Complex Numbers: Lecture 3

In today's lecture, you learnt two important concepts of complex numbers: moduli and argument.

1. Moduli

• Modulus of a complex number z, denoted by z, represents the distance between z and origin in the argand diagram. Hence, we have

- |z| is non-negative

- |z|=0 if and only if z=0

• z1-z2 or z2-z1 represents the distance between z1 and z2 in the argand diagram.

2. Argument

• If z is represented by point P in the argand diagram, argument of z is the angle between OP and the positive real axis.
• The principal argument of z is in the interval (-pi, pi]. Take note that -pi is not included by pi is included! (why not include both?)
• To find the argument of z=x+yi

- find the basic angle using \[tan^{-1}\frac{y}{x}\]

(If you don't know what basic angle is, please revise your secondary school work.)

- determine which quadrant z lies in and use the basic angle to get the argument

3. Properties of argument and modulus

• Properties of argument is similar to the properties of logarithm. This is not by coincidence. After you learnt the exponential form of complex numbers, you will know the reason.

After today's lecture, you can attempt Q1 and Q3 in tutorial 10b.