I Wish They'd Taught Me That - Robin Pemantle and Julian Gould ***

Subtitled 'overlooked and omitted topics in mathematics', the obvious concern is that there is a good reason these topics are overlooked and omitted. Thankfully, this is not the case, but it's fair to say that despite attempts to dress it up that way, this isn't a recreational maths book. There's a fair description in the blurb: 'the topics which every undergraduate mathematics student "should" know, but has probably never encountered... magnificent secrets that are beautiful, useful and accessible.'

As someone who many years ago did a degree with a fair amount of mathematics in it, I think it probably would have appealed back then - though to be honest a lot of it has disappeared from my memory, strongly reducing the entertainment value. Here's an example. The first real page contains the sentence:

'If you are handed a real number 𝓍 ∈ ⁠ $\mathbb {R}$ ⁠, one way to tell if 𝓍 is rational or irrational is to look at sequences of rational numbers q_n that approach (but don't touch!) 𝓍.'

This isn't a complicated sentence per se, it's just the way it is put across is almost designed to put off any reader that isn't doing a maths degree. Which really isn't necessary.

To all intents and purposes this is a light undergraduate textbook with rather more text than would be common in a maths book, introducing some genuinely interesting topics: irrationality, generating functions, polynomial systems, transfinite induction, Brownian motion (and its application to option pricing), approximation by polynomials, public key cryptography, asymptotic analysis, fractional dimensions, ultrafilters, scissors congruence, and Brownian motion take 2.

I can agree with the suggestion that these would be of real interest to maths students who treat mathematics as fun (in my experience, this doesn't apply to all of them). Some of the topics are reasonably obvious - for example what's going on behind the cryptography that protects the internet, which is rarely well explained in popular books. But others hide engaging concepts - for example, ultrafilters can be regarded as a mathematical methodology for solving logic puzzles.

For that tiny target market, this is a real, wonderful find. I'm afraid for the rest of us, this is not Martin Gardiner's Mathematical Puzzles and Diversions on steroids, as it's far too much of a textbook.

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