MATH 201-01, Transition to Advanced Mathematics, Fall 2012

Abstract

This class is intended to help you to make the leap from computational mathematics to proof-based mathematics. Computational mathematics often requires little more than the ability to commit a formula to memory and apply it to a particular case. Proving theorems (at least, the harder ones) requires creativity and the ability to see patterns and find unexpected connections. There is no tidy formula, no one size fits all approach, that will allow you to prove every theorem. There are many mathematical conjectures that remain unproved despite the best efforts of some of the world’s most brilliant minds. Examples of computational mathematics include factoring polynomials, computing derivatives or integrals, and finding equations of tangent lines. Examples of proof-based mathematics include showing that the square root of 2 is irrational, establishing that there is a prime number between n and 2n for all positive integers n, and proving that there are infinitely many primes p so that p+2 is also prime. We will first study the fundamental notions of mathematics, such as sets, rules of logical inference, relations, and functions. Then, we will explore standard proof techniques, including proofs by construction, cases, contradiction, and induction. We will study these methods in mathematical settings including algebra, analysis, number theory, and combinatorics. Along the way, we will learn about some facets of mathematical culture. We will discuss issues in the philosophy of mathematics, learn about some open problems, “meet” some famous mathematicians, and try to get a birds-eye view of the history and present condition of the mathematical enterprise. We will discuss the features that make for good mathematical writing. We will learn a bit about how to use LaTeX, a software package that is used to typeset mathematical writing. You will receive 5 percentage points of extra credit on each homework prepared using LaTeX.