Mathematical Thinking: Problem-Solving and Proofs

by John P. D'Angelo, Douglas B. West

This survey of both discrete and continuous mathematics focuses on the logical thinking skills necessary to understand and communicate fundamental ideas and proofs in mathematics, rather than on rote symbolic manipulation. Coverage begins with the fundamentals of mathematical language and proof techniques (such as induction); then applies them to easily-understood questions in elementary number theory and counting; then develops additional techniques of proofs via fundamental topics in discrete and continuous mathematics. Topics are addressed in the context of familiar objects; easily-understood, engaging examples; and over 700 stimulating exercises and problems, ranging from simple applications to subtle problems requiring ingenuity. ELEMENTARY CONCEPTS. Numbers, Sets and Functions. Language and Proofs. Properties of Functions. Induction. PROPERTIES OF NUMBERS. Counting and Cardinality. Divisibility. Modular Arithmetic. The Rational Numbers. DISCRETE MATHEMATICS. Combinatorial Reasoning. Two Principles of Counting. Graph Theory. Recurrence Relations. CONTINUOUS MATHEMATICS. The Real Numbers. Sequences and Series. Continuity. Differentiation. Integration. The Complex Numbers. For anyone interested in learning how to understand and write mathematical proofs, or a reference for college professors and high school teachers of mathematics.

Genres: Mathematics, Nonfiction, Textbooks

412 pages, Hardcover

First published August 1, 1996

Reviews

[Joseph Mirabelli]

The book had a lot of respect for the reader, and some excellent example problems. However, the definitions were often put so contextually as to give pause, and the material was very broad. Loved it overall, though, as a survey and a simple reference. Had I had the time at UIUC (the one semester I did, the waitlist was too long), I would have enjoyed the proofs class using the text. Alas.

[Alexander]

Decent problems, awful prose.

[Lucinda]

If you love math, this is a fun book