About these notes
These notes are an introductory exploration of complex numbers, written for first-year students at the University of Cape Town. They are structured to be both readable and thorough, so that working through them and doing all of the exercises should put you in a good place for understanding the foundation of the subject.
They begin somewhere slightly unusual. Before any complex number appears, the first two sections ask what a number is at all, and how much freedom we have to invent one. That is a historical and philosophical aside, but the hope is that it gets you thinking carefully about things you probably thought you knew.
Authors
Written by Jonathan Shock and Mashudu Mokhithi, with improvements by Ruach Slayen and Harry Wiggins.
Department of Mathematics & Applied Mathematics, University of Cape Town.
How to use the site
Each section is a page, and the arrows at the top and bottom move you through them in order. Along the way you will find:
- Worked examples in green, with solutions hidden behind a toggle. Try them before you look.
- Check your understanding questions in amber, with brief answers behind a toggle.
- Definitions, theorems and properties in blue, indigo and teal, numbered so they can be referred to across sections.
- Proofs in red, which you can read or skip on a first pass.
- A checklist at the end of most sections. Ticking a box saves it in your browser, so your progress is still there when you come back.
Corrections
These notes are a work in progress and there are likely still typos. If you find one, please email jon.shock@gmail.com and we will fix it as soon as we can.
The notes are released under a CC BY 4.0 licence: use them, adapt them, share them, with attribution.