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:
As mentioned above, for any negative number , we have
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
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 .
The Complex Plane
Fundamental Theorem of Algebra