# Complex Numbers

This chapter introduces complex numbers, beginning with factoring polynomials, and proceeding on to the complex plane and Euler's identity.

## Factoring a Polynomial

Remember ``factoring polynomials''? Consider the second-order polynomial

*monic*because its leading coefficient, the coefficient of , is . By the fundamental theorem of algebra (discussed further in §2.4), there are exactly two

*roots*(or

*zeros*) of any second order polynomial. These roots may be real or complex (to be defined). For now, let's assume they are both real and denote them by and . Then we have and , and we can write

*factored form*of the monic polynomial . (For a non-monic polynomial, we may simply divide all coefficients by the first to make it monic, and this doesn't affect the zeros.) Multiplying out the symbolic factored form gives

This is a system of two equations in two unknowns. Unfortunately, it is a
*nonlinear* system of two equations in two
unknowns.^{2.1} Nevertheless, because it is so small,
the equations are easily solved. In beginning algebra, we did them by
hand. However, nowadays we can use a software tool such as Matlab or
Octave to solve very large systems of linear equations.

The factored form of this simple example is

*first-order*monic polynomials, each of which contributes one zero (root) to the product. This factoring business is often used when working with

*digital filters*[68].

## The Quadratic Formula

The general second-order (real) polynomial is

where the coefficients are any real numbers, and we assume since otherwise it would not be second order. Some experiments plotting for different values of the coefficients leads one to guess that the curve is always a scaled and translated

*parabola*. The canonical parabola centered at is given by

where the magnitude of determines the width of the parabola, and provides an arbitrary vertical offset. If , the parabola has the minimum value at ; when , the parabola reaches a maximum at (also equal to ). If we can find in terms of for any quadratic polynomial, then we can easily factor the polynomial. This is called

*completing the square*. Multiplying out the right-hand side of Eq.(2.2) above, we get

Equating coefficients of like powers of to the general second-order polynomial in Eq.(2.1) gives

Using these answers, any second-order polynomial can be rewritten as a scaled, translated parabola

*quadratic formula*. It was obtained by simple algebraic manipulation of the original polynomial. There is only one ``catch.'' What happens when is negative? This introduces the square root of a negative number which we could insist ``does not exist.'' Alternatively, we could invent complex numbers to accommodate it.

## Complex Roots

As a simple example, let , , and , *i.e.*,

*complex numbers*, that is, numbers of the form

It can be checked that all algebraic operations for real
numbers^{2.2} apply equally well to complex numbers. Both real numbers
and complex numbers are examples of a
mathematical *field*.^{2.3} Fields are
*closed* with respect to multiplication and addition, and all the rules
of algebra we use in manipulating polynomials with real coefficients (and
roots) carry over unchanged to polynomials with complex coefficients and
roots. In fact, the rules of algebra become simpler for complex numbers
because, as discussed in the next section, we can *always* factor
polynomials completely over the field of complex numbers while we cannot do
this over the reals (as we saw in the example
).

## Fundamental Theorem of Algebra

This is a very powerful algebraic tool.

^{2.4}It says that given any polynomial

we can *always* rewrite it as

where the points are the polynomial roots, and they may be real or complex.

## Complex Basics

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

*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 .

## The Complex Plane

We can plot any complex number in a plane as an ordered pair
, as shown in Fig.2.2. A *complex plane* (or
*Argand diagram*) is any 2D graph in which the horizontal axis is
the *real part* and the vertical axis is the *imaginary
part* of a complex number or function. As an example, the number
has coordinates in the complex plane while the number has
coordinates .

Plotting as the point in the complex plane can be
viewed as a plot in *Cartesian* or
*rectilinear* coordinates. We can
also express complex numbers in terms of *polar coordinates* as
an ordered pair
, where is the distance from the
origin to the number being plotted, and is the angle
of the number relative to the positive real coordinate axis (the line
defined by and ). (See Fig.2.2.)

Using elementary geometry, it is quick to show that conversion from rectangular to polar coordinates is accomplished by the formulas

where denotes the arctangent of (the angle in radians whose tangent is ), taking the quadrant of the vector into account. We will take in the range to (although we could choose any interval of length radians, such as 0 to , etc.).

In Matlab and Octave, `atan2(y,x)` performs the
``quadrant-sensitive'' arctangent function. On the other hand,
`atan(y/x)`, like the more traditional mathematical notation
does not ``know'' the quadrant of , so it maps
the entire real line to the interval
. As a specific
example, the angle of the vector
(in quadrant I) has the
same tangent as the angle of
(in quadrant III).
Similarly,
(quadrant II) yields the same tangent as
(quadrant IV).

The formula
for converting rectangular
coordinates to radius , follows immediately from the
*Pythagorean theorem*, while the
follows from the definition of the tangent
function itself.

Similarly, conversion from polar to rectangular coordinates is simply

These follow immediately from the definitions of cosine and sine, respectively.

## More Notation and Terminology

It's already been mentioned that the rectilinear coordinates of a complex
number in the complex plane are called the *real part* and
*imaginary part*, respectively.

We also have special notation and various names for the *polar
coordinates*
of a complex number :

The *complex conjugate* of is denoted
(or ) and is defined by

In general, you can always obtain the complex conjugate of any expression
by simply replacing with . In the complex plane, this is a *vertical flip* about the real axis; *i.e.*, complex conjugation
replaces each point in the complex plane by its *mirror image* on the
other side of the axis.

## Elementary Relationships

From the above definitions, one can quickly verify

Let's verify the third relationship which states that a complex number multiplied by its conjugate is equal to its magnitude squared:

## Euler's Identity

Since is the algebraic expression of in terms of its rectangular coordinates, the corresponding expression in terms of its polar coordinates is

There is another, more powerful representation of in terms of its
polar coordinates. In order to define it, we must introduce *Euler's
identity*:

A proof of Euler's identity is given in the next chapter. Before, the only algebraic representation of a complex number we had was , which fundamentally uses Cartesian (rectilinear) coordinates in the complex plane. Euler's identity gives us an alternative representation in terms of polar coordinates in the complex plane:

*polar form*of the complex number , in contrast with the

*rectangular form*. Polar form often simplifies algebraic manipulations of complex numbers, especially when they are multiplied together. Simple rules of exponents can often be used in place of messier trigonometric identities. In the case of two complex numbers being multiplied, we have

A corollary of Euler's identity is obtained by setting to get

For another example of manipulating the polar form of a complex number, let's again verify , as we did above in Eq.(2.4), but this time using polar form:

*even*function ( ) while sine is

*odd*( ).

We can now easily add a fourth line to that set of examples:

Euler's identity can be used to derive formulas for sine and cosine in terms of :

Similarly, , and we obtain the following classic identities:

## De Moivre's Theorem

As a more complicated example of the value of the polar form, we'll prove
*De Moivre's theorem*:

## Conclusion

This chapter has covered just enough about complex numbers to enable us to talk about the discrete Fourier transform.

Manipulations of complex numbers in Matlab and Octave are illustrated in §I.1.

To explore further the mathematics of complex variables, see any textbook such as Churchill [15] or LePage [37]. Topics not covered here, but which are important elsewhere in signal processing, include analytic functions, contour integration, analytic continuation, residue calculus, and conformal mapping.

## Complex_Number Problems

See `http://ccrma.stanford.edu/~jos/mdftp/Complex_Number_Problems.html`

**Next Section:**

Proof of Euler's Identity

**Previous Section:**

Introduction to the DFT