How to Think Like a Mathematician: A Companion to Undergraduate Mathematics [Paperback]

Kevin Houston (Author)

Book Description

ISBN-10: 052171978X | ISBN-13: 978-0521719780 | Publication Date: February 23, 2009

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

Editorial Reviews

Review

"In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof... In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof."
Charles Ashbacher, Journal of Recreational Mathematics

"The author provides concise, crisp explanations, including definitions, examples, tips, remarks, warnings, and idea-reinforcing questions. Houston expresses thoughts clearly and concisely, and includes succinct remarks to make points, clarify arguments, and reveal subleties."
W.R. Lee, Choice Magazine

Book Description

Looking for a head start in your undergraduate degree in mathematics? This friendly companion eases beginning students into real mathematical thinking, unlocking important techniques for effective mathematics so you can communicate with clarity, solve problems, and explore the world of definitions, theorems and proofs with real confidence.

See all Editorial Reviews

Product Details

• Paperback: 278 pages
• Publisher: Cambridge University Press (February 23, 2009)
• Language: English
• ISBN-10: 052171978X
• ISBN-13: 978-0521719780
• Product Dimensions: 9.5 x 7.3 x 0.6 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #34,668 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

21 of 21 people found the following review helpful:

5.0 out of 5 stars Great Introduction to Mathematical Method, December 14, 2009

By 

Edmon Begoli (Knoxville, TN United States) - See all my reviews
(REAL NAME)   

This review is from: How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (Paperback)

I am reading a great book that is in author's words intended for undergraduate students of mathematics, but that in my opinion offers much more to a motivated reader.

"How to Think Like Mathematician" by Dr. Kevin Houston is a very engaging, readable and pragmatic text on mathematical "technique". It is a non-pompous, well written, valuable, easy to follow and understand valuable set of lessons and tips on understanding and adopting mathematical method, language, theorems, proofs and techniques.

It is an introductory text, so for more in depth treatment of the subjects such as proofs and number theory you may need to look further into books such as Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) orThe Princeton Companion to Mathematics but if you are looking to understand the mathematical method and how to be able to read and write "serious" math this is an ideal book.

I specially have to point out Dr. Houston's writing style - it is engaging, humorous, but substantial, pedagogical and never trivial.

6 of 6 people found the following review helpful:

5.0 out of 5 stars A systematic and gentle approach to explaining the main ideas of mathematics, July 18, 2010

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
(TOP 500 REVIEWER)    (VINE VOICE)    (HALL OF FAME REVIEWER)   

This review is from: How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (Paperback)

To be successful in mathematics, your mind must perform operations that are unlike the operations needed to do most other things. You must be able to hold abstract ideas, sometimes several at a time, as well as see the relationships between multiple concepts. Furthermore, those abstract ideas are built on other abstract ideas; for example, most of mathematics is built on the fundamental abstract idea of the use of a variable.
However, being different and at times being hard does not mean that the ideas of mathematics are incomprehensible. Humans excel at understanding abstract ideas, a strong argument can be made that such a skill is the very definition of human intelligence. In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof.
Many students are capable of leaping the comprehension hurdle; yet hit a wall when it comes time to apply the concepts in order to generate a proof. In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof.

Published in Journal of Recreational Mathematics, reprinted with permission

5 of 5 people found the following review helpful:

5.0 out of 5 stars Will be using this book for a looong time., August 2, 2010

By 

KHuereca - See all my reviews

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This review is from: How to Think Like a Mathematician: A Companion to Undergraduate Mathematics (Paperback)

Loved the book, It's definately going to be a go-to book for my other advanced classes. It is wrote in an extraordinarily user-friendly way and is actually somewhat fun to read.