How Things (Mathematical) Work

This book started life as a short treatise on the development and application of mathematical algorithms, using the Ruby programming language for demonstration purposes. However, it soon became clear that there are a number of interesting algorithms hidden in the electronic devices that are becoming commonplace. What started with mortgages, greatest common divisors, internal rate of return algorithms, the Kelly (betting) Criterion, and the traveling salesman problem, soon turned into a more diverse project with attention given to the mathematical innards of things like GPS units and MP3 players. Every non-trivial algorithm is accompanied by a Ruby program (or two or three) with a line-by-line analysis and examples. The programs range from fairly elementary ones, such as mortgage calculations, which demonstrate why the formulas banks use for this purpose, and have used for years, are just plain wrong, with the banks coming out on the short end! The attack on the traveling salesman problem, using simulated annealing, is more advanced. The book ends with a fairly advanced algorithm, the so-called Fast Fourier Transform, or FFT, which, while not being particularly long, is quite deep. Some of the chapters in this book, e.g., number theory, can be grasped with no more than a course in high school algebra while for others, e.g., the FFT, a course in calculus would be a great help. All the programs are written in Ruby, so chosen because it's first of all free, it's interpreted, and Ruby is also easy to learn. None of the programs, however, require more than a very basic knowledge of the Ruby language.