1. 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)
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2. 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

3. 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, )
‘Pure’ Mathematicians:
Look for patterns
Generalize and prove results
Inspire ‘applied’ mathematics
‘Applied’ Mathematicians:
Model reality
Solve problems
Inspire ‘pure’ mathematics
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4. 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?
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5. 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

6. 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 > (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), …
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7. 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

8. 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

9. 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

10. 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

11. 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