Introduction to Complex Numbers - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Dr Vicky Neale
Course Term: 
Michaelmas
Course Lecture Information: 

This course of two lectures will run in the first week of Michaelmas Term.

Course Overview: 

Generally, students should not expect a tutorial to support this short course. Solutions to the problem sheet will be posted on Monday of Week 2 and students are asked to mark their own problems and notify their tutor.

This course aims to give all students a common background in complex numbers.

Learning Outcomes: 

By the end of the course, students will be able to:

(i) manipulate complex numbers with confidence;
(ii) use the Argand diagram representation of complex numbers, including to solve problems involving the $n$th roots of unity;
(iii) know the polar representation form and be able to apply it in a range of problems.

Course Synopsis: 

Complex numbers and their arithmetic.
The Argand diagram (complex plane).
Modulus and argument of a complex number.
Simple transformations of the complex plane.
De Moivre's Theorem; roots of unity.
Euler's theorem; polar form $r\mathrm{e}^{\mathrm{i}\theta}$ of a complex number.
Polynomials and a statement of the Fundamental Theorem of Algebra.

Reading List: 

1) R. A. Earl, Complex numbers https://www.maths.ox.ac.uk/study-here/undergraduate-study/bridging-gap

2) D. W. Jordan & P Smith, Mathematical Techniques (Oxford University Press, Oxford, 2002), Ch.6.