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A complex number, in mathematics, is a number comprising a real number part and an imaginary number part; it is normally written in the form a + bi, where a and b are real numbers, and i is the square root of minus one.^{[1]}
Complex numbers are a field in mathematics, with specific notions of addition, subtraction, multiplication and division, satisfying certain axioms. These operations extend the corresponding operations on real numbers, mainly because the product of two imaginary numbers (or the square of one imaginary number) is a negative real number.
Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations.^{[2]} 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, a solution exists to every polynomial equation of degree one or higher.
The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.^{[3]} 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 a number of fields, including: 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.
Contents 
The set of all complex numbers is usually denoted by C, or in blackboard bold by $\backslash mathbb\{C\}$.
Although other notations can be used, complex numbers are usually written in the form
where a and b are real numbers, and i is the imaginary unit, which has the property i^{ 2} = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part.
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 a + 0i. Complex numbers with a real part of zero are called imaginary numbers; instead of writing 0 + bi, that imaginary number is usually denoted as just bi. If b equals 1, instead of using 0 + 1i 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 or a + jb.
plot of the function
$f(x)\; =\; \backslash tfrac\{(x^2\; \; 1)(x\; \; 2\; \; i)^2\}\{x^2\; +\; 2\; +\; 2\; i\}$
The hue represents the function argument, while the saturation and value represent the magnitude.]]
In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is −1. The definition must therefore be a little less intuitive, building on the knowledge of real numbers. Write C for R^{2}, the set of ordered pairs of real numbers, and define operations on complex numbers in C according to
Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. We can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). It is then just a matter of notation to express (a, b) as a + ib.
Two complex numbers are said to be equal if and only if their real parts are equal and their imaginary parts are equal. In other words, if the two complex numbers are written as a + bi and c + di with a, b, c, and d real, then they are equal if and only if a = c and b = d.
Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i^{ 2} = −1:
where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator with the complex conjugate of the denominator, that is (c − di).
Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in onetoone correspondence with points on a plane. This complex plane is described below.
There are also other elementary functions that can be applied to complex functions. The most important is perhaps the exponential function exp(z), defined in terms of the infinite series
The elementary functions are, loosely, those which can be built using exp and the arithmetic operations given above, as well as taking inverses; in particular, the inverse of the exponential function, the logarithm.
The realvalued logarithm over the positive reals is welldefined (written ln), and the complex logarithm generalises this idea. However, is a multivalued function, unique only up to a multiple of 2πi. so the principal value is often taken by restricting the complex part to the interval (−π,π], giving
where Arg is the principal argument.
The familiar trigonometric functions are composed of these, so are also elementary. For example,
Hyperbolic functions are similarly constructed.
A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers form a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess:
Examples of other fields are 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 twodimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001) named after JeanRobert 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 (or modulus or magnitude) of a complex number $z=re^\{i\backslash phi\}$ is defined as $z=r$. Algebraically, if $z=x+yi$, then $z=\backslash sqrt\{x^2+y^2\}.$
The absolute value has three important properties:
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 $xyi$, written as $\backslash bar\{z\}$ or $z^*\backslash ,$. As seen in the figure, $\backslash bar\{z\}$ is the "reflection" of z about the real axis, and so both $z+\backslash bar\{z\}$ and $z\backslash cdot\backslash bar\{z\}$ are real numbers. Many identities relate complex numbers and their conjugates:
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 distributes over all the algebraic operations (and many functions; e.g. $\backslash sin\backslash bar\; z=\backslash overline\{\backslash sin\; z\}$) is rooted in the ambiguity in choice of i (−1 has two square roots). It is important to note, however, that the function $f(z)\; =\; \backslash bar\{z\}$ is not complexdifferentiable (see holomorphic function).
The operations of addition, multiplication, and complex conjugation in the complex plane admit natural geometrical interpretations.
These geometric interpretations allow problems of algebra to be translated into geometry. And, conversely, geometric problems can be examined algebraically. For example, the problem of the geometric construction of the 17gon was by Gauss translated into the analysis of the algebraic equation x^{17} = 1. (See Heptadecagon).
Alternatively to the cartesian representation z = x+iy, the complex number z can be specified by polar coordinates. The polar coordinates are r = z ≥ 0, called the absolute value or modulus, and φ = arg(z), called the argument or the angle of z. The representation of a complex number by its polar coordinates is called the polar form of the complex number.
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).
(taking the sign appropriately so that z = e^{iφ}).
The value of φ can change by any multiple of 2π and still give the same angle. In applied contexts, such as computing, the principal value in the range (−π, +π] of the arg function may be written as atan2. The arg function is sometimes considered as multivalued taking as possible values Arg(z) + 2πk, where k is any integer.
The notation of the polar form as
is called trigonometric form. The notation cis φ is sometimes used as an abbreviation for cos φ + i sin φ:
Using Euler's formula it can also be written as
which is called exponential form.
In electronics it is common to use angle notation to represent a phasor with amplitude A and phase θ as:
where θ may be in either radians or degrees. In electronics j is used instead of i because i is used for electric current.
Multiplication, division, exponentiation, and root extraction have simple formulas in polar form.
Using sum and difference identities it follows that
= r_1\,r_2\,e^{i(\varphi_1 + \varphi_2)} \,
and that
= \frac{r_1}{r_2}\,e^{i (\varphi_1  \varphi_2)}. \,
Geometrically, multiplication and division of complex numbers is respectively a counterclockwise and clockwise rotation on the argand plane.
Exponentiation with integer exponents; according to De Moivre's formula,
from which it follows that
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 counterclockwise 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 nth root of c. If c is nonzero, there are exactly n distinct nth roots of c, which can be found as follows. Write c = re^{ iφ} with real numbers r > 0 and φ, then the set of nth roots of c is
where $\backslash sqrt[n]\{r\}$ represents the usual (positive) nth root of the positive real number r. If c = 0, then the only nth root of c is 0 itself, which as nth root of 0 is considered to have multiplicity n.
While usually not useful, alternative representations of the complex field can give some insight into its nature. One particularly elegant representation interprets each complex number as a 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form
\begin{bmatrix}
a & b \\ b & \;\; a
\end{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 nonzero 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
\begin{bmatrix}
a & b \\ b & \;\; a
\end{bmatrix} = a \begin{bmatrix}
1 & \;\; 0 \\ 0 & \;\; 1
\end{bmatrix} + b \begin{bmatrix}
0 & 1 \\ 1 & \;\; 0
\end{bmatrix} which suggests that we should identify the real number 1 with the identity matrix
\begin{bmatrix}
1 & \;\; 0 \\ 0 & \;\; 1
\end{bmatrix}, and the imaginary unit i with
\begin{bmatrix}
0 & 1 \\ 1 & \;\; 0
\end{bmatrix},
a counterclockwise 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.
\begin{vmatrix}
a & b \\ b & a
\end{vmatrix} = (a^2)  ((b)(b)) = a^2 + b^2. If the matrix is viewed as a transformation of the plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be represented by the transpose of the matrix corresponding to z.
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 CayleyDickson 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.
While the above is a representation of C in the real matrices (2 x 2), it is not the only one. Any matrix
has the property that its square is the negative of the identity matrix. Then $\backslash \{\; z\; =\; a\; I\; +\; b\; M:\; a,b\; \backslash in\; R\; \backslash \}$ is also isomorphic to the field C.
C is a twodimensional real vector space. Unlike the reals, the set of complex numbers cannot be totally ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. More generally, no field containing a square root of −1 can be ordered.
Rlinear maps C → C have the general form
with complex coefficients a and b. Only the first term is Clinear, and only the first term is holomorphic; the second term is realdifferentiable, but does not satisfy the CauchyRiemann equations.
The function
corresponds to rotations combined with scaling, while the function
corresponds to reflections combined with scaling.
A root of the polynomial p is a complex number z such that p(z) = 0. A surprising result in complex analysis is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and it shows that the complex numbers are an algebraically closed field. Indeed, the complex numbers are the algebraic closure of the real numbers, as described below.
One construction of C is as a field extension of the field R of real numbers, in which a root of x^{2}+1 is added. To construct this extension, begin with the polynomial ring R[x] 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 wellknown complex plane, but it only characterizes it algebraically. The field C is characterized up to field isomorphism by the following three properties:
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^{[citation needed]}). As described below, topological considerations are needed to distinguish these subfields from the fields C and R themselves.
As just noted, the algebraic characterization of C fails to capture some of its most important topological properties. These properties are key for the study of complex analysis, where the complex numbers are studied as a topological field.
The following properties characterize C as a topological field:^{[citation needed]}
Given a field with these properties, one can define a topology by taking the sets
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 Dedekindcomplete 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 threedimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Some applications of complex numbers are:
In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros 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 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
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) the phase.
If Fourier analysis is employed to write a given realvalued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.
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, frequencydependent 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.
In applied fields, complex numbers are often used to compute certain realvalued improper integrals, by means of complexvalued functions. Several methods exist to do this; see methods of contour integration.
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.
In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary. (This 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.
In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^{rt}.
In fluid dynamics, complex functions are used to describe potential flow in two dimensions.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.
The earliest fleeting reference to square roots of negative numbers perhaps occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when, apparently inadvertently, he considered the volume of an impossible frustum of a pyramid,^{[4]} though negative numbers were not conceived in the Hellenistic world.
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 x^{3} − x = 0:
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z^{3} = i has solutions –i, $\{\backslash scriptstyle\backslash frac\{\backslash sqrt\{3\}\}\{2\}\}+\{\backslash scriptstyle\backslash frac\{1\}\{2\}\}i$ and $\{\backslash scriptstyle\backslash frac\{\backslash sqrt\{3\}\}\{2\}\}+\{\backslash scriptstyle\backslash frac\{1\}\{2\}\}i$. Substituting these in turn for $\{\backslash scriptstyle\backslash sqrt\{1\}^\{1/3\}\}$ in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x^{3} – x = 0. 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.
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^{[citation needed]} (see imaginary number for a discussion of the "reality" of complex numbers). A further source of confusion was that the equation $\backslash sqrt\{1\}^2=\backslash sqrt\{1\}\backslash sqrt\{1\}=1$ seemed to be capriciously inconsistent with the algebraic identity $\backslash sqrt\{a\}\backslash sqrt\{b\}=\backslash sqrt\{ab\}$, 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 $\backslash scriptstyle\; 1/\backslash sqrt\{a\}=\backslash sqrt\{1/a\}$) 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 $\backslash sqrt\{1\}$ to guard against this mistake.
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 reexpressed by the following wellknown 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 existence of complex numbers was not completely accepted until the geometrical interpretation (see above) 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 $\backslash pm\backslash sqrt\{1\}$ should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year JeanRobert 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 $\backslash cos\; \backslash phi\; +\; i\backslash sin\; \backslash phi$ the direction factor, and $r\; =\; \backslash sqrt\{a^2+b^2\}$ the modulus; Cauchy (1828) called $\backslash cos\; \backslash phi\; +\; i\backslash sin\; \backslash phi$ the reduced form (l'expression réduite); Gauss used i for $\backslash sqrt\{1\}$, introduced the term complex number for a + bi, and called a^{2} + b^{2} the norm.
The expression direction coefficient, often used for $\backslash cos\; \backslash phi\; +\; i\; \backslash sin\; \backslash phi$, 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 $a\; +\; b\backslash omega$, where $\backslash omega$ 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.

Template:· Transcendental numbersTemplate:· Quaternions ($\backslash scriptstyle\backslash mathbb\{H\}$)Template:· Octonions ($\backslash scriptstyle\backslash mathbb\{O\}$)Template:· Sedenions ($\backslash scriptstyle\backslash mathbb\{S\}$)Template:· Cayley–Dickson constructionTemplate:· Splitcomplex numbers
group2= Complex
extensions  list2= Bicomplex numbersTemplate:· BiquaternionsTemplate:· SplitquaternionsTemplate:· TessarinesTemplate:· Hypercomplex numbersTemplate:· Musean hypernumbersTemplate:· Superreal numbersTemplate:· Hyperreal numbersTemplate:· Supernatural numbersTemplate:· Surreal numbers
group3=Other extensions  list3= Dual numbersTemplate:· Transfinite numbersTemplate:· Extended real numbersTemplate:· Cardinal numbersTemplate:· Ordinal numbersTemplate:· padic numbers }}
Contents 
Plural 
complex number (plural complex numbers)


A complex number is a number, but is different from normal numbers in many ways. A complex number is made up using two numbers combined together. The first part is a real number. The second part of a complex number is an imaginary number $i$, defined as $+\backslash sqrt\{1\}$. Using arithmetic, addition, subtraction, multiplication, and division can be used. They also follow commutative properties and associative properties, just like real numbers. But answers to equations that have exponents in them began to pose real problems for mathematicians. As a comparison, using negative numbers, it is possible to find the x in the equation $a\; +\; x\; =\; b$ for all real values of a and b.
With exponentiation, there is a problem.^{[1]} There is no real number that gives 1 when it is squared, multiplied by itself. In other words, 1 (or any other negative number) has no real square root. To solve this problem, mathematicians introduced an imaginary number called i. That imaginary number will give 1 when it is squared ("squared" means "multiplied by itself").
The first mathematicians to have thought of this were probably Gerolamo Cardano and Raffaele Bombelli. They lived in the 16th century. It was probably Leonhard Euler who introduced writing $\backslash mathrm\; i$ for that number.
A complex number can now be written as $a\; +\; bi$^{[1]} (or $a\; +\; b\; \backslash cdot\; i$ ), where a is called the real part of the number, and b is called the imaginary part. Usually, the complex number is written as the set (a, b). Both a and b are real numbers.
Any real number can simply be written as $a\; +\; 0\; \backslash cdot\; i$ or as the set (a, 0).^{[1]}
Addition, subtraction, multiplication, division as long as the divisor is not zero, and exponentiation (raising numbers to exponents) are all possible with complex numbers. Some other calculations are also possible with complex numbers.
The set of all complex numbers is usually written as $\backslash mathbb\{C\}$.
Here are sentences from other pages on Complex number, which are similar to those in the above article.
