Showing complex numbers as a point on an Argand diagram allows further properties to be found.

The modulus of a complex number is the length of the line which joins it to the origin (0, 0). From the diagram below, using Pythagoras' Theorem, it can be seen that:

The modulus of the number a + bi

r = √(a2 + b2)

The modulus of a complex number z is written IzI


The argument (θ) of a complex number is the angle formed at the origin between the positive x-axis and the line which indicates the modulus. This angle is usually measured in radians, and is given for values between -π < θ ≤ π.The argument of a complex number z is written arg(z)

From the diagram it can be seen that:

a = r.cos θ and b = r.sin θ

Thus the complex number a + bi can be written:

a + bi = r.cos θ + r.sin θ i

          = r (cos θ + i sin θ)

          = r cis θ

cis is an abbreviation for cos θ + i sin θ.


a + bi is called the RECTANGULAR or REAL-IMAGINARY form

r cisθ is called the POLAR or MODULUS-ARGUMENT form.


e.g. For the complex number z = 1 + i

Modulus IzI = √ (12 + 12) = √2

Arg(z) = cos-1 (1/√2) = 

Therefore 1 + i = √2 cis