Third Year Projects:

Dedekind sections, decimals and the real numbers
BA/BSc Project
Professor Maurice Dodson

The construction of the familiar set of real numbers form collections of rationals using Dedekind's approach of `cuts' in the line. Comparison with other approaches, such as Cauchy sequences or the familiar representation of real numbers by infinite decimal expansions.

Sources:
M Spivak, Calculus.
G L Isaacs, Real Numbers.
N McCoy, Introduction to Modern Algebra.

Diophantine approximation or how close irrationals are to rationals
BA/BSc or MMath Project
Professor Maurice Dodson

Diophantine approximation is a quantitative analysis of the fact that the rationals are dense in the reals. Dirichlet's Theorem for real numbers is a fundamental result which could be the starting point of a deeper study of the approximation of real numbers by rationals through continued fractions (related to Euclid's algorithm, it was used in the design of gears in ancient Greece and in Europe in the 17 and 18th centuries). Alternatively the higher dimensional theory or the metric theory could be developed.

Sources:
Hardy and Wright, Introduction to Number Theory.
W M Schmidt, Diophantine approximation.

Counting, numbers and sets
BA/BSc Project
Professor Maurice Dodson

Topics could include:
An account of the history of counting, the development of the natural numbers and the idea of place value.
A comparison of a modern view of the integers with the historical evolution of arithmetic.
The ideas of cardinal and ordinal numbers.
Irrational numbers in ancient Greece.
This project could be more historical than mathematical. In this case the historical material will be taken from books or other secondary sources and should be clearly referenced. You will be expected to add the mathematical knowledge you have acquired to the history. In particular, you should show an understanding and appreciation of the development of the number system since the 19th century.

Sources:
J Barrow, Pi in the Sky, Penguin.
D H Fowler, The Mathematics of Plato's Academy.

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