I love the “Very Short Introductions” series by Oxford University Press as generally they are very well written and accessible to a general audience but not patronising.

Number 260 in this series, “Numbers” by Peter M. Higgins is no exception to this rule. The text flows nicely, and (considering the brevity of the text) is a comprehensive introduction to the types of number – starting from the counting numbers, moving through rationals, reals and transcendentals before looking at different types of infinity and countable versus uncountable sets. The book ends with an introduction to complex numbers and operations on complex numbers.

Here are a few highlights from the book:

- When discussing primes there is the following aid to memory given “inequality signs always point to the smaller quantity”. It may seem stupid, but I don’t think I have ever said that.
- He often mentions things but doesn’t discuss them in full – this is really effective at encouraging people to go away and find out more – for example the Cattle Problem attributed to Archimedes.
- There is a great exploration of Cantor’s Middle Third Set where there is a suggestion that using a ternary representation of numbers instead of decimal makes it easier to see that there are infinitely many numbers in the Middle Third Set. Recall that to construct the Middle Third Set you start with the unit interval \([0,1] \) and delete the middle third (i.e remove all the numbers between \(\frac{1}{3}\) and \(\frac{2}{3}\).) This process is then repeated recursively on the remaining intervals. If you represent numbers in ternary then working in base 3 a third is given by 0.1 and 0.2 represents two thirds. So removing the first middle portion removes all the numbers whose ternary expansion starts with a 1, the next iteration removes all those numbers remaining which have a 1 in the second place in their ternary expansion and so on. So in the end, all numbers with a 1 anywhere in their ternary expansion have been removed. I really like this approach, as it makes it a lot clearer that Cantor’s Middle Third Set is uncountable.
- Introducing matrices by relating them to complex numbers is really nice. The number \(z = a+ i b \) can be represented by the matrix \( \begin{pmatrix} a & b \\ -b & a \end{pmatrix} \). If I do ever write an A-Level textbook I think I could be tempted to introduce matrices after complex numbers to make the link with matrix multiplication etc.

I’d encourage anyone to get this book and they are also all £4.99 instead of £7.99 at the moment if you buy them through Blackwells.