Reddit mentions of An Imaginary Tale: The Story of The Square Root of Minus One

Paul Nahin has published many good historical math books that don't skimp on the mathematical underpinnings. I particularly enjoyed An Imaginary Tale: http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004

Regarding Spivaks: I'm also working on it, and found that my proof technique was lacking. An Introduction to Mathematical Reasoning (Eccles) was helpful for me: http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.

Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.

Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.

Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.

Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.

Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.

The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.

An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.

Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)

You need it because it makes the math work, but the term imaginary is foolish. "Imainary" numbers are numbers just like real numbers. The only difference being that you cannot have an Imaginary quantity of something (just like you cannot have a negative quantity of something, but we still use negstives). Imaginary numbers are associated with rotations and periodicity (sine waves), and they even have the geometric interpretation of a problem being "unsolvable" with real lengths, but even if you construct these unsolvable problems with the complex numbers, lo-and-behold the complex solution gives you the geometric property you wanted to construct!

If you are at all interested, this is an excellent book written by an electrical engineering professor about the history and applications of imaginary numbers.

You and the other A/AS-level kids shouldn't worry about complex numbers. At university, you'll come to appreciate the usefulness and beauty of them. For example, see this post I made on /r/math earlier today. People have written entire books eulogising about them. For example, Paul Nahin has written two such books.

Not that I'm aware of. If you're interested in reading more, I recommend checking out Paul Nahin's An Imaginary Tale. It's midway between a textbook and a popular math book that explores the history and utility of imaginary numbers (believe me, there are a lot of applications of imaginaries)

You will absolutely enjoy:

An Imaginary Tale

Dr. Euler's Fabulous Formula

Euler: The Master of Us All

One of my favorites is An Imaginary Tale: The Story of √-1. I read it when I was a freshman and couldn't wait to take Complex Analysis. The author has a sort-of-sequel about Euler's equation, which I didn't like as much but was still enjoyable.