Mathematical Thinking: Problem-Solving and Proofs (2nd Edition) [Hardcover]

John P. D'Angelo (Author), Douglas B. West (Author)

Book Description

ISBN-10: 0130144126 | ISBN-13: 978-0130144126 | Publication Date: December 27, 1999 | Edition: 2

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.

Editorial Reviews

From the Publisher

Offering a survey of both discrete and continuous mathematics, Mathematical Thinking begins with the fundamentals of mathematical language and proof techniques such as induction. These are applied to easily-understood questions in elementary number theory and counting. Further techniques of proofs are then developed via fundamental topics in discrete and continuous mathematics. The text can be used for courses emphasizing discrete mathematics, continuous mathematics, or a balance between the two. It contains many engaging examples and stimulating exercises. --This text refers to an out of print or unavailable edition of this title.

From the Back Cover

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.

Product Details

• Hardcover: 412 pages
• Publisher: Prentice Hall; 2 edition (December 27, 1999)
• Language: English
• ISBN-10: 0130144126
• ISBN-13: 978-0130144126
• Product Dimensions: 9.1 x 6.1 x 1.1 inches
• Shipping Weight: 1.5 pounds (View shipping rates and policies)
• Average Customer Review: 3.2 out of 5 stars  See all reviews (11 customer reviews)
• Amazon Best Sellers Rank: #141,576 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

6 of 6 people found the following review helpful:

5.0 out of 5 stars Difficult but well worth it, October 10, 2007

By 

A. Townley - See all my reviews
(REAL NAME)   

This review is from: Mathematical Thinking: Problem-Solving and Proofs (2nd Edition) (Hardcover)

I'm using this in an undergraduate introduction to proofs class with a focus on analysis. As a freshman, it seems a bit overwhelming at times - I wouldn't recommend it to most freshmen or even sophomores.
I do feel like this does a more than adequate job preparing me for more advanced math, and goes far above and beyond similar "proofs and problem solving" style books.

5 of 5 people found the following review helpful:

3.0 out of 5 stars Good content, but needed a bit more, April 20, 2009

By 

Benjamin - See all my reviews

This review is from: Mathematical Thinking: Problem-Solving and Proofs (2nd Edition) (Hardcover)

This book was an excellent read, and provided some great information. However, as a math text, I feel like it should have provided a bit more examples, and perhaps even a solutions manual.

I understand that the books main focus was on the abstract discussions of mathematics, but I feel like that should merit the writer to put a bit more examples to drive home the theorems before copious amounts of problems are assigned at the end of each chapter. A ton of these problems are classical, and need to be understood, in light of this, why isn't there a solutions manual to better explain these problems?

I greatly appreciated the voices of the writers keeping themselves grounded in pragmatic language. Too often will mathematicians get lofty in their dictions and fuddle the material they claim to understand all too well. This book did a great job discussing simple concepts simply, meanwhile working the more difficult ones with more space.

9 of 11 people found the following review helpful:

5.0 out of 5 stars Extremely Useful - Great Read, February 2, 2008

By 

mathmaven "gary j" (Springfield, Va) - See all my reviews

This review is from: Mathematical Thinking: Problem-Solving and Proofs (2nd Edition) (Hardcover)

I ran into the first edition of this book ten years ago when taking courses at George Mason University, and really loved it. I still love it. It covers proofs from all basic "pieces" of mathematics and gives the reader a good feel for the "proofology," both in technique and fundamental nomenclature and results, that a student is expected to know when taking the first analysis and abstract algebra courses. It's not perfect though. I haven't bought the second edition, but in the first edition, Example 2.21, p.27 says: "An integer is even if and only if it is the sum of two odd integers." Obviously, it is easy to show the sum of two odd integers is even by forming the sum (2k+1) + (2l+1) = 2(k+l+1), which is twice an integer and thus even. But, if an integer is even, it can be the sum of two odd integers OR two even integers, so the statement is not complete. If small stuff like that doesn't bother you, this book is for you.

The author gives solutions or hints for one-third to one half the problems depending on the chapter, which is more than enough for self-study. I would disregard the whiny one star review that is posted for this book; it is typical of someone who wants to be spoonfed mathematics.