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672 CHAPTER 19 Complex Numbers and Functions

Inequalities
There are several inequalities that are useful in dealing with complex quantities. Let z and
w be complex numbers. Then
1. |Re(z)|≤|z| and | Im(z)|≤|z|.
2. |z + w|≤|z|+|w|.
3. ||z|−|w|| ≤ |z − w|.

2
2
2
2
Property (1) follows from the fact that |x|≤ x + y and |y|≤ x + y .
Property (2) is called the triangle inequality. It follows immediately from the vec-
tor interpretation of complex numbers, since we already know the triangle inequality for
vectors.
For property (3), use the triangle inequality to write
|z|=|(z + w) − w|≤|z + w|+|w|.
Therefore,
|z|−|w|≤|z + w|.

By interchanging z and w,
|w|−|z|≤|z + w|.
Upon multiplying this inequality by −1, which reverses the inequality, we have

−|z + w|≤|w|−|z|.
Combine inequalities to obtain

−|z + w|≤|z|−|w|≤|z + w|
and this implies that ||z|−|w||≤|z + w|.
Argument and Polar Form
Let z = a + ib be a nonzero complex number. The point (a,b) has polar coordinates (r,θ),
where r =|z| (Figure 19.3). We call θ an argument of z. Of course, given any argument θ, then
θ + 2nπ is also an argument for any integer n.
Using Euler’s formula, we can write
iθ
z = a + ib =r cos(θ) + ir sin(θ) =r(cos(θ) + i sin(θ)) =re .
iθ
The expression re is called the polar form of z.

y
r = z

θ
x

FIGURE 19.3 Polar form of z.

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October 15, 2010 18:5 THM/NEIL Page-672 27410_19_ch19_p667-694