Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. 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. The proofs in this collection are arranged by topic into five chapters: Geometry and algebra; Trigonometry, calculus and analytic geometry; Inequalities; Integer sums; and Sequences and series. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.
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Proofs without words are generally pictures or diagrams that help the reader see why a particular mathematical statement may be true, and how one could begin to go about proving it. Teachers will find that many of the proofs in this collection are well suited for classroom discussion and for helping students to think visually in mathematics.
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Most Helpful Customer Reviews
2 of 2 people found the following review helpful
Format:Paperback
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Format:Paperback
This delightful book presents algebraic statements of various elementary results in algebra together with geometric constructions that make those results patently obvious. I'd recommend it to any student of mathematics.
On occasion students may encounter mathematicians who favour exclusively axiomatic methods and who will disparage the kinds of diagrammatic proof that this book presents. Anyone coming up against such miserable scoffers should Google for: Arnold & teaching & mathematics, and find out how Russia's most distingushed mathematician puts the idiots in their place.
Divorcing mathematics from geometry is a dangerous mental affliction that should be stamped out. More power to this book for upholding sanity in the teaching of mathematics. We need more like it.
On occasion students may encounter mathematicians who favour exclusively axiomatic methods and who will disparage the kinds of diagrammatic proof that this book presents. Anyone coming up against such miserable scoffers should Google for: Arnold & teaching & mathematics, and find out how Russia's most distingushed mathematician puts the idiots in their place.
Divorcing mathematics from geometry is a dangerous mental affliction that should be stamped out. More power to this book for upholding sanity in the teaching of mathematics. We need more like it.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
3.9 out of 5 stars
8 reviews
28 of 30 people found the following review helpful
4.0 out of 5 stars
Engaging exercises to train your intuition
12 Sep 2000
By David Collins
- Published on Amazon.com
Format:Paperback
Famous mathematicians have often emphasized the role of visual intuition; e.g., Hilbert: "Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"? Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigor a difficult theorem on the continuity of functions or the existence of points of condensation?" (from his famous address at the 1900 International Congress). This book is a collection of well over 100 one-page proofs, collected from various sources. The topics range from number theory to calculus, and most of them require no advanced mathematics. Typically there is a statement of a result, with a labelled diagram showing how it is "proved"; in some cases there are a few auxiliary equations along with the picture. These are not simple, often requiring quite a bit of thought before the "Aha!" moment. Working through them is a valuable exercise for the student of mathematics--having seen, e.g., six different visual proofs of the Pythagorean theorem, one comes to really *understand* the result, not just "follow the logic". I have not encountered any better way than this book to "see" how mathematical truth is discovered and proved. It can be valuable as a supplement to courses through precalculus and elementary calculus. Perhaps one of its best uses is to inspire teachers to present results in a more lively way then "definition-theorem-proof" or "just memorize it".
10 of 10 people found the following review helpful
5.0 out of 5 stars
A picture is worth...
9 Sep 2007
By S. Lederman
- Published on Amazon.com
Format:Paperback
How many of you remember doing geometry proofs in High School? How many of you enjoyed writing them? I don't know about you but I've always preferred pictures to words when it comes to understanding how something works.
This is a wonderful book that provides visual insights into how one might go about proving mathematical theorems. The Pythagorean Theorem has always been a mystery to me. How are the squares of the sides of a right triangle related to its hypotenuse? "Proof Without Words" has five clever illustrations that guide readers in writing their own proofs.
If you ever doubted that algebra and geometry were related, the diagrams demonstrating how to compute sums of series will produce aha! experiences.
Writing proofs when one is guided by visual cues is a much more fulfilling endeavor than stringing together dry facts from memory. This book delivers much fulfillment in exploring theorems in geometry, algebra, trigonometry, sequences, and other aspects of Math.
This is a wonderful book that provides visual insights into how one might go about proving mathematical theorems. The Pythagorean Theorem has always been a mystery to me. How are the squares of the sides of a right triangle related to its hypotenuse? "Proof Without Words" has five clever illustrations that guide readers in writing their own proofs.
If you ever doubted that algebra and geometry were related, the diagrams demonstrating how to compute sums of series will produce aha! experiences.
Writing proofs when one is guided by visual cues is a much more fulfilling endeavor than stringing together dry facts from memory. This book delivers much fulfillment in exploring theorems in geometry, algebra, trigonometry, sequences, and other aspects of Math.
15 of 19 people found the following review helpful
4.0 out of 5 stars
Visual justification has a role in mathematics
20 Nov 2000
By Charles Ashbacher
- Published on Amazon.com
Format:Paperback
The first mathematical proofs were no doubt primarily diagrammatic in structure, and we all should appreciate the role they have played in the development of mathematics. Unfortunately, the figure is now somewhat maligned as a tool in mathematics. A symbol used in a proof is a representative of an abstract concept, and if a diagram is also considered in that way, then it should be just as acceptable. The proofs in this book are not truly without words, as most of the time there is a formula as well. However, they are easy to understand and cannot fail to be appreciated.
Proof by diagram does have a place in the mathematical educational experience as well. After all, the point of a proof is to convince us of the validity and also explain why the result must hold. Students who struggle their way through abstract formulas and symbols can be exposed to proofs like this and learn there is a place for visual thinking in mathematics.
Mathematics teachers face a difficult task and should use every tool that is available to present the wonder and greatness of mathematics as a form of human endeavor. Proofs without words will not work everywhere, but when they do, it can be the difference that makes the light bulb of understanding burn bright. This book should be read by all teachers of mathematics.
Proof by diagram does have a place in the mathematical educational experience as well. After all, the point of a proof is to convince us of the validity and also explain why the result must hold. Students who struggle their way through abstract formulas and symbols can be exposed to proofs like this and learn there is a place for visual thinking in mathematics.
Mathematics teachers face a difficult task and should use every tool that is available to present the wonder and greatness of mathematics as a form of human endeavor. Proofs without words will not work everywhere, but when they do, it can be the difference that makes the light bulb of understanding burn bright. This book should be read by all teachers of mathematics.
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3.9 out of 5 stars
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