3 Communications

8 Complex (Imaginary) Numbers

**
Imaginary numbers got a
foot hold intellectually because they showed up in solutions to quadratic
equations. Consider the quadratic equation x2+1=0 => x2=-1, x = square root
of (-1). There is no number that we can square to get (-1). This is why it is
imaginary.
**

Two complex numbers are equal if and only if their real parts are
equal *and* their imaginary parts are equal. That is, *a* + *bi* = *c* + *di*
if and only if *a* = *c* and *b* = *d*.

The set of all
complex numbers is usually denoted by **C**, or in blackboard bold by (Unicode ℂ). The real numbers, **R**, may be regarded as
"lying in" **C** by considering every real number as a complex: *a* = *a* + 0*i*.

Complex numbers are added, subtracted, and multiplied by formally applying
the associative,
commutative
and distributive
laws of algebra, together with the equation *i*^{2} = −1:

Division of complex numbers can also be defined (see below). Thus, the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed.

In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

Formally, the complex numbers can be defined as ordered pairs of real
numbers (*a*, *b*) together with the operations:

So defined, the complex numbers form a field, the complex
number field, denoted by **C** (a field is an algebraic structure
in which addition, subtraction, multiplication, and division are defined and
satisfy certain algebraic laws. For example, the real numbers form a field).

Since a complex number *a* + *bi* is uniquely specified by an
ordered pair (*a*, *b*) of real numbers, the complex numbers are in one-to-one
correspondence with points on a plane, called the complex plane.

We identify the real number *a* with the complex number (*a*, 0),
and in this way the field of real numbers **R** becomes a subfield of **C**.
The imaginary unit *i* is the complex number (0, 1).

In **C**, we have:

- additive identity ("zero"): (0, 0)
- multiplicative identity ("one"): (1, 0)
- additive inverse of (
*a*,*b*): (−*a*, −*b*) - multiplicative
inverse (reciprocal) of non-zero (
*a*,*b*):

**C** can also be defined 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 two-dimensional Cartesian
coordinate system called the **complex plane** or **Argand
diagram** (named after Jean-Robert Argand
– see figure at right). 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.

Alternatively, 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** of *z*. For *r* = 0 any value of φ
describes the same number. To get a unique representation, a conventional choice
is to set arg(0) = 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. The representation
of a complex number by its polar coordinates is called the *polar form* of
the complex number.

The previous formula requires rather laborious case differentiations. However, many programming languages provide a variant of the arctangent function which is often named atan2 and processes these internally. A formula that uses the arccos function requires fewer case differentiations:

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*.

Multiplication, division, exponentiation, and root extraction are much easier in the polar form than in the Cartesian form.

By simple trigonometric identities we see that

and that

Exponentiation with integer exponents is just as simple; according to De Moivre's formula,

Exponentiation with arbitrary complex exponents is discussed in the article on exponentiation.

The addition of two complex numbers is just the vector addition of two vectors, and multiplication by a fixed complex number can be seen as a simultaneous rotation and stretching.

Multiplication by *i* corresponds to a counter-clockwise rotation by ¼
turn (90 degrees
) or (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.

All the roots of any number, real or complex, may be found with a simple algorithm.
The *n*th roots are given by

for *k* = 0, 1, 2, …, *n* − 1,
where represents the principal *n*th root of *r*.