Maths – Advice & Reading

Dominic Rowland, Debate Chamber Maths Tutor:
Gauss called Mathematics the “Queen of the sciences” and indeed it underpins every aspect of how we try and model the world around us. For many people the way Mathematics allows us to predict real world phenomena is part of what makes the subject so fascinating. Have you ever noticed how the stream of water from a slowing running tap gets thinner? Have you ever wondered what kind of curve the stream follows? How might you model the situation? (If you know about constant acceleration you can solve this problem so go and try it.)

There are however, scores of Mathematicians who, quite frankly, couldn’t care less about the real world. They love maths for its precision, its intellectual challenge and its elegant solutions. Again we might start with an observation, for example the fact that some numbers (like 13 and 149) can be written as the sum of two squares, while others (like 6 and 11) cannot. How can we tell which category a number like 4165 belongs to? Any undergraduate Number Theory course should completely answer this question: that is to say not only determine exactly which numbers are the sums of two squares but, more importantly, prove rigorously that the assertions are correct. (Rather miraculously the proof involves the elusive square root of minus one.) Perhaps you can already think of ways to generalise this problem. If you can, and are interested in how you might turn questions like these into Theorem’s with water-tight proofs then you should think about pursuing some Mathematics for its own sake: there is some very, very cool stuff out there.

A) Books full of maths.

A concise introduction to Pure Mathematics by Martin Liebeck

Clear and accessible with 15 free standing chapters introducing a range of topics. Almost no background knowledge is assumed and Liebeck still manages to prove some interesting and important results.

Introductory Mathematics: Algebra and Analysis by Geoff Smith

An excellent bridge from school to university level mathematics introducing the two foundational areas of first year pure maths. This is a textbook but the style is less formal than usual and there is plenty of commentary (and even the occasional maths joke) unlike the relentless Theorem – Proof, Theorem – Proof format of many books.

B) Books about maths.

A Mathematician’s Apology by G.H.Hardy

This undoubtedly “the book” on what being a professional mathematician is like written by arguably the best English Mathematician of the last two hundred years. Completely non-technical, easy to read and short.

Fermat’s Last Theorem by Simon Singh

The story of one of the greatest maths problems of all time. Singh charts the various attempts to tackle Fermat’s conjecture posed in 16973 leading up to the celebrated proof by Andrew Wiles in 1995. He includes entertaining descriptions of some of the colourful characters involved and includes some maths in a worthwhile appendix.

C) Books somewhere between the two

Gamma by Julian Havil

This book also has a historical feel, starting with the invention of the logarithm and ending with the Zeta function but Havil doesn’t shy away from symbols and you’ll need a pencil and paper at times.

A challenging but immensely rewarding read.

Prime Obsession by John Derbyshire

Books about the Riemann Hypothesis, unquestionably the greatest unsolved problem in Mathematics, abound but this is one of the best, Derbyshire mixes readable history with enough maths to get a feel for the ideas involved yet