Good books on complex numbers
I searched on this site but I found posts asking for books on complex analysis which I guess is not what I want.
I want a book on problems concerning with roots of unity, argand plane, exponential form, inequalities etc. Some examples : 1, 2, 3.
I don't think these types of problems are covered in a analysis book but I don't know. I think I want a book on algebra of complex numbers.
Are these topics covered in analysis book ? If not, please suggests some books on this topic. I guess my question is bit vague, sorry for that. :).
As suggested in the comments I would like to have books similar to Complex Numbers from A to ...Z.
Solution 1:
A distinction needs to be made between purely geometric uses of complex numbers and uses in the theory of equations (polynomials, rational functions, etc.). Obviously, there is a good deal of overlap, but some books deal primarily with one aspect or the other.
Parsonson (1971). Pure Mathematics, Vol. 2 (both). The material on complex numbers and equations occupies roughly the first half of the book. Challenging problems, similar to STEP papers or old S-levels.
Ferrar (1943). Higher Algebra (both). About 60 pages on geometric/trigonometric applications and 100 on the theory of equations. Problems at or above the difficulty in Parsonson. Not to be confused with the same author's Higher Algebra for Schools.
Durell and Robson (1930, 1937). Advanced Algebra, Volume II and Advanced Trigonometry (both). I'm less familiar with these books, but I know they were the standard books on these subjects at higher certificate/scholarship level in England for many years. They can be downloaded here.
Hahn (1994). Complex Numbers and Geometry (geometry).
Andreescu and Andrica (2005). Complex Numbers from A to... Z (geometry). I don't know the last two books well, but they're recommended at imomath.com. They seem to be mostly about geometry and have little on the theory of equations in comparison with Parsonson and Ferrar. Andreescu and Andrica's book is very focused on using complex numbers to do coordinate geometry (including cases where this results in pages' worth of calculations), and it comes with solutions to the exercises.
Colin and Morvan (2011). Nombres complexes, polynômes et fractions rationnelles. After briefly introducing the theory, most of the book is devoted to presenting detailed solutions to exercises on these topics.
Gautier, Girard, Gerll, Thiercé, Warusfel (1971). Aleph 0. Algèbre, Terminale CDE: nombres réels, calcul numérique, nombres complexes (both). Most of this book is devoted to geometric and algebraic uses of complex numbers. Similar or slightly lower level to Parsonson, but more detailed treatment.
Engel (2009). Komplexe Zahlen und ebene Geometrie. In addition to the basic material, this book discusses the Riemann sphere and gives some computer visualizations in MAPLE.
Kretzschmar (2011). Komplexe Zahlen für Dummies. The title speaks for itself! I'm including this title just for fun, as it seems to be aimed at very elementary users such as those in electronics.
The book by Engel gives an analytic proof of the fundamental theorem of algebra. Unfortunately, I don't believe any of the other books proves it.
There is a book by Yaglom called Complex Numbers in Geometry, but it actually discusses topics that are far removed from what one usually thinks of with this title. The book Geometry of Complex Numbers by Schwerdtfeger deals with advanced topics.
Solution 2:
Books on complex analysis definitely use the topics that you mentioned, but usually assume that the reader is already familiar with some algebra and geometry of complex numbers. The book Visual Complex Analysis by Tristan Needham is a great introduction to complex analysis that does not skip the fundamentals that you mentioned. In particular, the first chapter includes detailed sections on the roots of unity, the geometry of the complex plane, Euler's formula, and a very clear proof of the fundamental theorem of algebra. The exercises for chapter one cover all of those topics and also develop many identities. Needham's book has a friendlier and more inviting tone and structure than any other analysis book that I've read; I loved studying from it in my undergraduate complex analysis course.