Modelling and control by Anthony Rossiter

# INTRODUCTION TO COMPLEX NUMBERS

This is a section in the chapter on mathematical skills. Some is revision of pre-university mathematics and some is syllabus mostly covered in year 1 of engineering programmes. Use the left hand toolbar to access the other chapters and themes.

This page 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, and thus 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 if they want too. It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, exponentials, logarithms and Laplace. This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and also a tutorial sheet for users to test themselves.

## Complex numbers 1 - the basics

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

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Multiplication of complex numbers is best done:
A. In cartesion form (that is using real and imaginary parts).
B. In modulus argument form.
C. None of the above.

## Complex numbers 2 - modulus and argument

Introduces the concepts of modulus and argument. Also spends some time showing how the modulus and argument of conjugates and inverses are related to the orginal complex number.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

For a complex number a+bj, which is true?
A. The argument is given as atan(b/a)
B. The modulus is given as sqrt(a2+b2).
C. The argument is atan(a/b).
D. Several but not all of the above.

## Complex numbers 3 - multiplication

Introduces fast methods of complex number multiplication using the modulus/argument form. Spends sometime 'demonstrating' the vailidity of the result, although viewers could focus on just the result should they choose.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## Complex numbers 4 - division

Introduces fast methods of complex number division using the modulus/argument form. Spends sometime 'demonstrating' the vailidity of the result, although viewers could focus on just the result should they choose.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## Complex numbers 5 - scaling and rotation

Shows how the rules for multiplication/division in modulus/argument form lead to a very useful interpretation of complex numbers as scaling and rotation operators. This is used to solve problems in many engineering scenarios. Obvious typo at 7min 30 where writing uses 110 instead of 100.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## Complex numbers 6 - exponential form

Introduces the exponential form for a complex number and demonstrates how this is consistent with the multiplication/division rules in modulus/argument form as well as interpretations as a scaling/rotation operator.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

#### Quick test question

Which are true?
A. The exponential form is interesting but of little engineering use..
B. The exponential form embeds modulus/argument information into routine engineering techniques.
C. The exponential form allows more convenient manipulation of complex numbers.
D. Several but not all of the above.

## Complex numbers 7 - square roots

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.

## Complex numbers 8 - using De Moivre for root computations

Shows how to solve for cube roots, 4th roots and so on of complex numbers. Makes use of De Moivre's Theorem, but the focus is on a simple presentation style which shows why the result works.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

## Complex numbers 9 - Tutorial sheet

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

• A summary of key facts and derivations is given in the pdf file.
• Talk through video 1 is on youtube.  