Top positive review
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An unusual and outstanding guide
on 5 November 2008
I received a copy of this book for my birthday this year, and was immediately drawn towards the material on algebra with which I was most familiar. The most striking thing about the book as a whole, both in the bits I know well, and the bits I don't know much about, is the way in which it combines large portions of accessible text with substantial mathematical content - it is the lack of actual serious content which frustrates me most about popular books about maths (Fermat's Last Theorem and the Riemann Hypothesis have received the popular treatment with limited content). But in this companion there are clues to enable an enthusiast to engage with the material.
It is right to say, as the introduction does, that there is material in this book which will require some knowledge beyond what you might learn at school. But do not let that put you off (it is no more daunting than the physical size of the book, which mirrors only its extensive coverage). For the genius of this companion is in the way that it engages with real mathematics and how it is actually done by real mathematicians. It is not simply a presentation of particular subject matter (as in a text book) nor is it an encyclopedia of mathematics, for it seeks to convey an understanding of the nature and importance of pure mathematics, rather than simply definitions of the component parts - and through disciplined editing and clear exposition it sets a standard which will not easily be matched.
There is a huge amount which will challenge and excite a good A-level student. My one observation is that the suggested further reading is sometimes a great deal more demanding than the component articles of the Companion (which is partly a reflection of just how good these articles are) - but more consistency, or a scheme for marking references accessible at or just beyond A-level is the only improvement I could think of.
If this book had been in my school library when I was 17 ... well every school with a 6th form should have one. And anyone with a true interest in mathematics should have it to browse (and will want to read it).