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670 CHAPTER 19 Complex Numbers and Functions

Complex arithmetic obeys many of the rules we are accustomed to from working with real
numbers. If z, w, and u are complex numbers, then
1. z + w = w + z (addition is commutative)
2. zw = wz (multiplication is commutative)
3. z + (w + u) = (z + w) + u (associative law for addition)
4. z(wu) = (zw)u (associative law for multiplication)
5. z(w + u) = zw + zu (distributive law)
6. z + 0 = 0 + z = z
7. z · 1 = 1 · z = z.

The Complex Plane
Complex numbers admit two geometric interpretations.
Any complex number z = x + iy can be identified with the point (x, y) in the plane
(Figure 19.1(a)). In this context, the plane is called the complex plane, the horizontal axis is
called the real axis, and the vertical axis is called the imaginary axis. Any real number x graphs
as a point (x,0) on the horizontal (or real) axis, and any pure imaginary number yi (with y real)
is a point (0, y) on the imaginary axis.
A complex number x + iy also can be identified with the vector xi + yj in the plane
(Figure 19.1(b)). This is consistent with addition, since we add two vectors by adding their
respective components, and we add two complex numbers by adding their real and imaginary
parts, respectively.
Magnitude and Conjugate

The magnitude of x + iy is the real number

2
2
|x + iy|= x + y .

y y y
x + iy xi + yj
(x, y)
(x, y) x + iy
x + iy
x x x

(a) (b) (c)
y
w

z – w
z x

(d)
FIGURE 19.1 x + iy as a point, a vector, and its
length.

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October 15, 2010 18:5 THM/NEIL Page-670 27410_19_ch19_p667-694