MATH5535 is a Mathematics Level V course. See the course overview below.

Units of credit: 6

Prerequisites:

Cycle of offering: Variable

Graduate attributes: The course to uses techniques from widely diverse areas of mathematics: number theory, calculus, set theory, complex analysis, linear algebra and the theory of computation.

More information: this recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus. (This pdf will usually be updated by the end of the first week of the semester.)

If you are currently enrolled in MATH5535, you can log into the My eLearning Vista instance of this course.

MATH5535 Irrationality and Transcendence is a course whose roots go back to about 500 B.C., when Pythagoras or one of his followers proved that, contrary to "common sense", some numbers cannot be expressed as a ratio of integers. While the Ancient Greeks succeeded in proving various surd expressions to be irrational, little further progress was made until the eighteenth century, when Euler and Lambert proved the irrationality of e, π and related numbers. We look first at more modern proofs of these results, deferring Lambert's work until later.

The question of transcendence is deeper, and harder, than that of irrationality. After giving a survey of the basic ideas regarding algebraic numbers, we shall prove the existence of transcendentals, firstly (following Cantor) without exhibiting any particular example! The simplest approach to showing that a specific number is transcendental is to study its approximations by rational numbers; continued fractions provide an important tool for doing so. Taking another look at e and π, we shall adapt Hermite's method to prove the transcendence of these numbers.

A recent and fascinating topic connects transcendence with deterministic finite automata, a kind of very elementary computing device. Ideas concerning such automata can be used to investigate the transcendence of numbers which display some sort of "pattern" in their decimal expansions or continued fractions.