``Geoff Smith's Introductory Mathematics: Algebra and Analysis is a delight. The book adopts a gentle style with discursive commentary throughout, and contains real humour in places. This is, though, a book with a serious purpose, namely to introduce the fundamental ideas of algebra and analysis with precision and rigour.
Taking the view that to ask where numbers come from is at least as dangerous a question as to ask where babies come from, the book begins with a treatment of sets. We learn that Sigma means `add this lot up', and are warned about `confusion opportunities'.
Complex numbers are introduced from first principles, and all through the treatment of exponential and hyperbolic functions, the reader is persuaded of doing and writing mathematics with due regard to rigour.
Elementary linear algebra is included, with a treatment of vectors, matrices and determinants. Linear independence is considered, but eigenvectors and eigenvalues are left out, and this is appropriate. Group theory is introduced via a discussion of permutation groups. The need for rigour is stressed again when sequences and series are considered, particularly the dangers arising from sloppy arguments with limits and infinite sums. Analysis is considered to be `calculus with attitude', and here is the introduction to the feared `epsilon-delta technology'. This is done with sympathy, and enables the concepts of continuity, limit and differentiability to be handled simply but with rigour.
There is a list of further reading, and the book is supported by a web site, which also contains more examples''
Reviewer: Nigel Steele, Head of Mathematics, University of Coventry.
This book is published in paperback by Springer. It costs £14-95 in Europe and $29-95 in North America. Its ISBN is 3-540-76178-0.