Complex Numbers: Lecture 5

In this lecture, there are two main concepts.

1. Exponential form of a complex number
2. De Moivre's Theorem

Exponential form

You have already learnt the polar form of a complex number, i.e. \[z=r(cos\theta+i sin\theta)\]

By using the Euler's Equation: \[e^{i\theta}=cos\theta+i sin\theta\]
We can change polar form to exponential form \[z=re^{i\theta}\]

The laws of exponents for real numbers still hold for complex numbers. Hence, when complex numbers are written in exponential form, it is comparatively easy to do the computation.

De Moivre's Theorem

Basically De Moivre's Theorem tells us that \[(cos\theta+i sin\theta)^{n}=cos n\theta+i sin n\theta\] for all n being rational numbers.

It is directly derived from the exponential form of complex numbers.

De Moivre's Theorem is very useful in terms of simplifying trigonometric calculation.

After today's lecture, you can do all the questions in tutorial 10b except Q13.