Mathematics went through quite a revolution around the turn of the 20th century. In particular, an axiomatic approach infiltrated the mathematical paradigm, both as a tool to ensure mathematical rigour and to abstract common principles working in a variety of different settings.
First year mathematics emphasizes computation over abstraction and rigour. Later year courses (and Pure Mathematics in general) reverse this, so students need to learn some new skills and some new ways of thinking about mathematical objects.
This course is designed to help you develop the ability to write rigorous mathematical proofs in a setting where the level of abstraction is still quite modest. As such it will serve as an excellent preparation for the third year Pure Mathematics courses.
The course consists of two halves, algebra and analysis, each taught for 6 weeks.
Analysis half. Most of the calculus you have seen involves equalities. Mathematical analysis however, is largely about inequalities, about suitably bounding quantities that cannot be calculated precisely. Many nice examples come from geometry and we will frequently use these to motiva????te our discussion in the first part of the analysis section. In the latter part we will look more closely at some aspects of the real numbers, such as how well one can approximate π by a rational p/?? (in terms of how large ?? is).
Algebra half. We will investigate various transformations on the plane and projective plane. We will first study several types of transformations such as translations, reflections, rotations etc. in terms of groups. We will then look at symmetries, i.e. transformations of geometric figures that preserve some property (such as distance or angles between lines), and projective geometry. Projective transformations can change a conic section of one type to another, e.g. an ellipse to a hyperbola.
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