Aims: University mathematics introduces progressively more and more abstract ideas and structures, and demands more and more in the way of proof, until by the end of a mathematics degree most of the student's time is occupied with understanding proofs and creating his or her own. This is not because university mathematicians are more pedantic than schoolteachers, but because proof is how one knows things in mathematics, and it is in its proofs that the strength and richness of mathematics is to be found.
But learning to deal with abstraction and with proofs takes time. This module aims to bridge the gap between school and university mathematics, by beginning with some rather concrete techniques where the emphasis is on calculation, and gradually moving towards abstraction and proof.
Number systems: Natural numbers, integers. Rationals and real numbers. Existence of irrational numbers. Complex Numbers.
Polar and exponential form of complex numbers. De Moivre's Theorem, $n$'th roots and roots of unity.
Euclidean algorithm; greatest common divisor and least common multiple.
Prime numbers, existence and uniqueness of prime factorisation. Infiniteness of the set of primes.
Modular arithmetic. Congruence, addition and multiplication modulo $n$.
- Language and Proof
Proof by induction.
Proof by contradiction.
Basic set theory:$\cap,\cup$ , Venn diagrams and de Morgan's Laws. Cartesian product of sets, power set.
Logical connectives$\wedge$ , $\vee$ , $\Rightarrow$ and their relation with $\cap$ , $\cup$ and $\subseteq$. Quantifiers $\forall$ and $\exists$.
Sets, functions and relations
Injective, surjective and bijective functions.
Relations: equivalence relations, order relations.
Multiplication and long division of polynomials.
Euclidean algorithm for polynomials.
Remainder theorem; a degree $n$ polynomial has at most $n$ roots.
Algebraic and transcendental numbers. Fundamental theorem of
Algebra (statement only).
Cardinalities, including infinite cardinalities.
Cardinality of the power set of X is greater than cardinality of X.
Countability of the rational numbers, uncountability of the reals.
Transcendental numbers exist!
Students will work with number systems and develop fluency with their properties; they will learn the language of sets and quantifiers, of functions and relations, and will become familiar with various methods and styles of proof.
None of these is the course text, but each would be useful, especially the first:
A.F.Beardon, Algebra and Geometry, CUP, 2005.
I.N. Stewart and D.O. Tall, Foundations of Mathematics, OUP, 1977.
J. A. Green, Sets and Groups; First Course in Algebra, Chapman and Hall, 1995.