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