Introductory Mathematics: Algebra and Analysis (Springer Undergraduate Mathematics Series) Paperback – 1998

Here is an author who brings a fresh vitality to basic undergraduate mathematics with his witty, discursive treatment of topics ranging from the nature of a proof to a kit for the construction of the real numbers.
Smith adopts a tone of camaraderie which lures the reader along a trail of mathematical discovery, all the while remaining true to the standards of exactitude which must characterise mathematics at this level. Never condescending, he blithely raises issues which other authors would sweep under the carpet in a beginners' textbook, treating them with his inimitable forthright professionalism. The author's exuberant interest and keen mind pervade the book, making it a perfect delight to read.
I would recommend that all young undergraduates/school-leavers are given this book before more traditional dusty texts blind them to the sheer pleasure of mathematics.

As a returning novice to university mathematics, (it is some 15 years since the last time I last took an introductory maths course) I found this book to be both refreshing and highly original. The material and layout is different to most textbooks. It is probably a book for people who want to grasp the idea of mathematics rather than just pass an exam. However, as the author notes in the preface, it is (other things being equal) both a 'gentle and relaxed introduction'. The mathematics is pure and the emphasis is on the idea rather than on how to solve particular problems in the life sciences or engineering. Topics covered include; Sets, functions and relations; Proofs; Complex numbers; Vectors and matrices; Group theory; Sequences and series; Real numbers; and Mathematical analysis. It is an excellent book for those interested in learning and understanding mathematics. The colloquial tone helps alot and it is a style that deserves to become more common in the future. The book also offers an interesting glimpse of the mathematical mind. The author has a remarkable sense of humour and the book is hilarious at times.

Geoff Smith's Introductory Mathematics: Algebra and Analysis provides a splendid introduction to the concepts of higher mathematics that students of pure mathematics need to know in upper division mathematics courses. Smith's explanations are clear and laced with humor. He gives the reader a sense of how mathematicians think about the subject, while making the reader aware of pitfalls such as notation that varies from book to book or country to country and subtleties that are hidden within the wording of definitions and theorems. Since the book is written for first-year British university students who are reading pure mathematics, Smith's approach is informal. He focuses on conveying the key concepts, while gradually building greater rigor into the exposition. The exercises range from straightforward to decidedly non-routine problems. Answers to all questions are provided in an appendix or on a website devoted to the book whose address is listed in the book's preface. That website also contains a list of known errata, extra, generally more difficult, exercises on the material in the book, and discussions of topics related to those in the book. The book is suitable for self-study. Students preparing to take or review advanced mathematics courses will be well-served by working through the text.

The text begins with material on set theory, logic, functions, relations, equivalence relations, and intervals that is assumed or briefly discussed in all advanced pure mathematics courses. Smith then devotes a chapter to demonstrating various methods of proof, including mathematical induction, infinite descent, and proofs by contradiction. He discusses counterexamples, implication, and logical equivalence. However, the chapter is not a tutorial on how to write proofs. For that, he suggests that you work through D. L. Johnson's text Elements of Logic via Numbers and Sets (Springer Undergraduate Mathematics Series).

Once this foundation is established, Smith discusses complex numbers. After describing the types of problems that can be solved using natural numbers, integers, rational numbers, and real numbers, he justifies the introduction of complex numbers by showing that they are necessary to solve quadratic equations. After deriving the Quadratic Formula, Smith describes the algebra of complex numbers, their rectangular and polar forms, and their relationship to trigonometric, exponential, and hyperbolic functions. Throughout the remainder of the book, he draws on the complex numbers as a source of examples.

The next portion of the book is devoted to algebra. Smith discusses key concepts from linear algebra, including vectors, the Cauchy-Schwarz and Triangle inequalities, matrices, determinants, inverses, vector spaces, linear independence, span, and basis, that are widely used in mathematics. In addition to looking at their algebraic properties, Smith examines their geometric interpretation. He continues this practice with permutation groups, which he uses to introduce group theory, the branch of mathematics in which he does his research. Group theory is a deep topic, on which Smith and his wife, Olga Tabachnikova, have written a text for advanced undergraduates, Topics in Group Theory (Springer Undergraduate Mathematics Series). In this text, he confines the discussion to subgroups, cosets, Lagrange's Theorem, cyclic groups, homomorphisms, and isomorphisms.

Smith introduces analysis with a chapter on sequences and series. After providing another proof of the Triangle Inequality, Smith focuses on limits, thereby giving the reader a first exposure to quantifiers. He also discusses some properties of the real numbers, introducing the concept of boundedness, the Completeness Axiom, and Cauchy sequences. The aforementioned exposure to quantifiers makes the subsequent definitions and proofs of theorems about continuity and limits of functions easier to grasp. He concludes the book with a discussion of how the real numbers can be constructed using Dedekind cuts and Cauchy sequences.

There is a book by Ian Stewart and David Tall, The Foundations of Mathematics, that covers similar ground. It is devoted to building up the properties of number systems, which is a useful foundation for courses in analysis. However, it will not prepare you as well for courses in algebra as Smith's text, which I recommend enthusiastically.

If you want to understand mathematics after high school you must read this book first before plunging into university level mathematics. It will make your journey in mathematics much easier.

...so take my review with a grain of salt.

What this book seems to be telling me is I have no business trying to teach myself math. I suspect those with a natural aptitude will have no trouble with it, but I need more hand-holding than this text offers. Though I'm not a total numb-nut - I've gone to Wikipedia while studying some sections and have found clearer, more enlightening explanations.

Truly great writing. I'm an undergraduate student and have recently came to love mathematics. The author's humor and clear passion for his book really shines through. I am using this book to teach myself analysis, basic proof writing and as a bridge to higher mathematics. Many undergrads experience difficulty in bridging to higher maths and analysis. I urge you to use the "look inside" feature of this book. I was sold doing just that, and am happy to report that the entire book is just as accessible, even to someone new to the field, such as me.

Mathematics has changed so much and is now so widely applied - and this is a great introduction to modern algebra and frankly, modern math. I have been reading this book at a leisurely pace - no hurry here, just a nice diversion for me personally - and couldn't be more satisfied with my purchase. Just the right combination of simplicity and rigor, and logical progression of ideas through the book. As a novice, I have needed to go on the web and clarify some things that the author glosses over but only in a couple of instances.

I have no idea how this book is for undergraduates or anyone that is pursuing math as a field of study - there are other reviewers that can attest to that better than I can. But for anyone curious about modern mathematics, this is likely an invaluable stepping stone.