What is Mathematics? The science (or art?) that deals with numbers, quantities, shapes, patterns and measurement An abstract symbolic communication system based on formal logic Learning (Greek mathematikoi, ‘educated people’) “Mathematics can be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true” (Bertrand Russell) Department of Mathematics

Why is Maths important? It describes the relationships of abstract, symbolic concepts It provides a good description of the physical universe It is at the heart of all scientific and technical development Every idea in mathematics results from an attempt to solve some problem Department of Mathematics

What do mathematicians do? “It is difficult to give an idea of the vast extent of modern mathematics” (Arthur Cayley, 1883) “Mathematics is a game played according to certain simple rules with meaningless marks on paper” (David Hilbert, 1862-1943) ‘Pure’ Mathematicians: Look for patterns Generalize and prove results Inspire ‘applied’ mathematics ‘Applied’ Mathematicians: Model reality Solve problems Inspire ‘pure’ mathematics Department of Mathematics

How do we know things? By PROVING them! To a mathematician, proof is an argument to be accepted beyond even unreasonable doubt Different from science, law, etc. What is your favourite proof? Department of Mathematics

How many primes are there? Suppose there are finitely many prime numbers: p1, p2, …, pn Let q = (p1 p2 … pn) + 1 Divide q by any of p1, p2, …, pn The remainder is 1, so none of p1, p2, …, pn is a factor of q But every integer is prime or a product of primes (this needs proof!) We have reached a contradiction, so there are infinitely many primes Department of Mathematics

Solved and unsolved The Four Colour theorem (Proved by computer analysis in 1977) Fermat’s last theorem: xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2 (Proved by Andrew Wiles in 1995) The twin primes conjecture: There are infinitely many pairs of prime numbers which differ by 2, e.g. (3, 5), (5, 7), (11, 13), … Department of Mathematics

How does Maths advance? Questions are raised by Observing patterns Trying to explain phenomena Formulating and trying to solve mathematical models If the necessary mathematics already exists, the problem can be solved. Otherwise, research is necessary Department of Mathematics

Research at Surrey includes … Dynamical systems Partial differential equations Weather prediction Population modelling Sleep cycles Medical statistics Water waves String theory Quantum field theory Department of Mathematics

What’s in a Maths degree? Three strands: Mathematical techniques, e.g. Calculus. Like A-level but harder Theory and proof. Why are results valid? Abstract Pure Mathematics Applications. Modelling the real world, e.g. Mathematical Biology. Statistical analysis Need a really good knowledge of the A- level Core Department of Mathematics

Infinity behaves strangely! The set of natural numbers is N = {1, 2, 3, 4, 5, 6, 7, …} How many natural numbers are there? An infinite set is countable if it can be put into one-to-one correspondence with N The set of integers is Z = {…, -3, -2, -1, 0, 1, 2, 3, …} Is Z the same size as N? (Is Z countable?) The set of rational numbers is Q = {m/n}, where m is in Z and n is in N Is Q bigger than Z? (Is Q countable?) What about the set of real numbers R? Z and Q are countable. R is not Department of Mathematics

Another counting problem On a 12-hour clock face, the hour hand is pointing to 12. Where will it point a million hours later? 12 Department of Mathematics