epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Preface
This book is addressed to those interested in learning more about how
and why proofs of mathematical statements work, and it has been written
keeping in mind the following remark by George Polya. "A great discovery
solves a great problem but there is a grain of discovery in the solution of any
problem. Your problem may be modest; but... if you solve it by your own
means, you may experience the tension and enjoy the triumph of discovery."
Since the only background required is the material covered in a first
semester Calculus class, most of the statements considered will deal with
basic properties of numbers and functions. The fact that a statement might
seem "easy" to understand does not imply that proving it will be an effortless
task, as Fermat's Last Theorem^ has so clearly shown!
^Fermat's Last Theorem states that it is impossible to find three (nonzero) integer numbers x, y,
and z, such that
x" + y" = z"
when « > 3. Pierre de Fermat (1601-1665) claimed he had a proof of this statement, but he
never pubHshed it and it has not been found. Through the years several mathematicians were
able to prove the truth of the statement only for some values of the exponent n. Finally Andrew
Wiles, with some help from his colleagues and using several results developed since Fermat's
time, offered a complete and lengthy proof of the theorem in 1993. Quite a few years and some
very advanced mathematical ideas were needed to prove a theorem that can be stated in one
sentence!
XI