A
complex number, in
mathematics, is a number comprising a
real number and an
imaginary number; it can be written in the
form
a +
bi, where
a and
b are real numbers, and
i is the standard
imaginary unit, having the property
that
i ^{2} = −1. The complex numbers
contain the ordinary real numbers, but extend them
by adding in extra numbers and correspondingly expanding the
understanding of addition and multiplication. This is in order to
form a
closed field, where any
polynomial equation has a
root, including examples such as
x^{2} = −1.
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. 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. 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.
Definitions
Notation
The
set of all complex numbers is
usually denoted by
C, or in
blackboard bold by \mathbb{C}.
Although other notations can be used, complex numbers are usually
written in the form
 a + bi \,
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 +
2
i is a
complex number, with real part 3 and
imaginary part 2. If
z =
a +
bi, the
real part
a is denoted Re(
z) or ℜ(
z),
and the imaginary part
b is denoted Im(
z) or
ℑ(
z).
The real numbers,
R, may be regarded as a
subset of
C by considering every
real number a complex number with an imaginary part of zero; that
is, the real number
a is identified with the complex
number . Complex numbers with a real part of zero are called
imaginary numbers; instead of writing , that imaginary
number is usually denoted as just
bi. If
b equals
1, instead of using or 1
i, 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.
Two complex numbers are said to be equal
if and only if their real parts are equal
and their imaginary parts are equal.
Formal development
In a
rigorous setting, it is not acceptable to
simply assume that there exists a number whose square is −1. There
are various ways of defining
C, building on the
knowledge of real numbers. Firstly, write
C for
R^{2}, the set of
ordered pairs of real numbers, and define
operations on complex numbers in
C according to
 (a, b) + (c, d) =
(a + c, b + d)
 (a, b)·(c, d) =
(a·c − b·d, b·c + a·d)
It is then just a matter of notation to express
(
a,
b) as
a +
ib. This means we can associate the
numbers (
a, 0) with the real numbers, and write
i = (0, 1). Since (0, 1)·(0, 1) =
(−1, 0), we have found
i by constructing it, not
postulating it. Using these formal operations on
R^{2}, it is easy to check that we satisfy
the
field axioms (associativity,
commutativity, identity, inverses, distributivity). In particular,
R is a subfield of
C.
Though this lowlevel construction does accurately describe the
structure of the complex numbers, the definitions seem arbitrary,
so secondly
C can be considered algebraically. In
algebra (the theory of grouplike structures), this explicit
definition of operations in fact turns out to be the mechanism
behind the idea of constructing the algebraic closure of the reals,
that is, adding in some elements to
R to make a
new field, of which
R is a subfield, where every
nonconstant polynomial has a root. Finally, yet another way of
characterising
C is in terms of its
topological properties. Details of these are given
below.
Operations
Complex numbers can be intuitively added, subtracted, multiplied,
and divided by applying the
associative,
commutative and
distributive laws of algebra, together with the
equation
i^{ 2} = −1. Start by
defining the two basic operations:
 {width="100%"
The notation is welldesigned, since proceeding to manipulate the
symbols purely intuitively gives
 (a + bi) (c + di) = ac + bci + adi + bd i^2 = (ac  bd) + (bc +
ad)i.
The fact that this agrees with the definition justifies this sort
of manipulation, by indirectly deriving the distributive law, and
so on.
Subtraction and division are found from these two by working
backwards. In the case of subtraction, uwmeans "
uplus the
number which cancels out
w(its
additive
inverse)", giving
 {width="100%"
Similarly for multiplication, the inverse of c+diis found to be

\left(\frac{c}{c^2+d^2}\right)\left(\frac{d}{c^2+d^2}\right)i
when
cand
dare not both zero (by solving for
zin z(c+di)=1), which then yields
 {width="100%"
If the symbols are manipulated intuitively, so that
 \begin{align}
\frac{a + bi}{c + di} &= \frac{(a+bi)(cdi)}{(c+di)(cdi)} =
\frac{ac + bci  adi + bd}{(c+di)(cdi)}\\&=\left({ac + bd
\over c^2 + d^2}\right) + \left( {bc  ad \over c^2 + d^2}
\right)i,\end{align}the formula agrees with the one just derived.
In fact, once the gaps are filled in, all the usual rules of simple
arithmetic on real numbers can be shown to work when the symbols
are appropriately swapped for complex numbers.
Finally, square (and higher) roots can now be defined. However,
every nonzero number has two square roots, so there is an
ambiguity. Where the radical symbol (√) is used, it is normally
used to denote the root with smallest nonnegative phase (see
section below on polar representation).
Conjugation
Define the
conjugateof
z=x+iyto be xiy, written as \bar{z}or z^*. Both z+\bar{z}and
z\cdot\bar{z}are real numbers. Conjugation distributes over all the
algebraic operations and many derived functions, for example
\sin\bar z=\overline{\sin z}; this is as expected, since the
labelling of one of the two roots of −1 as
iis arbitrary,
so many situations should exhibit symmetry about the real axis. It
is important to note, however, that the function f(z) = \bar{z}is
not complexdifferentiable (see
Holomorphic function).
Elementary functions
One of the most important functions on the complex numbers is
perhaps the
exponential
functionexp(
z), also written
e^{z}, defined in terms of the
infinite series
 \exp(z):=\sum_{n=0}^{\infty} \frac{z^n}{n!} =
1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots.
The elementary functions are those which can be finitely 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, and the
complex logarithmgeneralises this idea.
The inverse of exp is shown to be
 \log(x+iy)=\tfrac{1}{2}\ln(x^2+y^2)+i\arg(x+iy),
where arg is the
argumentdefined
below, and ln the real logarithm. As arg
is a
multivalued function,
unique only up to a multiple of 2
π, log is also
multivalued. The
principal valueof
log is often taken by restricting the imaginary part to the
interval(−π,π].
The familiar
trigonometric
functionsare composed of these, so they are also elementary.
For example,
 \sin(z)=\frac{e^{iz}  e^{iz}}{2i}.
Hyperbolic functionssuch as
sinhare similarly constructed.
Exponentiation
Raising numbers to positive integer powers is done using the
operation of multiplication:
 z^n = \underbrace{z\cdot z \cdots z}_{n\text{ factors}}.
\,
Negative integer powers also are defined just as for real numbers,
since 1/
z^{n}is the only way of
interpreting
z^{−n}such that the familiar
rules of indices still work
(
z^{−n} =
z^{−n}(
z^{n}/
z^{n})
=
z^{−n+n}/
z^{n} = 1/
z^{n}).
Similar considerations show that we can define rational real powers
just as for the reals, so
z^{1/n}is the
nth root of
z. Roots are not unique, so it is
already clear that complex powers are multivalued, thus careful
treatment of powers is needed; for example
(8
^{1/3})
^{4} ≠ 16, as there are three
cube roots of 8, so the given expression, often shortened to
8
^{4/3}, is the simplest possible.
For arbitrary complex powers, the general meaning of
z^{ω}must be multivalued, since it is in
the case of
ωrational. To agree with the definitions so
far, this suggests
 z^\omega := \exp(\omega \log z), \,
which is the general extension of exponentiation to the complex
numbers.
The complex plane
A complex number can be viewed as a point or
position vectorin a twodimensional
Cartesian coordinate
systemcalled the
complex planeor
Argand diagram (see and ), named after
JeanRobert 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.
Geometric interpretation of the operations
The operations described algebraically above can be visualised
using Argand diagrams.
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
17gonwas by
Gausstranslated into the analysis of the
algebraic equation
x^{17}= 1 (see
Heptadecagon).
Polar form
Figure 2: The argument
φand modulus
rlocate a
point on an Argand diagram; r(\cos \phi + i \sin \phi)or r
e^{i\phi}are
polarexpressions of the point.
The diagrams suggest various properties. Firstly, the distance of a
point
zfrom the origin (shown as
rin Figure 2) is
known as the
modulus,
absolute value, or
magnitude, and written z. By
Pythagoras' theorem,
 x+iy=\sqrt{x^2+y^2}.
In general, distances between complex numbers are given by the
distance function d(z,w)=zw, which turns the complex numbers
into a
metric spaceand introduces the
ideas of
limitsand
continuity. All of the standard
properties of two dimensional space therefore hold for the complex
numbers, including important properties of the modulus such as
nonnegativity, and the
triangle
inequality( z + w  \leq  z  +  w for all
z,
w).
Secondly, the
argumentor
phaseof a complex number z=x+yiis the angle to the real
axis (shown as
φin Figure 2), and is written as \arg(z).
As with the modulus, the argument can be found from the rectangular
form x+iy:
 \varphi = \pm\arctan\frac{y}{x} (taking the sign appropriately
so that x+iy=\cos \phi + i \sin \phi ).
The value of
φcan change by any multiple of 2
πand
still give the same angle (note that
radiansare being used). Hence, the arg function is
sometimes considered as
multivalued, but often the value is
chosen to lie in the interval (\pi,\pi], or [0,2\pi)(this is the
principal value).
Together, these 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 (confirmed by
recovering the original rectangular coordinates (x,y)=(r
\cos\varphi,r\sin\varphi)from the polar pair
(
r,
φ)). This can be notated in various ways,
including
 z = r(\cos \varphi + i\sin \varphi )\,
called
trigonometric form, and sometimes abbreviated
rcis
φ, or using
Euler's formula
 z = r e^{i \varphi},
which is called
exponential form. In
electronicsit is common to use
angle notationto represent a
phasorwith amplitude
Aand phase
θas
 A \ang \theta = A e ^ {j \theta }.
In angle notation
θmay be in either radians or degrees. In
electronics it is also common to use
jinstead of
i.
Operations in polar form
Multiplication and division have simple formulas in polar
form:
 (r_1e^{i\varphi_1}) \cdot (r_2e^{i\varphi_2}) = r_1 r_2
e^{i(\varphi_1 + \varphi_2)}
and
 \frac{r_1\,e^{i\varphi_1}}{r_2\,e^{i\varphi_2}} =
\left(\frac{r_1}{r_2}\right)\,e^{i (\varphi_1  \varphi_2)}.
This form demonstrates that multiplication can be visualised as a
simultaneous stretching and rotation of one of the multiplicands,
adding to its angle the phase of the other and scaling its length.
For example, multiplying by
icorresponds to a
quarterrotation counterclockwise, from which it is clear why
i^{ 2} = −1. In particular,
multiplication by any number on the unit circle around the origin
is a pure rotation. Division is the same, in reverse.
Exponentiation is also simple; with integer exponents:
 {width=100%
Arbitrary complex exponents are discussed in
Exponentiation.
Finally, polar forms are also useful for finding roots. Any complex
number
zsatisfying
z^{n} =
c(for
na positive integer) is called an
nth root of
c. If
cis nonzero, there are exactly
ndistinct
nth roots of
c(by the
fundamental theorem of
algebra). Let
c =
re^{ iφ}with
r > 0; then the set of
nth roots of
cis
 \left\{ \sqrt[n]r\,e^{i\left(\frac{\varphi+2k\pi}{n}\right)}
\mid k\in\{0,1,\ldots,n1\} \, \right\},
where \sqrt[n]{r}represents the usual (positive)
nth root
of the positive real number
r. If
c = 0, then the only
nth root of
cis 0 itself, which as
nth root of 0 is
considered to have
multiplicityn, hence these do
represent all the
nroots. Note that the roots differ only
by the rotations
e^{2kπi/n}, the
nth roots of unity, so all the roots of
clie on a
circle about the origin.
Some properties
Matrix representation of complex numbers
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
matrixwith
realentries which stretches and rotates the
points of the plane. Every such matrix has the form\begin{bmatrix}
a & b \\
b & \;\; a
\end{bmatrix}
where
aand
bare 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,
isomorphicto 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
iwith\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
determinantof
that matrix.
 z^2 =
\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
zcorresponds to the transformation which rotates through
the same angle as
zbut in the opposite direction, and
scales in the same manner as
z; this can be represented by
the
transposeof 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 constructionof
algebras.
It should also be noted that the two
eigenvaluesof the 2x2 matrix representing a
complex number are the complex number itself and its
conjugate.
While the above is a representation of
Cin the
2 × 2 real matrices, it is
not the only one. Any matrix
 M = \begin{pmatrix}p & q \\ r & p \end{pmatrix}, \quad
p^2 + qr + 1 = 0
has the property that its square is the negative of the identity
matrix.Then \{ z = a I + b M : a,b \in R \}is also isomorphic to
the field
C.
Real vector space
Cis a twodimensional real
vector space.Unlike the reals, the set of
complex numbers cannot be
totally
orderedin any way that is compatible with its arithmetic
operations:
Ccannot be turned into an
ordered field. More generally, no field
containing a square root of −1 can be ordered.
Rlinearmaps
C→
Chave the general form
 f(z)=az+b\overline{z}
with complex coefficients
aand
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
 f(z)=az\,
corresponds to rotations combined with scaling, while the function
 f(z)=b\overline{z}
corresponds to reflections combined with scaling.
Solutions of polynomial equations
A
rootof the
polynomialpis a complex number
zsuch that
p(
z) = 0. A surprising result
in complex analysis is that all polynomials ofdegree
nwith
real or complex coefficients have exactly
ncomplex roots
(counting
multiple
rootsaccording 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 closureof the real
numbers, as described below.
Construction and algebraic characterization
One construction of
Cis as a
field extensionof the field
Rof real numbers, in which a root of
x^{2}+1 is added. To construct this extension,
begin with the
polynomial
ringR[
x] of the real numbers in the
variable
x. Because the polynomial
x^{2}+1 is
irreducibleover
R,
the
quotient
ringR[
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
Cfactors 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
Cis
characterizedup tofield
isomorphismby
the following three properties:
One consequence of this characterization is that
Ccontains many proper subfields which are
isomorphic to
C(the same is true of
R, which contains many subfields isomorphic to
itself ). As described below, topological considerations are needed
to distinguish these subfields from the fields
Cand
Rthemselves.
Characterization as a topological field
As just noted, the algebraic characterization of
Cfails 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
Cas a
topological field:
 C is a field.
 C contains a subset P of nonzero
elements satisfying:
 P is closed under addition, multiplication and taking
inverses.
 If x and y are distinct elements of P, then either
xy or yx is in P
 If S is any nonempty subset of P, then
S+P=x+P for some x in C.
(a + bi) + (c + di) := (a + c) + (b + d)i 
[addition] 

(a + bi) (c + di) := (ac  bd) + (bc + ad)i. 
[multiplication] 
(a + bi)  (c + di) = (a  c) + (b  d)i. 
[subtraction] 
\frac{a + bi}{c + di} =\left({ac + bd \over c^2 +
d^2}\right) + \left( {bc  ad \over c^2 + d^2} \right)i. 
[division] 


X A + B: The sum of
two points A and B of the complex plane is the
point X A + B such that the triangles with vertices 0, A, B,
and X, B, A, are congruent. Thus the addition of two
complex numbers is the same as vector
addition of two vectors. 


X AB: The product of two
points A and B is the point X
AB such that the triangles with vertices 0, 1, A,
and 0, B, X, are similar. 


X A*: The complex conjugate
of a point A is the point X A* such that
the triangles with vertices 0, 1, A, and 0, 1, X,
are mirror images of each other. 
(r(\cos\varphi + i\sin\varphi))^n = r^n\,(\cos
n\varphi + i \sin n \varphi). 
[De Moivre's formula] 
 C has a nontrivial involutive automorphism
x→x*, fixing P and such that xx* is in
P for any nonzero x in C.
Given a field with these properties, one can define a topology by
taking the sets
 B(x,p) = \{y  p  (yx)(yx)^*\in P\}
as a
base, where
x ranges
over the field and
p ranges over
P.
To see that these properties characterize
C as a
topological field, one notes that
P ∪ {0} ∪
P is an ordered
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.
Complex analysis
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.
Applications
Some applications of complex numbers are:
Control theory
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.
Signal analysis
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
 f ( t ) = z e^{i\omega t} \,
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.
Improper integrals
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.
Quantum mechanics
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.
Relativity
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.
Applied mathematics
In
differential equations, it
is common tofirst find all complex roots
r of the
characteristic equation of a
linear differential equation
and then attempt to solve the systemin terms of base functions of
the form
f(
t) =
e^{rt}.
Fluid dynamics
In
fluid dynamics, complex functions
are used to describe
potential flow in two
dimensions.
Fractals
Certain
fractals are plotted in the complex
plane, e.g. the
Mandelbrot set and
Julia sets.
History
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, 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:

\frac{1}{\sqrt{3}}\left(\sqrt{1}^{1/3}+\frac{1}{\sqrt{1}^{1/3}}\right).
At first glance this looks like nonsense. However formal
calculations with complex numbers show that the equation
z^{3} =
i has solutions
–i,
{\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i and
{\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i.
Substituting these in turn for {\scriptstyle\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 (see
imaginary number
for a discussion of the "reality" of complex numbers). A further
source of confusion was that the equation
\sqrt{1}^2=\sqrt{1}\sqrt{1}=1 seemed to be capriciously
inconsistent with the algebraic identity
\sqrt{a}\sqrt{b}=\sqrt{ab}, which is valid for positive real
numbers
a and
b, and which was also used in
complex number calculations with one of
a,
b
positive and the other negative. The incorrect use of this identity
(and the related identity \scriptstyle 1/\sqrt{a}=\sqrt{1/a}) in
the case when both
a and
b are negative even
bedeviled
Euler. This difficulty eventually
led to the convention of using the special symbol
i in
place of \sqrt{1} to guard against this mistake. 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
reexpressed by the following wellknown formula which bears his
name,
de Moivre's formula:
 (\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n
\theta. \,
In 1748
Leonhard Euler went further
and obtained
Euler's formula of
complex analysis:
 \cos \theta + i\sin \theta = e ^{i\theta } \,
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 \pm\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 \cos \phi + i\sin \phi the
direction
factor, and r = \sqrt{a^2+b^2} the
modulus; Cauchy
(1828) called \cos \phi + i\sin \phi the
reduced form
(l'expression réduite); Gauss used
i for \sqrt{1},
introduced the term
complex number for
a +
bi, and called
a^{2} +
b^{2} the
norm.
The expression
direction coefficient, often used for \cos
\phi + i\sin \phi, is due to Hankel (1867), and
absolute
value, for
modulus, is due to Weierstrass.
Following Cauchy and Gauss have come a number of contributors of
high rank, of whom the following may be especially mentioned:
Kummer (1844),
Leopold Kronecker (1845),
Scheffler (1845, 1851, 1880),
Bellavitis (1835, 1852), Peacock (1845), and
De Morgan (1849).
Möbius must also be mentioned
for his numerous memoirs on the geometric applications of complex
numbers, and
Dirichlet for the
expansion of the theory to include primes, congruences,
reciprocity, etc., as in the case of real numbers.
A complex
ring or
field is a set of complex numbers which
is
closed under addition,
subtraction, and multiplication.
Gauss studied complex numbers of the
form
a +
bi, where
a and
b are integral, or rational (and
i is one of the
two roots of
x^{2} + 1 = 0).
His student,
Ferdinand
Eisenstein, studied the type a + b\omega, where \omega is a
complex root of
x^{3} − 1 = 0.
Other such classes (called
cyclotomic
fields) of complex numbers are derived from the
roots of unity
x^{k} − 1 = 0 for
higher values of
k. This generalization is largely
due to
Kummer, who also invented
ideal numbers, which were expressed as
geometrical entities by
Felix Klein in
1893. The general theory of fields was created by
Évariste Galois, who studied the fields
generated by the roots of any polynomial equation in one
variable.
The late writers (from 1884) on the general theory include
Weierstrass,
Schwarz,
Richard
Dedekind,
Otto Hölder,
Henri Poincaré,
Eduard Study, and
Alexander
MacFarlane.
See also
Notes
 A brief history of complex numbers
References
Mathematical references
Historical references
 {{citationtitle=An Imaginary Tale: The Story of
\sqrt{1}first=Paul J.last=Nahinpublisher=Princeton University
Pressisbn=0691027951year=1998edition=hardcover}}
 :A gentle introduction to the history of complex numbers and
the beginnings of complex analysis.
 :An advanced perspective on the historical development of the
concept of number.
Further reading
 The Road to Reality: A Complete Guide to the Laws of the
Universe, by Roger Penrose;
Alfred A. Knopf, 2005; ISBN 0679454438. Chapters 47 in
particular deal extensively (and enthusiastically) with complex
numbers.
 Unknown Quantity: A Real and Imaginary History of
Algebra, by John Derbyshire; Joseph Henry Press; ISBN
030909657X (hardcover 2006). A very readable history with
emphasis on solving polynomial equations and the structures of
modern algebra.
 Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN
0198534477 (hardcover, 1997). History of complex numbers and
complex analysis with compelling and useful visual
interpretations.
External links