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

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

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.

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.

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.

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.