BodhiAI

Nov. 11, 2019
Modulus of Complex Numbers

The modulus of any complex number z = a+ is denoted by |z| and is defined as

|z| = a2+b2

|z| =

Clearly |z|0

Properties of Modulus

Let a = , and be three imaginary numbers, such that

P-1 If |z| = 0, then Re (z) = 0 and Im (z) = 0 simultaneously.

Reasoning: Since, |z| =

a = 0 and b = 0 both simultaenously { a, b R}

P-2

Reasoning: Since,

Also,

Again,

P-3

Reasoning: For any complex number z = a+, we have

where a, bR

P-4

p-5

Reasoning: Since, and we have

P-6 where

P-7 |zn| = |z|n

P-8

Reasoning: Since,

P-9

Reasoning: Since,

P-10

Reasoning: Since, {Reference P-3}

Replace z by we have

Adding throughout the inequality, we have

P-11