How and Why we do Mathematical Proofs

**As taught in Autumn Semester 2009/10**

Dr Joel Feinstein, School of Mathematical Sciences

The aim of this short unit is to motivate students to understand *why
*we might want to do proofs (why proofs are important and how they can
help us) and to help students with some
of the relatively routine aspects of *doing *proofs.

In particular, the student will learn the following:

- proofs
can help you to really see
*why*a result is true;

- problems that are easy
to state can be hard to solve (
*e.g.*Fermat's Last Theorem);

- sometimes
statements which appear to be intuitively obvious may turn out to be
false (
*e.g.*Simpson's paradox);

- the answer to a question will often depend crucially on the definitions you are working with;

- how to start proofs;

- how and when to use definitions and known results.

The module is organised into three sections: Why; How (Part I); How (Part II) - see navigation panel on the left of the screen.

With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs. A practice sheet is included after students have completed all three sections. Each section is suitable for a different level of audience, as described below:

Suitable for: **Foundation, undergraduate year one and undergraduate year two students **

- Section 1: Why: Anyone with a knowledge of elementary algebra and prime numbers, as may be obtained by studying A level mathematics. (Foundation)

- Section 2: How (Part I) – Suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions). (Undergraduate year one)

- Section 3: How (Part II) – Requires some background knowledge of convergence and divergence of series of real numbers. A revision sheet is available. (Undergraduate year two)