Most helpful positive review
8 of 9 people found the following review helpful
5.0 out of 5 starsClear and logically presented
ByC. R. Longon 27 August 2012
This book is very approachable and spans the school to university mathematics. It presents in the first chapter 18 "test" statements that can be proved mathematically. The rest of the book takes the statements progressively and by explanation, further examples and student exercises it helps the reader to first clearly understand what each statement is saying then working through the proof.
Different wording of statements requires different approaches. For example:
Statement 4: IF 2^67 - 1 is prime then 2^67 + 1 is divisible by 3.
and
Statement 15: There exists a unique positive integer t such that t + 2 and t + 4 are all primes.
Most of the examples are from (simple) number theory since these are the most accessible and familiar to most readers.
I particularly like the discussion before the working of a proof that tries to work through the thinking that leads to the choice of a particular approach that is valid. Also it helps you read a mathematical statement so you can understand what is (are) the given(s) and what result you have to establish. I like example 1:
"300 000 067 110 605 737 is not a perfect square"
What are you being asked to prove? Is this statement actually true? The book clearly tackles such uncomfortable questions.
The exercises help to clarify and reinforce the approach in the worked example. The book is finished off by a discussion of fallacies and mistakes in proofs.
That's what the book contains and I think it does that job well. Beyond that, I think this is an important book because it introduces and develops skill in proof, a skill that is much neglected in school maths which focusses on applying methods without questioning if the approach is valid. If ever you are slightly bothered as to whether the mathematics you are writing is actually valid or are not clear as to what a mathematical question is asking you to show, then this may be the book for you.