This section introduces various notation and terms associated with
complex
numbers. As discussed above, complex numbers arise by introducing
the square-root of

as a primitive new algebraic object among
real
numbers and manipulating it symbolically as if it were a real number
itself:

Mathematicians and physicists often use

instead of

as

.
The use of

is common in engineering where

is more often used for
electrical current.
As mentioned above, for any negative number

, we have
where

denotes the absolute value of

. Thus, every square
root of a negative number can be expressed as

times the square
root of a positive number.
By definition, we have
and so on. Thus, the sequence

,

is a
periodic sequence with
period 
, since

. (We'll
learn later that the sequence

is a sampled complex
sinusoid having
frequency equal to one fourth the
sampling rate.)
Every
complex number 
can be written as
where

and

are real numbers.
We call

the
real part and

the
imaginary part.
We may also use the notation
Note that the real numbers are the subset of the complex numbers having
a zero imaginary part (

).
The rule for
complex multiplication follows directly from the definition
of the imaginary unit

:
In some mathematics texts, complex numbers

are defined as ordered pairs
of real numbers

, and algebraic operations such as multiplication
are defined more formally as operations on ordered pairs,
e.g.,

. However, such
formality tends to obscure the underlying simplicity of complex numbers as
a straightforward extension of real numbers to include

.
It is important to realize that complex numbers can be treated
algebraically just like real numbers. That is, they can be added,
subtracted, multiplied, divided, etc., using exactly the same rules of
algebra (since both real and complex numbers are mathematical
fields). It is often preferable to think of complex numbers as
being the true and proper setting for algebraic operations, with real
numbers being the limited subset for which

.
Next Section: The Complex PlanePrevious Section: Fundamental Theorem of Algebra