20 CHAPTER 2. ALGEBRA AND NOTATION

Definition 2.3.3 Define the absolute value of a complex number as follows.

|a+ ib| ≡√

a2 +b2.

Thus, denoting by z the complex number z = a+ ib,

|z|= (zz)1/2 .

Also from the definition, if z = x+ iy and w = u+ iv are two complex numbers, then|zw|= |z| |w| . You should verify this. ▶

Notation 2.3.4 Recall the following notation.

n

∑j=1

a j ≡ a1 + · · ·+an

There is also a notation which is used to denote a product.

n

∏j=1

a j ≡ a1a2 · · ·an

The triangle inequality holds for the absolute value for complex numbers just as it doesfor the ordinary absolute value.

Proposition 2.3.5 Let z,w be complex numbers. Then the triangle inequality holds.

|z+w| ≤ |z|+ |w| , ||z|− |w|| ≤ |z−w| .

Proof: Let z = x+ iy and w = u+ iv. First note that

zw = (x+ iy)(u− iv) = xu+ yv+ i(yu− xv)

and so |xu+ yv| ≤ |zw|= |z| |w| .

|z+w|2 = (x+u+ i(y+ v))(x+u− i(y+ v))

= (x+u)2 +(y+ v)2 = x2 +u2 +2xu+2yv+ y2 + v2

≤ |z|2 + |w|2 +2 |z| |w|= (|z|+ |w|)2 ,

so this shows the first version of the triangle inequality. To get the second,

z = z−w+w, w = w− z+ z

and so by the first form of the inequality

|z| ≤ |z−w|+ |w| , |w| ≤ |z−w|+ |z|

and so both |z| − |w| and |w| − |z| are no larger than |z−w| and this proves the secondversion because ||z|− |w|| is one of |z|− |w| or |w|− |z|. ■

With this definition, it is important to note the following. Be sure to verify this. It is nottoo hard but you need to do it.