Rational Trigonometry

2005-09-17 21:05

Today I discovered a math related post over at Slashdot that definitely sparked my interest. N.J. Wildberger a professor of mathematics at the University of New South Wales in Sydney Australia recently published a book called Divine Proportions: Rational Trigonometry to Universal Geometry:

This text introduces a new and simplified approach to trigonometry and a major restructuring of Euclidean geometry. It replaces cos, sin, tan and all those other transcendental trig functions with rational functions and elementary arithmetic. It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms’.

The new approach is based on the dual concepts of spread and quadrance rather than distance and angles. The first chapter is available online in pdf form and gives a pretty good break down of how the approach builds on those two concepts. Despite the nearly 10 years since my last math class I was able to grasp the material on a basic level pretty quickly and it looks promising. If the book wasn’t priced like a typcial text book I would seriously consider purchasing it when it becomes available.

I’m really curious to see how the mathematics community around the world will react to these ideas. As Dr. Wildberger puts it in his introduction:

In the Roman period, which saw the beginnings of classical trigonometry, arithmetic used Roman numerals (such as the page numbers in this introduction). Cities were built, students were taught, and an empire was administered, with an arithmetic that was cumbersome and hard to learn, at least when compared to the one we now use built from the Arabic-Hindu numerical system. Today we understand that the difficulty with arithmetic in Roman times was largely due to the awkward conceptual framework.

Much the same holds, in my opinion, for classical trigonometry‚ it has been such a hurdle to generations of students not because of the essential intractability of the subject, but rather because the basic notions used to study it for the last two thousand years are not the right ones.

By the time you have finished this book, you should be comfortable with the fact that geometry is a quadratic subject, requiring quadratic mathematics. Using more or less linear ideas, such as distance and angle, may be initially appealing but is ultimately inappropriate. With the natural approach of rational trigonometry, many more people should be able to appreciate the rich patterns of geometry and perhaps even experience the joy of mathematical discovery.

Definitely non traditional stuff, but it looks pretty promising to me and definitely something I want to keep an eye on.