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Number Systems: An Intoduction to Algebra and Analysis

Sergei Ovchinnikov
American Mathematical Society
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts 23
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Bollman
, on

Algebra and analysis?

In the same book?

It’s not the madness that it might seem at first.

Since we often use “algebra/analysis/topology” as a coarse organization structure for pure mathematics, it’s easy to overlook the considerable overlap between these branches of mathematics. If you’ve had the recent opportunity or pleasure to teach an introductory course in both of these subjects, you understand that both rely at the start on certain properties of certain number systems — and that’s the point of this book.

From the natural numbers through the integers, rationals, and reals to the complex numbers, the necessary and interesting properties of numbers are rigorously developed, in a style that is consistently attractive and engaging. The interplay between algebra and analysis is particularly well-done in the chapter on rational numbers, where the transition from ordered fields to Cauchy sequences seems entirely natural. The interplay recurs nicely when considering complex numbers as a normed division algebra before moving on to convergent sequences in the complex plane.

This book could find a niche as a supplemental text for both abstract algebra and real analysis classes. The temptation in doing this would be to get so caught up in its ideas and their exposition that the other content would be diminished.

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Mark Bollman ( is professor of mathematics and chair of the department of mathematics and computer science at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.