Proofs without Words: Exercises in Visual Thinking (Classroom Resource Materials Series, No.1) / Edition 1

Paperback (Print)
Used and New from Other Sellers
Used and New from Other Sellers
from $26.32
Usually ships in 1-2 business days
(Save 34%)
Other sellers (Paperback)
  • All (3) from $26.32   
  • Used (3) from $26.32   
Sort by
Page 1 of 1
Showing All
Note: Marketplace items are not eligible for any coupons and promotions
Seller since 2005

Feedback rating:



New — never opened or used in original packaging.

Like New — packaging may have been opened. A "Like New" item is suitable to give as a gift.

Very Good — may have minor signs of wear on packaging but item works perfectly and has no damage.

Good — item is in good condition but packaging may have signs of shelf wear/aging or torn packaging. All specific defects should be noted in the Comments section associated with each item.

Acceptable — item is in working order but may show signs of wear such as scratches or torn packaging. All specific defects should be noted in the Comments section associated with each item.

Used — An item that has been opened and may show signs of wear. All specific defects should be noted in the Comments section associated with each item.

Refurbished — A used item that has been renewed or updated and verified to be in proper working condition. Not necessarily completed by the original manufacturer.

1993 Paperback Fair Noticeable wear, but still very usable. Clean, mark-free interior! SHIPS W/IN 24 HOURS! Processed by DHL with USPS delivery for an average of 3-5 Day ... Standard Shipping & 2-3 Day Expedited Shipping! ! FREE INSURANCE! Fast & Personal Support! Careful Packaging. No Hassle, Full Refund Return Policy! Read more Show Less

Ships from: Alton, IL

Usually ships in 1-2 business days

  • Canadian
  • International
  • Standard, 48 States
  • Standard (AK, HI)
  • Express, 48 States
  • Express (AK, HI)
Seller since 2014

Feedback rating:


Condition: Acceptable
1993 Paperback Fair

Ships from: Blue Springs, MO

Usually ships in 1-2 business days

  • Canadian
  • Standard, 48 States
  • Standard (AK, HI)
  • Express, 48 States
  • Express (AK, HI)
Seller since 2006

Feedback rating:


Condition: Good
1993-10-01 Paperback UsedGood Good Condition item. We strive to provide the best shopping experience with every item we sell. Satisfaction guaranteed! ! Ships from US. Please ... allow 1-3 weeks for delivery outside US. Read more Show Less

Ships from: Appleton, WI

Usually ships in 1-2 business days

  • Canadian
  • International
  • Standard, 48 States
  • Standard (AK, HI)
  • Express, 48 States
  • Express (AK, HI)
Page 1 of 1
Showing All
Sort by


Just what are "proofs without words?" First of all, most mathematicians would agree that they certainly are not "proofs" in the formal sense. Indeed, the question does not have a simple answer. But, as you will see in this book, proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement is true, and also to see how one could begin to go about proving it true. While in some proofs without words an equation or two may appear to help guide that process, the emphasis is clearly on providing visual clues to stimulate mathematical thought. Proofs without words bear witness to the observation that often in the English language to see means to understand, and in "to see the point of an argument."

Proofs without words have a long history. In this collection you will find modern rendition of proofs without words from ancient China, classical Greece, twelth-century India- even one based on a published proof by a former President of the United States! However, most of the proofs are relatively more recent creations, and many are taken from the pages of MAA journals.

The proofs in this collection are arranged by topic into six chapters: Geometry and Algebra, Trigonometry, Calculus and Analytic Geometry; Inequalities; Integer Sums; Sequences and Series, and Miscellaneous. Teachers will find that many of the proofs without words in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.

Read More Show Less

Editorial Reviews

"By acquiring a knack for explaining in words what they already understand, students will be better prepared to seek their own explanations.... The book offers models of the sort of visual thinking that is often woefully neglected in the classroom."
Crux Mathematicorum
"This book is a nice collection of results which certainly will enhance one's geometric sense and stimulate mathematical thought."
The Matheamtics Teacher
"Teachers who desire to give their students visual mathematical insights will find a great deal of value in this book... I found this book to be a great source for both teachers and their students at the high school level. For those teachers who are striving to satisfy the NCTM's call to reach all students, including those with different writing styles, this book will be an outstanding tool to help students who are visual learners."
The Mathematical Gazette
"Every time I have shown this book to a colleague, I have been asked how to get hold of it. It is an absolute delight.... Buy a copy today."
Although not considered proper proofs, this collection of pictures helps students to visualize why a particular theorem is true and how one might go about proving it. Dr. Nelsen has gathered his pictures from many sources, including former President James Garfield, and Nicomachus of Gerasa (A.D. 100). Theorems are taken from geometry and algebra, trigonometry, calculus and analytical geometry, inequalities, integer sums, and sequences and series. Annotation c. Book News, Inc., Portland, OR (
Read More Show Less

Product Details

  • ISBN-13: 9780883857007
  • Publisher: Mathematical Association of America
  • Publication date: 1/28/1993
  • Series: Classroom Resource Materials Series, #1
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 152
  • Age range: 15 - 17 Years
  • Product dimensions: 6.90 (w) x 9.90 (h) x 0.40 (d)

Table of Contents

Geometry & Algebra
Trigonometry, Calculus & Analytic Geometry
Integer Sums
Sequences & Series
Index of Names
Read More Show Less


See (se) v. saw, seen, seeing. - v.t.
5. to perceive (things) mentally; discern; understand: to see the point of an argument.
- The Random House Dictionary of the English Language (2nd Ed.) Unabridged

"Proofs without words" (PWWs) have become regular features in the journals published by the Mathematical Association of America- notably Mathematics Magazine and the College Mathematics Journal. PWWs began to appear in Mathematics Magazine about 1975, and, in an editor's note in the January 1976 issue of the Magazine, J. Arthur Seebach and Lynn Arthur Steen encouraged further contributions of PWWs to the Magazine. Although originally solicited for "use as end-of-article fillers," the editors went on to ask "What could be better for this purpose than a pleasing illustration that made an important mathematical point?"

 A few years earlier Martin Gardner, in his popular "Mathematical Games" column in the October 1973 issue of the Scientific American, discussed PWWs as "look-see" diagrams. Gardner points out that "in many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance." This dramatically illustrates the dictionary quote above: in English "to see" is often "to understand."

 In the same vein, the editorial policy of The College Mathematics Journal throughout most of the 1980s stated that, in addition to expository articles, "The Journal also invites other types of contributions, most notably: proofs without words, mathematical poetry, quotes,..." (their italics). But PWWs are not recent innovations- they have long history. Indeed, in this volume you will find modern renditions of proofs without words from ancient China, classical Greece, and India of the twelfth century.

 Of course, "proofs without words" are not really proofs. As Theodore Eisenberg and Tommy Dreyfus note in their paper "On the Reluctance to Visualize in Mathematics" [in Visualization in Teaching and Learning Mathematics, MAA Notes Number 19], some consider such visual arguments to be of little value, and "that there is one and only one way to communicate mathematics, and 'proofs without words' are not acceptable." But to counter the viewpoint, Eisenberg and Dreyfus go on to give us some quotes on the subject:

[Paul] Halmos, speaking of Solomon Lefshetz (editor of Annals), stated: "He saw mathematics not as logic but as pictures." Speaking of what it takes to be a mathematician, he stated: "To be a scholar of mathematics you must be born with... the ability to visualize" and most teachers try to develop this ability in their students. [George] Polya's "Draw a figure..." is classic pedagogic advice, and Einstein and Poincare's views that we should use our visual intuitions are well known.

So, if "proofs without words" are not proofs, what are they? As you will see from this collection, this question does not have a simple, concise answer. But generally, PWWs are pictures or diagrams that help the observer see why a particular statement may be true, and also to see how one might begin to go about proving it true. In some an equation or two may appear in order to guide the observer in this process. But the emphasis is clearly on providing visual clues to the observer to stimulate mathematical thought.

 I should note that this collection is not intended to be complete. It does not include all PWWs which have appeared in print, but is rather a sample representative of the genre. In addition, as readers of the Association's journals are well aware, new PWWs appear in print rather frequently, and I anticipate that this will continue. Perhaps some day a second volume of PWWs will appear!

 I hope that the readers of this collection will find enjoyment inn discovering or rediscovering some elegant visual demonstrations of certain mathematical ideas; that teachers will want to share many of them with their students; and that all will find stimulation and encouragement to try to create new "proofs without words."

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star


4 Star


3 Star


2 Star


1 Star


Your Rating:

Your Name: Create a Pen Name or

Barnes & Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation


  • - By submitting a review, you grant to Barnes & and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Terms of Use.
  • - Barnes & reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)