Lancaster University

University mathematics has a rather different feel from that encountered at school; the emphasis is placed far more on proving general theorems than on performing calculations. This is because a result which can be applied to many different cases is clearly more powerful than a calculation that deals only with a single specific case.

For this reason we begin by taking a look at the language and structure of mathematical proofs in general, emphasizing how logic can be used to express mathematical arguments in a concise and rigorous manner.

We then apply these ideas to the study of number theory, establishing several fundamental results such as Bezout's Theorem on highest common factors and the Fundamental Theorem of Arithmetic on prime factorizations.

Next, we introduce the concept of congruence of integers. This, on the one hand, gives us a simplified form of integer arithmetic that enables us to answer with ease certain questions which would otherwise seem impossibly difficult; and on the other it leads naturally to the abstract idea of an equivalence relation which has applications in many areas of mathematics.

Finally, we show how the idea of a highest common factor can be generalized from the integers to the polynomials.

Set theory: understand and be able to use basic set-theoretic notation.

Logic: understand the use of truth tables, and be able to set them up; be able to express mathematical statements symbolically, using quantifiers and connectives, and to negate them; master the three main methods of proof (direct, contraposition and contradiction); be able to present simple mathematical proofs.

Number theory: be able to perform division with remainder, and prove the underlying theorem; know what is meant by the highest common factor of a pair of integers, and understand how to compute it using the Euclidean algorithm and prime factorization; be able to state and prove Bezout's theorem, and know how to apply it to derive related results; know what the lowest common multiple of a pair of integers is, its relation to the highest common factor, and how to compute it; know what is meant by a prime number, and how to find prime numbers using the Sieve of Eratosthenes; be able to state and prove the Fundamental Theorem of Arithmetic, and to prove that there are infinitely many prime numbers; be familiar with various applications of prime factorization.

Congruences: know what it means that two integers are congruent modulo a given number; be able to solve linear congruences, and understand the proofs of the underlying theorems; be able to state, prove, and apply the Chinese Remainder Theorem.

Relations: know what is meant by a relation, be able to decide whether or not a relation is reflexive, symmetric, or transitive; know what is meant by an equivalence relation; know what is meant by an equivalence class and a congruence class; be able to define the sum and the product of two congruence classes, and show that it is well-defined; be familiar with various applications of the arithmetic of congruence classes; be able to construct the integers from the natural numbers, the rational numbers from the integers, and the complex numbers from the real numbers.

Polynomials: know the division algorithm; know what is meant by the highest common factor of a pair of polynomials, and how to compute it using the Euclidean algorithm.

The student will be familiar with some fundamental concepts in logic and elementary number theory, will be able to understand and present simple mathematical proofs, and will know how to apply the theory to calculate highest common factors and solve equations involving congruences. 

Assessment will be through:
(i) weekly coursework, aimed at testing and consolidating understanding of the basic elements of the course;

(ii) a test at the end of the module which assesses the students' understanding through short questions;

(iii) an examination in the Summer which assesses more fully the students' understanding and summative knowledge of the topics.