A
complex number is a
number consisting of a
real and
imaginary part. It can be written in the form
a +
bi, where
a and
b are real numbers, and
i is the standard
imaginary unit with the property
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.
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 in radicals (without trigonometric functions) 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. Complex numbers thus form an
algebraically closed field, where any
polynomial equation has a
root.
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.
Complex numbers are plotted on the
complex plane, on which the real part is on the horizontal axis, and the imaginary part on the vertical axis.
A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram, representing the
complex plane.
Introduction and definition
Complex numbers have been introduced to allow for solutions of certain equations that have no real solution: the equation
has no real solution
x, since the square of
x is 0 or positive, so
x^{2} +
1 cannot be zero. Complex numbers are a solution to this dilemma. The idea is to enhance the real numbers by adding a number
i whose square is −1, so that
x =
i and
x = 
i are the two solutions to the preceding equation.
Definition
A
complex number is an expression of the form
Here
a and
b are
real numbers, and
i is a
mathematical symbol which is called
imaginary unit. For example, 3.5 + 2
i is a complex number.
The real number
a of the complex number
z =
a +
bi is called the
real part of
z and the real number
b is the
imaginary part.
^{} They are denoted Re(
z) or ℜ(
z) and Im(
z) or ℑ(
z), respectively. For example,
Some authors write
a+
ib instead of
a+
bi. 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 written as
a +
bj or
a +
jb.
A real number is thus a special case of a complex number: every real number
a can be regarded as a complex number with an imaginary part of zero, that is to say,
a + 0
i. Complex numbers whose real part is zero, that is to say, those of the form 0 +
bi, are called
imaginary numbers. It is common to write
a for
a + 0
i and
bi for
0 + bi. Moreover, when
b is negative, it is common to write
a − (−
b)i instead of
a +
bi, for example 3 − 4
i instead of 3 + (−4)
i.
The
set of all complex numbers is denoted by
C or
.
Addition and subtraction
Complex numbers are
added by adding the real and imaginary parts of the summands. That is to say:
Similarly,
subtraction is defined by
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers
A and
B, interpreted as points of the complex plane, is the point
X obtained by building a
parallelogram three of whose vertices are 0,
A and
B. Equivalently,
X is the point such that the
triangles with vertices 0,
A,
B, and
X,
B,
A, are
congruent.
Multiplication and division
The multiplication of two complex numbers is defined by the following formula:
In particular, the square of the imaginary unit is −1:
The preceding definition of multiplication of general complex numbers is the natural way of extending this fundamental property of the imaginary unit. Indeed, treating
i as a variable, the formula follows from this
 (distributive law)

 (commutative law of addition—the order of the summands can be changed)
 (commutative law of multiplication—the order of the factors can be changed
 (fundamental property of the imaginary unit).
The division of two complex numbers is defined by the following formula:
The real and imaginary part (
c and
d, respectively) of the
denominator must not both be zero for the division to be defined. Division is defined in this way in order because the product of the right hand expression with
c +
di (using the previous formula for multiplication) is
a +
bi. Thus, dividing
a +
bi by
c +
di and then multiplying it with
c +
di again gives back
a +
bi, as is familiar from real or rational numbers.
Square root
The
square roots of a + bi (with
b ≠ 0) are
, where
and
This can be seen by squaring
to obtain
a +
bi.
^{}^{} Here
is called the
modulus of
a + bi, and the square root with nonnegative real part is called the
principal square root.
Conjugation
The
complex conjugate of the complex number
z = x + yi is defined to be
x −
yi. It is denoted
or
. Geometrically,
is the "reflection" of
z about the real axis. In particular, conjugating twice gives the original complex number:
.
The real and imaginary parts of a complex number can be extracted using the conjugate:
Moreover, a complex number is real if and only if it equals its conjugate.
Conjugation distributes over the standard arithmetic operations:
The
reciprocal of a nonzero complex number
z = x + yi is given by
This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates.
Inversive geometry, a branch of geometry studying more general reflections than ones about a line, can be expressed in terms of complex numbers, too.
Polar form
Absolute value and argument
Another way of encoding points in the complex plane than using the
x and
ycoordinates is to use the distance of a point
P to
O, the point whose coordinates are (0, 0) (
origin), and the angle of the line through
P and
O. This idea leads to the polar form of complex numbers.
The
absolute value (or
modulus or
magnitude) of a complex number
z =
x+
yi is
If
z is a real number (i.e.,
y = 0), then
r = 
x. In general, by
Pythagoras' theorem,
r is the distance of the point
P representing the complex number
z to the origin.
The
argument or
phase of
z is the angle to the real axis, and is written as
arg(z). As with the modulus, the argument can be found from the rectangular form
x + iy:
^{}
The value of
φ can change by any multiple of 2
π and still give the same angle (note that
radians are being used). Hence, the arg function is sometimes considered as
multivalued. Normally, as given above, the
principal value in the interval
( − π,π] is chosen. Values in the range
[0,2π) are obtained by adding
2π if the value is negative. The polar angle of the origin is undefined but the value 0 is commonly used.
Together,
r and
φ give another way of representing complex numbers, the
polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular coordinates from the polar form is done by the formula called
trigonometric form
Using
Euler's formula this can be written as
Using the
cis function, this is sometimes abbreviated to
In
angle notation, often used in
electronics to represent a
phasor with amplitude
r and phase
φ it is written as
^{}
Figure 2: The argument
φ and modulus
r locate a point on an Argand diagram;
r(cosφ + isinφ) or
re^{iφ} are
polar expressions of the point.
Multiplication, division and exponentiation in polar form
The relevance of representing complex numbers in polar form stems from the fact that the formulas for multiplication, division and exponentiation are simpler than the ones using Cartesian coordinates. Given two complex numbers
z_{1} = r_{1}(cos φ_{1} + isin φ_{1}) and
z_{2} =r_{2}(cos φ_{2} + isin φ_{2}) the formula for multiplication is
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by
i corresponds to a quarterrotation counterclockwise, 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 π/4 (in
radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangle are
arctan(1/3) and arctan(1/2), respectively. Thus, the formula
holds. As the
arctan function can be approximated highly efficiently, formulas like this—known as
Machinlike formulas—are used for highprecision approximations of π.
Similarly, division is given by
This also implies
de Moivre's formula for exponentiation of complex numbers with integer exponents:
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} =
x there is no natural way of distinguishing one particular complex
nth root of a complex number. Therefore, the
nth root of
z is considered as a
multivalued function (in
z), as opposed to a usual function
f, for which
f(
z) is a uniquely defined number. Formulas such as
(which holds for positive real numbers), do in general not hold for complex numbers.
References :
wikipedia.com
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