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Geometry of Complex Numbers, Continued

1.4 The Geometry of Complex Numbers, Continued

In Secion 1.3 we saw that a complex number    could be viewed as a vector in the xy-plane whose tail is at the origin and whose head is at the point (x,y).  A vector can be uniquely specified by giving its magnitude (i.e., its length) and direction (i.e., the angle it makes with the positive x-axis). In this section, we focus on these two geometric aspects of complex numbers.

Let r be the modulus of z (i.e., r = |z|), and let    be the angle that the line from the origin to the complex number z makes with the positive x-axis. (Note: The number    is undefined if z=0).  We make the following definition.

Definition 1.9, (Polar Representation). The identity

(1-27)

is known as a polar representation of z, and the values r and    are called polar coordinates of z, and is illustrated in Figure 1.11 (a).

Figure 1.11  Polar representation of  complex numbers.

As Figure 1.11(b) shows, the angle   can be any value for which the identities    hold true.  Given ,  the collection of all values of   for which    is denoted by  .  Formally, we have the following definitions.

Example 1.7.  If  ,  then    and    is a polar representation of z.  The polar coordinates in this case are , and .
Explore Solution 1.7.

Definition 1.10, (Argument, arg z). If  ,  we denote    by

(1-28)                         .

If  ,  we say that is an argument of z.

Notice that we write    as opposed to  .  This is because    is a set, and the designation indicates that belongs to that set.  Notice also that, if  ,  then there exists some integer n such that

(1-29)                         .

Example 1.8.  Since , we have

.
Explore Solution 1.8.

Mathematicians have agreed to single out a special choice of  .  It is that value of for which as the following definition indicates.

Definition 1.11, (The principal value of argument, Arg z). Let  ,  be a complex number.  Then

(1-30)                         ,    provided and .

If  ,  we call the argument of z.

Example 1.9.  .
Explore Solution 1.9.

Remark.  Clearly if , where , then

.

where  .  Note that, as with arg, arctan is a set (as opposed to Arctan, which is a number).  We specifically identify as a proper subset of because has period , whereas cos and sin have period  .  In selecting the proper values for   , we must be careful in specifying the choices of so that the poin z associated with r and lies in the appropriate quadrant.

Example 1.10.  If  ,  then

and

.

It would be a mistake to use as an acceptable value for , as the point z associated with and is in the first quadrant, whereas is in the third quadrant.  A correct choice for is .  Thus

,  and

,

where n is any integer.  In this case,

,  and

.

Remark.  Note that    is indeed a proper subset of  .

Explore Solution 1.10.

If    is on the y-axis, it would be a mistake to attempt to find    by looking at  ,  as ,  so is undefined.  We emphasize these special cases:

If  ,  then  ,  and

If  ,  then  .

Example 1.11.  If  , it would be a mistake to attempt to find    by looking at  ,  as ,  so is undefined.  In this case

,   and   .

As we shall see in Section 2.4, is a discontinuous function of z because it "jumps'' by an amount of as z crosses the negative real axis.

In Section 5.1 we define for any complex number z.  You will see that this complex exponential has all the properties of real exponentials that you studied in earlier mathematics courses.  That is,  ,  and so forth. You will also see, amazingly, that if , then

(1-31)                         .

We will establish this result rigorously in Chapter 5, but there is a plausible explanation we can give now.

If    has the normal properties of an exponential, it must be that  .  Now, recall from Calculus the values of three infinite series:

,    ,   and   .

Substituting for in the infinite series for    gives  .  At this point, our argument loses rigor because we have not talked about infinite series of complex numbers, let alone whether such series converge.  Nevertheless, if we merely take the last series as a formal expression and split it into two series according to whether the index k is even (k=2n) or odd (k=2n+1), we get

Exploration

Thus, it seems the only possible value for is  .  We will use this result freely from now on, and, as stated, supply a rigorous proof in Chapter 5.

If we set x=0 and let take the role of y in the above equation, we get a famous result known as Euler Formula:

(1-32)                         .

If is a real number, will be located somewhere on the circle with radius 1 centered at the origin.  This assertion is easy to verify because

(1-33)                         .

Figure 1.12 illustrates the location of the points for various values of .

Figure 1.12  The location of for various values of .

Notice that, when , we get  ,  so

(1-34)                        .

Euler was the first to discover this relationship;  it is referred to as Euler's identity.  It has been labeled by many mathematicians as the most amazing relation in analysis---and with good reason.  Symbols with a rich history are miraculously woven together---the constant  used by Hippocrates as early as Hippocrates as early as 400 B.C.;   the base of the natural logarithms;  the basic concepts of addition (+) and equality (=);  the foundational whole numbers 0 and 1;  and , the number that is the central focus of this book.

Euler's formula is of tremendous use in establishing important algebraic and geometric properties of complex numbers.  You will see shortly that it enables you to multiply complex numbers with great ease. It also allows you to express a polar form of the complex number z in a more compact way.  Recall that if   and  ,  then  .  Using Euler's formula we can now write z in its exponential form:

(1-35)                         .

Example 1.12.  In example 1.10 we investigated the polar form of  ,  and saw that    and  .  Now we have

.

Explore Solution 1.12.

Together with the rules for exponentiation that we will verify in Chapter 5, the exponential form has interesting applications.  If and , then

(1-36)

Figure 1.13 illustrates the geometric significance of this equation.

Figure 1.13  The product of two complex numbers .

We have already seen that the modulus of the product is the product of the moduli;  that is, .  The above identity establishes that an argument of    is an argument of plus an argument of .  It also answers the question posed at the end of Section 1.3 regarding why the product was in a different quadrant than either or .  It further offers an interesting explanation as to why the product of two negative real numbers is a positive real number.  The negative numbers, each of which has an angular displacement of radians, combine to produce a product that is rotated to a point with an argument of    radians, coinciding with the positive real axis.

Using exponential form, if , we can write  ,  a little more compactly as

(1-37)                         .

Doing so enables us to see a nice relationship between the sets , , and .

Theorem 1.3.  If    and  ,  then as sets,

(1-38)                         .

Before proceeding with the proof, we recall two important facts about sets.  First, to establish the equality of two sets, we must show that each is a subset of the other.  Second, the sum of two sets is the sum of all combinations of elements from the first and second sets, respectively.  In this case,  .

Proof  Let  .  Because  ,  it follows that  .  Hence, there is some integer n such that  .  Further, as    we have  .  Likewise,    gives  .  But if  ,  then  .  This result shows that  .  Thus,  .  The proof that    is left as an exercise.

Using the exponential form,  ,  we see that  .  In other words,

Recalling that and , we also have

and

If z is in the first quadrant, the positions of the numbers    are as shown in Figure 1.14 when  .  Figure 1.15 depicts the situation when  .

Example 1.13.  If  ,  then    and  .  Therefore

the modulus is  ,  and the argument is  .
Explore Solution 1.13.

Example 1.14.  Given  ,  compute    using polar computations.

If    and  ,  then representative polar forms for these numbers are    and  .  Hence

the modulus is  ,  and the argument is .
Explore Solution 1.14.

Caveat.  The formula   does not hold for all complex numbers .  Whereas, the formula    does hold for all complex numbers , and is left for the reader to verify.

Complex Numbers

History of Complex Numbers

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