Proofs without Words: Exercises in Visual Thinking

by Roger B. Nelsen

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.

Genres: Mathematics, Science, Nonfiction, Academic, Stem, Technology

157 pages, Paperback

First published January 1, 1993

Reviews

[Nattawut Phetmak · Rating 5 out of 5]

เปิดโลกคณิตศาสตร์มาก กลายเป็นเรื่องสนุก+มีศิลปะไปเลย

ใครไม่กินเส้นกับคณิตศาตร์ลองพลิกดู อาจจะเปลี่ยนความคิดไปเป็นว่า คณิตศาสตร์นอกจากจะไม่ได้ยากเกินความเข้าใจ มันยังสวยงามหยดย้อยดั่งน้ำค้างบนยอดหญ้าอีกด้วย

[Natik · Rating 4 out of 5]

This book is full of geometric derivations/proofs of not strictly geometric theorems.

Unfortunately, it looks like this book is trying to be a coffee table curiosity rather than seriously help people understand something new. You can't even really "read" it (there are truly zero words in there) without already being very familiar with each theorem and how the proof is done symbolically.

However, you can make it into a really good resource by trying to map each part of the figure transformations/manipulations onto the relevant bits of math notation and annotating each step. Yes, you also have to already understand the symbolic proof (and have one printed out-this book won't give it to you), but this whole "translating" operation gives you some great mental skills.

If you're an extremely visual thinker who wants to learn to "read math", this is one way I've found really helps!

[Robert · Rating 3 out of 5]

Collection of Mathematical visualizations - illustrated examples of math concepts. Hit or miss in terms of usefullness - while all are accurate, some resonate better than others, and several of the topics are, to put it mildly, obscure.

[Justin · Rating 5 out of 5]

too many pictures

[Sadip · Rating 4 out of 5]

Goood book

[Đỗ Minh Triết · Rating 5 out of 5]

This is one of the best part of Mathematic.
The proof without words, just figure and all you have to do is think-think-and think.

[Pang Sukanya · Rating 5 out of 5]

ดีมาก เป็นหนังสือที่เปิดโลกเกี่ยวกับการพิสูจน์คณิตศาสตร์ที่ใช้รูปภาพในการพิสูจน์ ทำให้เข้าใจง่าย

[Petr · Rating 2 out of 5]

Eng/Cz

If you take this as a mathematical puzzle-book, then you might really enjoy it. However, the book cover promises a book that could help teachers explain mathematics or show new angles how to understand mathematical concepts. The book, in my view, lacks some kind of explaining text in order to be able to stand by itself. Sadly, in one puzzle there actually is a short explanation and suddenly the graphical proof is a lot more helpful!

Nenechte se zmást obalem knihy, který slibuje možnost pro učitele využít materiál pro výklad matematiky nebo pro vizualizaci různých matematických problémů. Přitom knize chybí, dle mého názoru, nějaký výklad či alespoň malé vysvětlení. Smutné je, že v jednom případě vskutku je grafický důkaz doplněn malým výkladem a hned lze vidět o kolik je poučnější, když chápeme motivace a matematické koncepty v pozadí. Jinak čtenář hledá v obrázku důkaz a vskutku to zkoumá jako nějaký puzzle.

[Tom Schulte · Rating 3 out of 5]

This slim collection of varied visual “proofs” (of course the term can be argued to be liberally applied here) is entertaining and enlightening. I personally find such representations engaging and stimulating aids to that “a-ha” moment when symbolic argument serves not to clarify. Since such pictures can be found sprinkled in the pages of many mathematics periodicals and even the occasional textbook, others obviously feel the same way. A collection of a gross of them in a single volume is really a delight to peruse.

[R.W. Clark · Rating 4 out of 5]

Inventive and compelling demonstration that words can cloud, but images bring meaning.