# A Concise Introduction to Pure Mathematics

###### Martin Liebeck
Publisher:
Chapman & Hall/CRC
Publication Date:
2016
Number of Pages:
301
Format:
Paperback
Edition:
4
Price:
64.95
ISBN:
9781498722926
Category:
Textbook
[Reviewed by
Fernando Q. Gouvêa
, on
05/27/2016
]

See our review of the second edition. For the fourth edition, the author has added two new chapters on groups, going as far as Lagrange’s Theorem and its applications. These chapters are intended to serve as an introduction to abstract algebra. There is also new material on inequalities, counting methods, the inclusion-exclusion principle, and Euler’s $\phi$ function.

These changes put the book firmly in the position of a “transition’ or “bridge’ course, assuming that students have taken calculus and linear algebra and preparing them for the next steps. Liebeck’s book stands out from the crowd of similar books by being short (as the title says, it is concise) and by trying to expose students to mathematical ideas beyond the basics of sets and logic. In addition to the pre-Analysis and pre-Algebra chapters, there are chapters on complex numbers, inequalities, some number theory and combinatorics, and the Platonic solids. Students are taught how to understand and create proofs, but they are also given a glimpse of what it is all for.

My one regret is the title. It is already very easy for students whose interest is mostly in applied mathematics to conclude that all the proofs in their Algebra and Analysis courses are of no value. To label the transition course an introduction to “pure mathematics’ is to reinforce that feeling. I suspect that if I were asked to teach this course, that problem would be enough to keep me from adopting the book. Which is a pity. After all, applied mathematicians also need to know about proofs, counting, inequalities, bounds, and even groups — and this book could help them learn all that.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.

Sets and Proofs
Number Systems
Decimals
Inequalities
nth Roots and Rational Powers
Complex Numbers
Polynomial Equations
Induction
Euler’s Formula and Platonic Solids
The Integers
Prime Factorization
More on Prime Numbers
Congruence of Integers
More on Congruence
Secret Codes
Counting and Choosing
More on Sets
Equivalence Relations
Functions
Permutations
Infinity
Introduction to Analysis: Bounds
More Analysis: Limits
Yet More Analysis: Continuity
Introduction to Abstract Algebra: Groups
Introduction to Abstract Algebra: More on Groups
Solutions to Odd-Numbered Exercises