1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1.4 • THE GEOMETRY OF COMPLEX NUMBERS, CONTINUED 27

y

Figure 1.13 The product of two complex numbers z3 = z1z2.

Together with the rules for exponentiation that we will verify in Chapter 5,

Equation (1-35) has interesting applications. If z 1 = r 1 ei^9 i and z 2 = r2e^19 2, then

z1z 2 = r1 ei^9 ' r2ei^9 • = r1 r2ei(^9 i +^9 •)

= r1r2 [cos(8 1 +82) + isin(81+82)]. (1-36)

Figure 1.13 illustrates the geometric significance of this equation.
We have already shown that the modulus of the product is the product of the
moduli; that is, lziZzl = jz 11 lz2I· Identity (1-36) establishes that an argument of
z 1 z 2 is an argument of z 1 plus an argument of z 2. It also answers the question
posed at the end of Section 1.3 regarding why the product z 1 z 2 was in a different
quadrant than either z 1 or z2. 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 rr radians,
combine to produce a product that is rotated to a point with an argument of
rr + rr = 2rr radians, coinciding with the positive real axis.
Using exponential form, if z °f' 0, we can write arg z a bit more compactly as

arg z = { 8 : z = rei^9 }. (1-37)

Doing so enables us to see a nice relationship between the sets arg (z 1 z2), arg zi,

and arg z2.

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