Appendix A - Some mathematical tools

Summary

This appendix collects mathematical tools that are needed in the main text. In addition, it gives a brief description of some essential background topics. It is assumed that the reader knows elementary calculus. The topics are grouped in four sections. First, we consider some useful methods of indirect proofs. Second, we introduce elementary results for complex numbers and polynomials. The third topic concerns series expansions. Finally, some further calculus is presented.

Some methods of indirect proof

Perhaps the most fundamental of all mathematical tools is the construction of a proof. When a direct proof is hard to obtain, there are indirect methods that can often help. In this section, we will denote a statement by p (such as “I like this book”), and another by q (such as “matrix algebra is interesting”). The negation of p will be denoted by ¬p. The statement “p and q” is denoted by p ∧ q, and the statement “p or q (or both)” is denoted by p ∨ q. The statements ¬(p ∨ q) and ¬p ∧ ¬q are equivalent: the negation transforms p, q into ¬p, ¬q and ∨ into ∧. This is the equivalent of De Morgan's law for sets, where p and q would be sets, ¬p the complement of p, q ∨ q the union of the sets, and p ∧ q their intersection.