## Polar Form

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.

Summary

 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  