# Chapter one

Mathematical skills

## Section two: Complex Numbers

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

This section gives an introduction to complex numbers and key attributes. It is assumed that students have been introduced to the definition of complex numbers before, albeit briefly. Therefore, the focus is more on identifying and understanding the key properties that will be used in engineering problem solving.

The videos do spend a little time demonstrating that some key results are true, but students could skip those parts and concentrate on the key results.

### 1. The basics

A rapid review of complex numbers as would be covered in introductory lectures including definitions, the Argand diagram, simple multiplication, division and addition and complex conjugates.

A talk through video is on YouTube.

### 2. Modulus and argument

This video introduces the concepts of modulus and argument. It also spends some time showing how the modulus and argument of conjugates and inverses are related to the original complex number.

A talk through video is on YouTube.

### 3. Multiplication

This video introduces fast methods of complex number multiplication using the modulus/argument form. We spend some time 'demonstrating' the validity of the result, although viewers could focus on just the result, should they choose.

A talk through video is on YouTube.

### 4. Division

This video introduces fast methods of complex number division using the modulus/argument form. We spend some time 'demonstrating' the validity of the result, although viewers could focus on just the result, should they choose.

A talk through video is on YouTube.

### 5. Scaling and rotation

This video shows how the rules for multiplication and division in modulus and argument imply that complex numbers are scaling and rotation operators. This is used to solve problems in many engineering scenarios.

There is a typing error at 7 minutes 30 seconds. The writing uses 110 instead of 100.

A talk through video is on YouTube.

### 6. Exponential form

This video introduces the exponential form for a complex number. It demonstrates how this is consistent with the multiplication and division rules in modulus and argument form, as well as interpretations as a scaling and rotation operator.

A talk through video is on YouTube.

### 7. Square roots

This video looks at the computation of square roots using complex number algebra. This uses simple problems to establish an approach to such problem solving which subsequently can be applied to more complex root problems.

A talk through video is on YouTube.

### 8. Using De Moivre for root computations

This video shows how to solve cube roots, fourth roots and so on of complex numbers. It makes use of De Moivre's Theorem, but the focus is on a simple presentation style which shows why the result works.

A talk through video is on YouTube.

### 9. Tutorial sheet

A number of questions for students to test their understanding of the videos in this series, followed by worked solutions.

A talk through video is on YouTube.