It is denoted by either or *z**.^{[22]} 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 z about the real axis. Conjugating twice gives the original complex number

which makes this operation an involution. 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

- and

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

The product of a complex number and its conjugate is always a positive 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 z can be extracted using the conjugation:

- and

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 is looked for.

Two complex numbers and are most easily added by separately adding their real and imaginary parts of the summands. That is to say:

Similarly, subtraction can be performed as

Using the visualization of complex numbers in the complex plane, the 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. A visualization of the subtraction can be achieved by considering addition of the negative subtrahend.

Since the real part, the imaginary part, and the indeterminate in a complex number are all considered as numbers in themselves, two complex numbers, given as and are multiplied under the rules of the distributive property, the commutative properties and the defining property in the following way

Using the conjugation, the reciprocal of a nonzero complex number *z* = *x* + *yi* 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

Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers *z*_{1} = *r*_{1}(cos ?_{1} + *i* sin ?_{1}) and *z*_{2} = *r*_{2}(cos ?_{2} + *i* sin ?_{2}), because of the trigonometric identities

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 *i* corresponds to a quarter-turn 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 + 5*i* are equal, the argument of that number is 45 degrees, or ?/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula

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

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*.^{[23]}^{[24]} Here is called the modulus of *a* + *bi*, and the square root sign indicates the square root with non-negative real part, called the **principal square root**; also where ^{[25]}

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

- ,

where *e* is the base of the natural logarithm. This can be proved through induction by observing that

and so on, and by considering the Taylor series expansions of *e*^{ix}, cos *x* and sin *x*:

The rearrangement of terms is justified because each series is absolutely convergent.

It follows from Euler's formula that, for any complex number *z* written in polar form,

where *r* is a non-negative real number, one possible value for the complex logarithm of *z* is

Because cosine and sine are periodic functions, other possible values may be obtained. For example, , so both and are two possible values for the natural logarithm of .

To deal with the existence of more than one possible value for a given input, the complex logarithm may be considered a multi-valued function, with

Alternatively, a branch cut can be used to define a single-valued "branch" of the complex logarithm.

We may use the identity

to define complex exponentiation, which is likewise multi-valued:

When *n* is an integer, this simplifies to de Moivre's formula:

The nth 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 *c*^{n} = *r*, 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.

The set **C** of complex numbers is a field.^{[26]} Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number z, its additive inverse -*z* is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers *z*_{1} and *z*_{2}:

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 *z*_{1} < *z*_{2} that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so *i*^{2} = -1 precludes the existence of an ordering on **C**.^{[27]}

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.

Given any complex numbers (called coefficients) *a*_{0}, ..., *a*_{n}, the equation

has at least one complex solution *z*, provided that at least one of the higher coefficients *a*_{1}, ..., *a*_{n} is nonzero.^{[28]} This is the statement of the *fundamental theorem of algebra*, of Carl Friedrich Gauss and Jean le Rond d'Alembert. Because of this fact, **C** is called an algebraically closed field. This property does not hold for the field of rational numbers **Q** (the polynomial *x*^{2} - 2 does not have a rational root, since is not a rational number) nor the real numbers **R** (the polynomial *x*^{2} + *a* does not have a real root for *a* > 0, since the square of x is positive for any real number x).

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

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

The field **C** has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ? + 1 ? 0 for any number of summands (all of which equal one). Second, its transcendence degree over **Q**, the prime field of **C**, is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to **C**. For example, the algebraic closure of **Q**_{p} also satisfies these three properties, so these two fields are isomorphic (as fields, but not as topological fields).^{[29]} Also, **C** is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that **C** contains many proper subfields that are isomorphic to **C**.

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

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

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

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

The only connected locally compact topological fields are **R** and **C**. This gives another characterization of **C** as a topological field, since **C** can be distinguished from **R** because the nonzero complex numbers are connected, while the nonzero real numbers are not.^{[30]}

William Rowan Hamilton introduced the approach to define the set **C** of complex numbers^{[31]} as the set **R**^{2} of of real numbers, in which the following rules for addition and multiplication are imposed:^{[32]}

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 addition, subtraction, multiplication and division operations that 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 *a*_{0}, ..., *a*_{n} are real numbers. The usual addition and multiplication of polynomials endows the set **R**[*X*] of all such polynomials with a ring structure. This ring is called the polynomial ring over the real numbers.

The set of complex numbers is defined as the quotient ring **R**[*X*]/(*X* ^{2} + 1).^{[33]} This extension field contains two square roots of -1, namely (the cosets of) *X* and -*X*, respectively. (The cosets of) 1 and *X* form a basis of **R**[*X*]/(*X* ^{2} + 1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (*a*, *b*) of real numbers. The quotient ring is a field, because *X*^{2} + 1 is irreducible over **R**, so the ideal it generates is maximal.

The formulas for addition and multiplication in the ring **R**[*X*], modulo the relation *X*^{2} = -1, correspond to the formulas for addition and multiplication of complex numbers defined as ordered pairs. So the two definitions of the field **C** are isomorphic (as fields).

Accepting that **C** is algebraically closed, since it is an algebraic extension of **R** in this approach, **C** is therefore the algebraic closure of **R**.

Complex numbers *a* + *bi* 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 product being:

The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:

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 the article on 2 × 2 real matrices for other representations of complex numbers.

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.

The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (?, ?)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, **C**, endowed with the metric

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

for any two complex numbers *z*_{1} and *z*_{2}.

Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the *exponential function* exp(*z*), also written *e*^{z}, is defined as the infinite series

The series defining the real trigonometric functions sine and cosine, as well as the hyperbolic functions sinh and cosh, also carry over to complex arguments without change. For the other trigonometric and hyperbolic functions, such as tangent, things are slightly more complicated, as the defining series do not converge for all complex values. Therefore, one must define them either in terms of sine, cosine and exponential, or, equivalently, by using the method of analytic continuation.

*Euler's formula* states:

for any real number *?*, in particular

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

for any complex number *w* ? 0. It can be shown that any such solution z - called complex logarithm of w - satisfies

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

Complex exponentiation *z*^{?} is defined as

and is multi-valued, except when is an integer. For *?* = 1 / *n*, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.

Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy

Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.

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

with complex coefficients a and b. This map is holomorphic if and only if *b* = 0. The second summand is real-differentiable, but does not satisfy the Cauchy-Riemann equations.

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

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

Three non-collinear points in the plane determine the **shape** of the triangle . Locating the points in the complex plane, this shape of a triangle may be expressed by complex arithmetic as

The shape of a triangle will remain the same, when the complex plane is transformed by translation or dilation (by an affine transformation), corresponding to the intuitive notion of shape, and describing similarity. Thus each triangle is in a similarity class of triangles with the same shape.^{[34]}

The Mandelbrot set is a popular example of a fractal formed on the complex plane. It is defined by plotting every location where iterating the sequence does not diverge when iterated infinitely. Similarly, Julia sets have the same rules, except where remains constant.

Every triangle has a unique Steiner inellipse - an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:^{[35]}^{[36]} Denote the triangle's vertices in the complex plane as *a* = *x*_{A} + *y*_{A}*i*, *b* = *x*_{B} + *y*_{B}*i*, and *c* = *x*_{C} + *y*_{C}*i*. Write the cubic equation , take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.

As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in **C**. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers - they are a principal object of study in algebraic number theory. Compared to **Q**, the algebraic closure of **Q**, which also contains all algebraic numbers, **C** has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge - a purely geometric problem.

Another example are Gaussian integers, that is, numbers of the form *x* + *iy*, where x and y are integers, which can be used to classify sums of squares.

Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function ?(*s*) is related to the distribution of prime numbers.

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

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form *f*(*t*) = *e*^{rt}. Likewise, in difference equations, the complex roots r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form *f*(*t*) = *r*^{t}.

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

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

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

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

Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value |*z*| of the corresponding z is the amplitude and the argument arg(*z*) is the phase.

If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form

and

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

This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

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

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

In electrical engineering, the imaginary unit is denoted by *j*, to avoid confusion with I, which is generally in use to denote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.

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

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

The complex-valued signal is called the analytic representation of the real-valued, measurable signal .
^{[37]}

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

The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics - the Schrödinger equation and Heisenberg's matrix mechanics - make use of complex numbers.

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

The process of extending the field **R** of reals to **C** is known as the Cayley-Dickson construction. It can be carried further to higher dimensions, yielding the quaternions **H** and octonions **O** which (as a real vector space) are of dimension 4 and 8, respectively.
In this context the complex numbers have been called the **binarions**.^{[38]}

Just as by applying the construction to reals the property of ordering is lost, properties familiar from real and complex numbers vanish with each extension. The quaternions lose commutativity, i.e.: *x*·*y* ? *y*·*x* for some quaternions *x*, *y*, and the multiplication of octonions, additionally to not being commutative, fails to be associative: (*x*·*y*)·*z* ? *x*·(*y*·*z*) for some octonions *x*, *y*, *z*.

Reals, complex numbers, quaternions and octonions are all normed division algebras over **R**. By Hurwitz's theorem they are the only ones; the sedenions, the next step in the Cayley-Dickson construction, fail to have this structure.

The Cayley-Dickson construction is closely related to the regular representation of **C**, thought of as an **R**-algebra (an **R**-vector space with a multiplication), with respect to the basis (1, *i*). This means the following: the **R**-linear map

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

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

has the property that its square is the negative of the identity matrix: *J*^{2} = -*I*. Then

is also isomorphic to the field **C**, and gives an alternative complex structure on **R**^{2}. This is generalized by the notion of a linear complex structure.

Hypercomplex numbers also generalize **R**, **C**, **H**, and **O**. For example, this notion contains the split-complex numbers, which are elements of the ring **R**[*x*]/(*x*^{2} - 1) (as opposed to **R**[*x*]/(*x*^{2} + 1)). In this ring, the equation *a*^{2} = 1 has four solutions.

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

The fields **R** and **Q**_{p} and their finite field extensions, including **C**, are local fields.

- Algebraic surface
- Circular motion using complex numbers
- Complex-base system
- Complex geometry
- Eisenstein integer
- Euler's identity
- Geometric algebra (which includes the complex plane as the 2-dimensional spinor subspace )
- Root of unity
- Unit complex number

**^**For an extensive account of the history, from initial skepticism to ultimate acceptance, See (Bourbaki 1998), pages 18-24.**^**In modern notation, Tartaglia's solution is based on expanding the cube of the sum of two cube roots: With , , , u and v can be expressed in terms of p and q as and , respectively. Therefore, . When is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugate of the first one.

**^**Penrose, Roger (2016).*The Road to Reality: A Complete Guide to the Laws of the Universe*(reprinted ed.). Random House. pp. 72-73. ISBN 978-1-4464-1820-8.Extract of p. 73: "complex numbers, as much as reals, and perhaps even more, find a unity with nature that is truly remarkable. It is as though Nature herself is as impressed by the scope and consistency of the complex-number system as we are ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales."**^**Burton, David M. (1995),*The History of Mathematics*(3rd ed.), New York: McGraw-Hill, p. 294, ISBN 978-0-07-009465-9**^**Bourbaki, Nicolas. "VIII.1".*General topology*. Springer-Verlag.**^**Axler, Sheldon (2010).*College algebra*. Wiley. p. 262.**^**Spiegel, M.R.; Lipschutz, S.; Schiller, J.J.; Spellman, D. (14 April 2009),*Complex Variables (2nd Edition)*, Schaum's Outline Series, McGraw Hill, ISBN 978-0-07-161569-3**^**Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), "Chapter P",*College Algebra and Trigonometry*(6 ed.), Cengage Learning, p. 66, ISBN 978-0-618-82515-8**^**Bourbaki, Nicolas. "VIII.1".*General topology*. Springer-Verlag.**^**See (Ahlfors 1979).**^**Brown, James Ward; Churchill, Ruel V. (1996),*Complex variables and applications*(6th ed.), New York: McGraw-Hill, p. 2, ISBN 978-0-07-912147-9,In electrical engineering, the letter

*j*is used instead of*i*.**^**Pedoe, Dan (1988),*Geometry: A comprehensive course*, Dover, ISBN 978-0-486-65812-4**^**See (Solomentsev 2001): "The plane $\R^2$ whose points are identified with the elements of $\C$ is called the complex plane"... "The complete geometric interpretation of complex numbers and operations on them appeared first in the work of C. Wessel (1799). The geometric representation of complex numbers, sometimes called the "Argand diagram" , came into use after the publication in 1806 and 1814 of papers by J.R. Argand, who rediscovered, largely independently, the findings of Wessel".**^**See (Apostol 1981), page 18.**^**Kasana, H.S. (2005), "Chapter 1",*Complex Variables: Theory And Applications*(2nd ed.), PHI Learning Pvt. Ltd, p. 14, ISBN 978-81-203-2641-5**^**Nilsson, James William; Riedel, Susan A. (2008), "Chapter 9",*Electric circuits*(8th ed.), Prentice Hall, p. 338, ISBN 978-0-13-198925-2**^**Kline, Morris.*A history of mathematical thought, volume 1*. p. 253.**^**Katz, Victor J. (2004), "9.1.4",*A History of Mathematics, Brief Version*, Addison-Wesley, ISBN 978-0-321-16193-2**^**Nahin, Paul J. (2007),*An Imaginary Tale: The Story of*, Princeton University Press, ISBN 978-0-691-12798-9, retrieved 2011**^**Descartes, René (1954) [1637],*La Géométrie | The Geometry of René Descartes with a facsimile of the first edition*, Dover Publications, ISBN 978-0-486-60068-0, retrieved 2011**^**Caparrini, Sandro (2000), "On the Common Origin of Some of the Works on the Geometrical Interpretation of Complex Numbers", in Kim Williams (ed.),*Two Cultures*, Birkhäuser, p. 139, ISBN 978-3-7643-7186-9Extract of page 139**^**Hardy, G.H.; Wright, E.M. (2000) [1938],*An Introduction to the Theory of Numbers*, OUP Oxford, p. 189 (fourth edition), ISBN 978-0-19-921986-5**^**Extracted quotation from "A Short History of Complex Numbers", Orlando Merino, University of Rhode Island (January, 2006) http://www.math.uri.edu/~merino/spring06/mth562/ShortHistoryComplexNumbers2006.pdf**^**For the former notation, See (Apostol 1981), pages 15-16.**^**Abramowitz, Milton; Stegun, Irene A. (1964),*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, Courier Dover Publications, p. 17, ISBN 978-0-486-61272-0, Section 3.7.26, p. 17**^**Cooke, Roger (2008),*Classical algebra: its nature, origins, and uses*, John Wiley and Sons, p. 59, ISBN 978-0-470-25952-8, Extract: page 59**^**See (Ahlfors 1979), page 3.**^**See (Apostol 1981), pages 15-16.**^**See (Apostol 1981), page 25.**^**Bourbaki, Nicolas. "VIII.1".*General topology*. Springer-Verlag.**^**Marker, David (1996), "Introduction to the Model Theory of Fields", in Marker, D.; Messmer, M.; Pillay, A. (eds.),*Model theory of fields*, Lecture Notes in Logic,**5**, Berlin: Springer-Verlag, pp. 1-37, ISBN 978-3-540-60741-0, MR 1477154**^**Bourbaki, Nicolas. "VIII.4".*General topology*. Springer-Verlag.**^**Corry, Leo (2015).*A Brief History of Numbers*. Oxford University Press. pp. 215-16.**^**See (Apostol 1981), pages 15-16.**^**Bourbaki, Nicolas. "VIII.1".*General topology*. Springer-Verlag.**^**Lester, J.A. (1994), "Triangles I: Shapes",*Aequationes Mathematicae*,**52**: 30-54, doi:10.1007/BF01818325**^**Kalman, Dan (2008a), "An Elementary Proof of Marden's Theorem",*American Mathematical Monthly*,**115**(4): 330-38, doi:10.1080/00029890.2008.11920532, ISSN 0002-9890**^**Kalman, Dan (2008b), "The Most Marvelous Theorem in Mathematics",*Journal of Online Mathematics and its Applications***^**Grant, I.S.; Phillips, W.R. (2008),*Electromagnetism*(2 ed.), Manchester Physics Series, ISBN 978-0-471-92712-9**^**Kevin McCrimmon (2004)*A Taste of Jordan Algebras*, pp 64, Universitext, Springer ISBN 0-387-95447-3 MR2014924

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*Complex analysis*(3rd ed.), McGraw-Hill, ISBN 978-0-07-000657-7 - Apostol, Tom (1981).
*Mathematical analysis*. Addison-Wesley. - Solomentsev, E.D. (2001) [1994], "Complex number", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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

- Ahlfors, Lars (1979),
*Complex analysis*(3rd ed.), McGraw-Hill, ISBN 978-0-07-000657-7 - Conway, John B. (1986),
*Functions of One Complex Variable I*, Springer, ISBN 978-0-387-90328-6 - Joshi, Kapil D. (1989),
*Foundations of Discrete Mathematics*, New York: John Wiley & Sons, ISBN 978-0-470-21152-6 - Pedoe, Dan (1988),
*Geometry: A comprehensive course*, Dover, ISBN 978-0-486-65812-4 - Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.5 Complex Arithmetic",
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 - Solomentsev, E.D. (2001) [1994], "Complex number", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

- Bourbaki, Nicolas (1998), "Foundations of mathematics § logic: set theory",
*Elements of the history of mathematics*, Springer - Burton, David M. (1995),
*The History of Mathematics*(3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9 - Katz, Victor J. (2004),
*A History of Mathematics, Brief Version*, Addison-Wesley, ISBN 978-0-321-16193-2 - Nahin, Paul J. (1998),
*An Imaginary Tale: The Story of*, Princeton University Press, ISBN 978-0-691-02795-1- A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

- Ebbinghaus, H. D.; Hermes, H.; Hirzebruch, F.; Koecher, M.; Mainzer, K.; Neukirch, J.; Prestel, A.; Remmert, R. (1991),
*Numbers*(hardcover ed.), Springer, ISBN 978-0-387-97497-2- An advanced perspective on the historical development of the concept of number.

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