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

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
.

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University