Pristine Landscapes in Elementary Mathematics

“Pristine Landscapes in Elementary Mathematics” is an interesting book. The topics and problems are carefully selected, though with a focus on algebra and number theory. I like the way that the topics and materials are organized and presented such that fresh insights and surprisingly elegant properties are exposed to the reader upon seemingly familiar concepts. For example, Chapter 3 on Squares covers Pell’s Equation and discusses representations with sums of squares, both beautiful results in number theory. I also like the list of challenging problems together with detailed solutions for each chapter. Sometimes multiple solutions from different perspectives are given. Those problems and solutions will certainly help the reader improve skills in logical thinking and formulating proofs.

Nice book which covers many different areas of elementary mathematics, yet has a decidedly algebraic flavor! Even the chapters in number theory verge towards algebraic methods (mostly inequalities and approximations). The book would thus be very suitable for students who want to get better at solving problems that involve more than a 'trick' and in which a bit of technique and maturity are essential.

Most chapters are fairly self-contained, with the first half of the book (ch. 1-7) accessible to all students (there are some problems involving groups, but a student not familiar with groups can easily skip them). The latter half (ch. 8-14) is to some extent also self-contained, but it might be hard to approach for a student who has no knowledge whatsoever of vectors, complex numbers, linear algebra and analysis. Not much is needed, but it shouldn't all look new.

For teachers, the book can provide problems of all levels (save, perhaps, very difficult problems) and so can be very useful for familiarizing students with a given subject.