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N J Wildberger

Divine Proportions: Rational Trigonometry to Universal Geometry Hardcover – September 20, 2005

by N J Wildberger (Author)

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This revolutionary book establishes new foundations for trigonometry and Euclidean geometry. It shows how to replace transcendental trig functions with high school arithmetic and algebra to dramatically simplify the subject, increase accuracy in practical problems, and allow metrical geometry to be systematically developed over a general field. This new theory brings together geometry, algebra and number theory and sets out new directions for algebraic geometry, combinatorics, special functions and computer graphics. The treatment is careful and precise, with over one hundred theorems and 170 diagrams, and is meant for a mathematically mature audience. Gifted high school students will find most of the material accessible, although a few chapters require calculus. Applications include surveying and engineering problems, Platonic solids, spherical and cylindrical coordinate systems, and selected physics problems, such as projectile motion and Snell's law. Examples over finite fields are also included.

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About the Author

N J Wildberger (PhD Yale 1984) is currently Associate Professor at UNSW in Sydney Australia, and has taught at Stanford University and the University of Toronto. His main research interests are Lie theory, representation theory and hypergroups. He has also done work in number theory, combinatorics and mathematical physics. He is married, and lives in Sydney with his wife Kim and daughter Ali. This is his first book.

Product details

Publisher : Wild Egg Books (September 20, 2005)
Hardcover : 320 pages
ISBN-10 : 097574920X
ISBN-13 : 978-0975749203
Item Weight : 1.9 pounds

Best Sellers Rank: #440,218 in Books (See Top 100 in Books)
- #105 in Trigonometry (Books)

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Bradley A. Rigg

Fascinating alternative to trigonometry and projective geometry.

Reviewed in the United States on November 23, 2019

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I enjoyed this book as I was trying to find alternatives to using trig functions in higher dimension division algebras. Using quadrances (square of a side of a triangle) and spreads allows one to solve many geometric problems by paper and pencil as well as right clear algorithms.
The author took to long getting to concrete examples and I almost put the book down but I kept on plugging away through his proofs until he finally got to practical examples.
This author puts out great you-tube videos on on a variety of math subjects that are extremely clear and succinct. I would start with his videos on Ration Trigonometry first then tackle this book.

3 people found this helpful

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Paul Fitzsimons

Revolutionary system of distances and angles

Reviewed in the United States on April 6, 2018

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This is a comprehensive review of a brand new type of trigonometry. I never had too much problem with classic trigonometry, sin, cos, and tan, but this system is just as accurate and complete, using quadrance and spread. I have not read every page yet, and still I wonder if I can fully transition to this system without too much shock to my classical training. The main concept starts with the fact that when you divide, say, a^2/b^2 you get either a terminating decimal number or a long or short repeating decimal. I do not know why and don't know if there exists a proof as to why this is. Wildberger has a video on how there are decimal numbers which go on forever, hence they are always only approximations. The video showed that these numbers are so inaccessible as to be nearly lost to infinities and not "rational" at all. Not so with Rational Trig as when you divide two squares you get an exact value. This is nearly magical and quite a robust way of doing angles and distances. I think this is a revolutionary system that would be a boon to computer science as well as engineering and general science.

2 people found this helpful

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Ivan Alim

A new way of learning trigonometry

Reviewed in the United States on June 12, 2015

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This book is really thought-provoking. It really explains how to develop the theory of trigonometry without any transcendental function like sine and cosine. At first, I thought that it is impossible to learn trigonometry without using sine and cosine as the fundamental concepts. However, it is possible to do that and it turns out to be a big improvement. I found no flaws in the mathematics, although I did not read every argument in depth. One thing that I want to suggest is that it is better if the author uses the linear algebra approach such as the dot or inner product to derive the main laws of Rational Trigonometry, as the derivation becomes dramatically simpler, instead of using algebraic manipulation only. It also helps the reader to appreciate the proofs of the theorems without resorting to the computer, especially in the latter part of this book. I also want to suggest that the author provide the answer to each exercise in his book so that the reader knows whether he or she has fully understood the material or not.

One thing that I like about this book is that we only need to have a strong understanding in arithmetic and algebra to appreciate this book. I also like the way the author gives precise definitions in each concept, which makes it relatively straightforward to the reader. However, I am still unsure about the claim that this Rational Trigonometry should ultimately replace Classical Trigonometry. I do hope that this Rational Trigonometry will ultimately replace the usual trigonometry that we learn in school, since the content is quite elementary, even though it requires mathematical maturity to fully understand and appreciate this book.

Nevertheless, this book provides many interesting and thought-provoking discussions to the reader. I hope that this book will get serious attention from mathematicians and educators to consider the implication of teaching and incorporating Rational Trigonometry to mathematics education.

12 people found this helpful

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Rex Henderson

Great concepts

Reviewed in the United States on September 21, 2018

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I've no natural inclination toward math. But I've been struggling with an idea for 20 years that involves a lot of geometry and the standard approaches were presenting results that were difficult to work with. I think this book just might be the answer I was looking for! I have to go through sections several times to grasp the concepts. I don't know if it is the way that it is written or because I'm just a lay person with no substantial math background so I'm missing some of the context that a math grad would automatically pick up on. Otherwise I'd give it 5 stars.

2 people found this helpful

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Yehoshua Valencia

Classic book for the future

Reviewed in the United States on September 1, 2016

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I hope I can find more books like this one. This is truly a classic for the future. People need to start realizing that mathematics taught at schools and universities are very old and they are not in alignment with the current state of the art in science and technology. This book emphasizes the importance of rational trigonometry which can be applied without approximations or normalizations to processes in our physical world.

6 people found this helpful

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Acuomai

Why worry if 16.999986634255 is closer to the right answer than the square root of 289?

Reviewed in the United States on August 25, 2014

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Professor Wildberger, in this book and in his numerous YouTube videos, has laid out a fascinating way to understand trigonometry and geometry. Since he makes a special point of avoiding transcendental functions, the formulas he shows -- all traceable back through the centuries of mathematical thought -- will be of use to programmers who want to avoid trig function lookup tables and irrational numbers. His linear algebra approach describes lines, polygons and curves in the plane as well as on surfaces of positive and negative curvature, and extends to arbitrary number fields.

7 people found this helpful

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Byron

A new view of trigonometry

Reviewed in the United States on November 7, 2019

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This is a very interesting perspective on trigonometry. At first I thought it was a textbook for students with problem sets, but it is a philosophy text

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Amazon Customer

A serious study

Reviewed in the United States on February 18, 2018

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This book hits the subject matter straight up. It is very comprehensive for tackling such a divers mathematical spectrum.

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木村弘一（こういち）

離散解析の基礎のバイブル。

Reviewed in Japan on December 9, 2020

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　星４つにした理由は以下の２つ。：
１．アメリカ人のレビュアーも指摘して居られる通り、また、実際に、 Charles G. Gunn の 'Rational trigonometry via projective geometric algebra' という論文で為されている通り、行列と同一視できるマルチ・ベクトルで理論構成する方が、遥かに簡素な理論になる。（なお、その行列で、小林吹代先生の御本でも展開してある離散三角法が、また、それ故に、中井晶也先生が解析として正当化できる唯一のものであることを証明なさった離散解析が、展開できる。それを、更に、和田博先生の近刊の御新著で展開してあるという、行列を多元数と解釈する理論に基づいて、多元数の計算で以て展開できる。）
２．本書では、正当にも、実無限小の概念を批判しただけでなく、不当にも、実無限大まで否認してしまっている。が、飯高茂先生の「数学的帰納法の原理の定理（公理ではない！）」から分かる通り、実無限大は実在する。
　以上の２つの論点について、小弟の他のレビュー（書評）どもに於ける詳論を御高覧ください。

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