Allenby R. Numbers and proofs

Oxford: Butterworth Heinemann, 2003. — 289 p. — ISBN: 9780340676530, 0340676531.Numbers and Proofs presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any attempt proof is made. Established proofs which the student is in a better position to follow then follow.
Presented in the author's entertaining and informal style, and written to reflect the changing profile of students entering universities, this book will prove essential reading for all seeking an introduction to the notion of proof as well as giving a definitive guide to the more common forms. Stressing the importance of backing up "truths" found through experimentation, with logically sound and watertight arguments, it provides an ideal bridge to more complex undergraduate maths.The need for proof
Statements and connectives
True or false?
Sets, negations, notations and functions
Proofs...for all
There exists...proofs
The principle of mathematical induction
The integers and rational numbers
The rational numbers and the real numbers
Real numbers and complex numbers
Guessing, analogy and transformation
Generalisation and specialisation
Fallacies and paradoxes
A mixed bag
Solutions appendix
Answers to exercises