Rational trigonometry

Divine Proportions: Rational Trigonometry to Universal Geometry, published in 2005, is a book by Norman J. Wildberger, a critic of traditional mathematics, presenting a reformulation of trigonometry. Wildberger holds a Ph.D. in mathematics from Yale University, and taught at Stanford University from 1984 to 1986 and at the University of Toronto from 1986 to 1989; he is currently an associate professor of mathematics at the University of New South Wales, Australia.

Quadrance and spread

Instead of distance and angle, rational trigonometry uses as its fundamental units quadrance (square of distance) and spread (square of sine of angle). This choice of variables enables calculations to produce output results whose complexity matches that of the input data. For instance, in a typical trigonometry problem if rational numbers are assigned to all quadrances and spreads, then the calculated results will be rational numbers (or roots of rational numbers).

This rationality is obtained at the expense of linearity. Unlike the traditional distance and angle units, doubling or halving a quadrance or spread does not double or halve a length or a rotation. Similarly, the sum of two lengths or rotations is not the sum of their individual quadrances or spreads.

For distinction, Wildberger refers to the traditional trigonometry as classical trigonometry.It is otherwise broadly based on Cartesian analytic geometry, with a point defined as an......
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