Dad's A-Level Mathematics: Complex Numbers. The Fantastic Numbers

Complex Numbers. Fantastic Numbers

This book is an introduction to the fantastic Complex numbers for high school students (A-Level, International Baccalaureate etc) who want to have a more formal introduction to the subject. It is aimed at the pupil who wants to dig deeper, or wants to follow a career in Mathematics or Mathematical Physics and as such it is sometimes rather more demanding. The content of the book and the level of exercises can also serve those who prepare for specialized entry exams that many Universities now demand. It also makes a good entry point for those who want to enter into mathematical competitions or simply want to have fun.
The book covers all the standard topics, starting from how Complex numbers were first discovered, the algebra of Complex numbers to the geometry of the Argand plane, Euler's famous identity, the nth-root of unity and complex functions like e^z, sin z and ln z. I have also included topics, like the Matrix representation of Complex numbers and the Riemann stereographic projection. It includes more than 200 exercises with solutions. Most of these questions make regular appearance in high school exams.
The book emphasizes proof and rigor together with many explanations and examples that illustrate the "why'' of the many theorems. Proofs and definitions can sometimes be very dry. One of the many enigmas that students of mathematics have is the "why this or that definition''. Many mathematical concepts are introduced by definitions that seem to come out of a rabbit-hat. The truth of course is that mathematical definitions of objects have a long evolutionary history before they assume the final form in a textbook. I tried, wherever I could, to illuminate that process by adding historical facts about them.
The Pandemic
This book grew out of the Covid pandemic. During the multiple lock-downs home schooling became very important and like many parents, I decided to help my children with their mathematics. I was not completely clueless, as some 30 years ago, one of my first jobs after completing my PhD was teaching A-level mathematics. I was very rusty but I had time and enthusiasm.
Why don't you write a book?
As any parent would testify, teaching your own kids is very hard. After many unsuccessful attempts, I decided that the best way was to write notes in LaTeX and email them. LaTeX is the unsurpassed software for scientific writing. It produces beautifully type-setted equations and beauty is very important in Mathematics. That strategy worked better. It was then that my son suggested of turning the notes into a book. I fancied the idea, but writing a textbook is not easy. It is a bit like people saying ``You cook well, why don't you open a restaurant?''. Writing a book and especially a mathematics book is not an easy task for seasoned mathematicians. Much harder for a rusty amateur like me. I nevertheless decided to do it and call it Dad's mathematics.
Maths versus Procedural Maths
I surveyed some of the recommended textbooks but found most of them not to my taste. After all, I am rather old fashioned. Many books resemble recipe books. Just a set of rules to be followed. A sort of of Haynes owners workshop manual for mathematics. This serves well students that want to pass an exam, but this aim is not equivalent to comprehending the subject. For instance, I always found hard to fathom how one can do derivatives but skip the concept of the limit. Anyway, the intention was not just to write another recipe book for solving problems. There are plenty of those around. I wanted the book to serve both the purpose of passing the exam but also to pass on the mathematical ideas and concepts.
I wanted exam success, but more importantly, I wanted the reader to get at least a glimpse of the beauty of mathematics. I hope I have succeeded in that aim.