Students must normally have A-level Mathematics, Grade A, or equivalent, in order to undertake this unit.
Aims & Learning Objectives:
Aims: This course is designed to cater for first year students with widely different backgrounds in school and college mathematics. It will treat elementary matters of advanced arithmetic, such as summation formulae for progressions and will deal with matters at a certain level of abstraction. This will include the principle of mathematical induction and some of its applications. Complex numbers will be introduced from first principles and developed to a level where special functions of a complex variable can be discussed at an elementary level.
Students will become proficient in the use of mathematical induction. Also they will have practice in real and complex arithmetic and be familiar with abstract ideas of primes, rationals, integers etc, and their algebraic properties. Calculations using classical circular and hyperbolic trigonometric functions and the complex roots of unity, and their uses, will also become familiar with practice.
Natural numbers, integers, rationals and reals. Highest common factor. Lowest common multiple. Prime numbers, statement of prime decomposition theorem, Euclid's Algorithm. Proofs by induction. Elementary formulae. Polynomials and their manipulation. Finite and infinite APs, GPs. Binomial polynomials for positive integer powers and binomial expansions for non-integer powers of a+b. Finite sums over multiple indices and changing the order of summation. Algebraic and geometric treatment of complex numbers, Argand diagrams, complex roots of unity. Trigonometric, log, exponential and hyperbolic functions of real and complex arguments. Gaussian integers. Trigonometric identities. Polynomial and transcendental equations.