What is Rational Trigonometry (RT)?
Rational trigonometry is a novel, completely algebraic alternative to classical geometry set out in the 2005 book Divine Proportions: Rational Trigonometry to Universal Geometry by N J Wildberger. The theory does not use angles, and avoids transcendental circular functions, such as sin x, or cos x or tan x, or their inverse functions. It relies on basic algebra, meaning arithmetic with rational numbers and polynomials, to recast planar metrical geometry in a more elementary, elegant, powerful and general way.
Rational trigonometry relies on quadrance and spread as the two main metrical notions, replacting distance and angle. Quadrance is the square or distance, or rather distance is the "square root" of quadrance. Spread on the other hand is a number between 0 and 1 that measure how far apart two lines are. Rational trig on Wikipedia / Rational trig on Cut-the-Knot / Rene Grothmann, creator of C.a.R. on Rational trig
/ Rational trig via projective Geometric Algebra by C. Gunn
Why hasn't rational trigonometry been discovered before?
This is best understood by a brief look at the history of trigonometry, which has its origins in astronomy. The Alexandrian Greeks, Hindus, Persians, Arabs, Chinese and indeed most major civilizations were fascinated by the night sky, and in particular the motion of the planets. Trigonometry was developed to study the celestial sphere--an imaginary sphere containing the stars with us at the center, on which the planets move. The basic notion here is the spherical (angle) between two stars, defined as the distance between the stars on the surface of this distant sphere divided by the radius of the sphere.
For 1500 of the roughly 2000 years of trig, spherical trigonometry was the main interest, and the planar case was hardly of significance. Now of course the situation is totally reversed, but tradition still keeps in place that relic from astronomy--the angle. So by the time people began investigating triangles seriously, they simply took the spherical notions and modified them to the planar case. Thus circles are nowadays important in discussing triangles, and the subject is more complicated than it needs to be.
Does rational trigonometry define an angle in a new way?
No. Rational trigonometry replaces the notion of an angle with a related but simpler notion--that of the spread. A big difference that you must get used to is that the spread is truly defined between lines, not rays. This turns out to be more natural algebraically, and allows the theory to extend to more general fields.
Isn't angle a natural concept?
Actually it is not. The ancient Greeks, who were keen and careful geometers, struggled mightily with the concept. Euclid does not use angles in his classic treatment (right angles and a few special related cases excepted).
Here is a simple test to see for yourself how complicated the notion of angle REALLY IS. Take any three random points in the plane (say A=[1,2], B=[5,3] and C=[4,0]). Now without a calculator or tables, see if you can determine the angle <ABC to say 4 decimal places in degrees. My bet: even if you are a professional mathematician, this will take you at least several hours, and possibly several days!
The next time someone tries to convince you that the idea of an angle is completely elementary, give them this little challenge, and watch them wriggle and squirm!
What is the spread between two lines?
The spread between two lines turns out to be the square of the sine of the usual angle between them, but that's not how you should think about it. The definition doesn't involve the notion of angle, nor the understanding of the trig function sine.
The spread between two lines is defined informally in terms of the ratio of two quadrances: the numerator is the quadrance from a point B on one of the lines to the closest point C on the other line, and the denominator is the quadrance between B and the intersection A of the two lines. All you need to define the spread is the notion of quadrance and the idea of dropping a perpendicular from B to the other line to get C.
More formally it is defined in terms of an algebraic expression involving the equations of the two lines. A theorem asserts the equivalence of these two approaches.
How do you replace adding adjacent angles?
An angle of 35 degrees beside an angle of 2 degrees gives an angle of 37 degrees. Or perhaps 33 degrees--it depends on the relative positions of the rays. So mathematically there are still generally two solutions, even if you only see one in any given picture. That suggests that there is really a quadratic equation around. With spreads, this quadratic equation is one of the five main laws (the Triple spread formula) and allows you to calculate the spread s(l,n) if you know s(l,m) and s(m,n) for lines l,m and n.
However if you have a problem involving lots of 'angle chasing' then you are better off with angles. The vast majority of trig problems that arise in practise are not of this kind.
Why is quadrance more natural than distance?
Distance is important from a practical point of view. A carpenter will still want to know how long his next beam should be cut, and a bush walker (hiker) will still want to know the distance from one place on the map to another.
The point is rather that the mathematical theory becomes simpler when you work with quadrance rather than distance. Pythagoras' theorem uses squares of distance, as do many, many theorems of Euclidean geometry. Given two points in the plane, finding the quadrance between them is easy, while the distance involves a square root.
The recommended approach is thus:
- convert the problem to quadrances and spreads
- solve the problem with rational trigonometry
- convert back to distances, and if necessary angles.
What about rotations and circular motion?
It turns out that rotations can be handled perfectly with rational trigonometry (the book Divine Proportions shows how). But uniform circular motion cannot be handled with rational trigonometry--the trig functions are necessary for it. The point is that trigonometry (the study of the measurement of triangles) and circular/periodic motion are two separate subjects. It is not appropriate to have a mathematical technology that covers both topics at once, as the current approach does. The reason is that the theory becomes unnecessarily complicated to incorporate two overlapping but disjoint subjects.
Here is an analogy: you might have a lawn mower and a barbecue in your back yard. How about a machine that both mowed your lawn and cooked your steaks? Wouldn't that be a great idea? Well, it probably would make both jobs more onerous, and cost a lot more than two separate machines specially designed for the separate tasks.
The knowledge of the trig functions required to understand circular motion is in fact rather minimal. Most of the identities students learn are unnecessary. In fact (horrible secret) once you do a bit more advanced mathematics you realize that cos and sin are just shadows of the complex exponential function. Use of this more natural and elegant function at one stroke simplifies almost all trig identities and integrals.
Won't students have to learn the old trig to pass exams?
For a while, sure. But slowly people will start to examine the effectiveness of the new theory, engineers may start using it in graphics packages and robotics, surveyors may change their instrumentation software to accomodate quadrance and spreads, NASA might start thinking about its uses for real time calculations, and lo and behold! Gradually it will trickle down from academics to teachers to students.
It won't happen overnight, and until then students will still have to grin and bear the current curriculum. At first perhaps rational trigonometry will appear as an enrichment subject for gifted students, then possibly it will be tried on very weak students who can't hope to understand the usual trig. Then slowly more and more students and parents will start demanding change, and after a while it will be inevitable.
The simplicity and power of rational trigonometry will sink in. You just can't argue with a theory that is at least three times as easy to learn! And more accurate and powerful to boot! My guess is that in twenty five years classical trigonometry as a way of studying triangles will have gone the way of the horse and buggy.
Is Divine Proportions a high school text?
No. It is a thorough, careful laying out of this new theory for academics to be able to judge its merits. The treatment is concise, requires mathematical maturity, and builds up Euclidean geometry over a general field. While a very talented high school student will be able to read much of it and gain a lot from it, for most high school students it is rather hard.
The book attempts to present a lot of evidence in one spot to convince mathematicians and teachers to rethink their current understandings.
How did you stumble upon rational trigonometry?
It was an accidental by-product of some more arcane investigations I was making 4 years ago in relativistic geometry. Although some of the ideas had been bubbling away in the back of my mind for many years before that. It all came together very quickly when I realized that I didn't really HAVE to consider angle and distance as the fundamental concepts.
The real surprise to me was that this somewhat simple minded theory (ie rational trigonometry) ended up not only solving all the standard trigonometric problems I investigated, but often did so more elegantly, accurately and quickly. That is a very important point.