1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

1.3 • THE GEOMETRY OF COMPLEX NUMBERS 17

y

4 +3i

4

Figure 1.5 The difference z1 - z2. Figure 1.6 The real and imaginary

parts of a complex number.

Definit ion 1.8: M odulus

The modulus, or absolute value, of the complex number z = x + iy is a
nonnegative real number denoted by lz l and defined by the relation

(1-20) I

The number lzl is the distance between the origin and the point z = (x, y).
The only complex number with modulus zero is the number O. The number
z = 4+3i has modulus 14 + 3il = v14^2 + 33 = v125 = 5 and is depicted in Figure
1.6. The numbers IRe(z)I, IIm(z)I, and lzl are the lengths of the sides of the right
triangle OPQ shown in Figure 1.7. The inequality lz 11 < lz2I means that the
point z 1 is closer to the origin than the point z 2. Although obvious from Figure
1.7, it is still profitable to work out algebraically the standard results that

lxl = IRe(z)I ~ lzl and IYI = llm(z)I ~ lz l , (1-21)

which we leave as an exercise.

The difference z1 -z2 represents the displacement vector from z2 to z1, so

the distanc e between z1 and z2 is given by lz1 - z2I-We can obtain this distance

by using Definitions (1.2) and (1.3) to obtain the familiar formula

lzi -z2I = V(x1 - x2)^2 + (Y1 -y2)^2 •

If z = (x, y) = x + iy, then - z = (- x, - y) = - x -iy is the reflection of z

through the origin, and z = (x, -y) = x - iy is the reflection of z through the

x-axis, as illustrated in Figure 1.8.

We can use an important algebraic relationship to establish properties of the

absolute value that have geometric applications. Its proof is rather straightfor-

ward, and we as k you to give it in the exercises for this section.

lzl2 = zz. (1-22)