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Rational Trigonometry and a new Foundation For Math

Posted by pwl on August 11, 2009

Two excellent video series by Norman J Wildberger. Very well done and designed to be easy to comprehend. While they start out very simple and basic they get quite deep yet remain as simple as possible. (The bonus series even gets into Relativity, very cool). Wildberger is the math teacher I wish I had all through high school and since, and now through his videos he can be our math teacher! Gotta love online education.

The first series is on the amazing new (and very old) way of doing Rational Trigonometry without using sin and cosine of Classical Trigonometry. It’s Fantastic.

“The new form of trigonometry developed here is called rational trigonometry, to distinguish it from classical trigonometry, the latter involving [cosine, sin and their fellow] functions and the many trigonometric relations currently taught to students. An essential point of rational trigonometry is that quadrance and spread, not distance and angle, are the right concepts for metrical geometry (i.e. a geometry in which measurement is involved).

Quadrance and spread are quadratic quantities, while distance and angle are almost, but not quite, linear ones. The quadratic view is the more general and powerful one. At some level, this is known by many mathematicians. When this insight is put into practice, as it is here, a new foundation for mathematics and mathematics education arises which simplifies Euclidean and non-Euclidean geometries, changes our understanding of algebraic geometry, and often simplifies difficult practical problems.

Rational trigonometry deals with many practical problems in an easier and more natural way than classical trigonometry, and often ends up with answers that are demonstrably more accurate. In fact rational trigonometry is so elementary that almost all calculations may be done by hand. Tables or calculators are not necessary, although the latter certainly speed up computations. It is a shame that this theory was not discovered earlier, since accurate tables were for many centuries not widely available.”

Exerpt from the Introduction to “Divine Proportions: Rational Trigonometry to Universal Geometry” by Associate Professor Norman J Wildberger

Here are a few videos from the Rational Trigonometry series with the full series of 46+ episodes here. These first episodes lay the corner stone of Rational Trig, and it only gets better from there!!! Watch them all a number of times and do the examples. Amazing!

Why classical trigonometry is hard:

The problem is that defining an angle correctly [in Classical Trigonometry] requires calculus [whereas Rational Trigonometry doesn’t]. This is a point implicit in Archimedes’ derivation of the length of the circumference of a circle, using an infinite sequence of successively refined approximations with regular polygons. It is also supported by the fact that The Elements [Euclid] does not try to measure angles, with the exception of right angles and some related special cases. Further evidence can be found in the universal reluctance of traditional texts to spell out a clear definition of this supposedly ‘basic’ concept.” – NJ Wildberger

“Quadrance measures the separation of two points. The easiest definition is that quadrance is distance squared.

… quadrance is the more fundamental quantity, since it does not involve the square root function. The relationship between the two notions is perhaps more accurately described by the statement that distance is the square root of quadrance.

In diagrams, small rectangles along the sides of a triangle indicate that quadrance, not distance, is being measured … .” – NJ Wildberger


Spread measures the separation of two lines. This turns out to be a much more subtle issue than the separation of two points.” – NJ Wildberger

The spread s between the two lines is the ratio of quadrances. The spread also works out to “square of sine of angle”.

“The spread between two lines is a dimensionless quantity, and in the rational or decimal number fields takes on values between 0 and 1, with 0 occurring when lines are parallel and 1 occurring when lines are perpendicular. Forty-five degrees becomes a spread of 1/2, while thirty and sixty degrees become respectively spreads of 1/4 and 3/4. What could be simpler than that? Another advantage with spreads is that the measurement is taken between lines, not rays. As a consequence, the two range of angles from 0◦ to 90◦ and from 90◦ to 180◦ are treated symmetrically.” – NJ Wildberger

You can find a couple freely available pdf chapters of the book “Divine Proportions: Rational Trigonometry to Universal Geometry” here. Make sure to check out Divine Proportions Overview Chapter 1 as it covers the basics with examples.

You will find some additional pdf papers by the professor here.


The second series is on the Foundations of Math. An excellent review and introduction for everyone at any age.

Here are a few videos from the Foundations of Math series with the full series of 35+ episodes here.


There is also a series of advanced Math Seminars on the topic of Hyperbolic Geometry is Projective Relativistic Geometry.

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12 Responses to “Rational Trigonometry and a new Foundation For Math”

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