An Imaginary Tale: The Story of "i" by Paul J Nahin
This was a very interesting read on the history and application of “i”, the square root of –1.
It starts with the history of the solution of the polynomial equations and explains the confusion that the mathematicians of the time had when they found they could manipulate SQRT(-1) to get valid solutions of equations with real roots, despite them not believing that such a value could exist. The book covers the geometrical justifications of allowing such a root, and them moves on to Argand diagrams. There are then several chapters giving various uses of the complex numbers, showing their use in deriving various power series expansions, Kepler’s laws and some properties of various circuits from electronics. The final chapter covers complex function theory and Cauchy’s Integral theorems which can be used to calculate various complicated integrals. Some of these results are almost magical in the simplicity of their derivation.
The book takes some work in places, with sometimes a page or two of algebra to derive a result. I think this is one of the appeals of the book, in that the author tries to make things fairly rigorous, at the expense of giving the reader some work to do.