New trigonometry is a sign of the time

physorg.com ^ | September 16, 2005

[cloud8]

Mathematics students have cause to celebrate. A University of New South Wales academic, Dr Norman Wildberger, has rewritten the arcane rules of trigonometry and eliminated sines, cosines and tangents from the trigonometric toolkit.

What's more, his simple new framework means calculations can be done without trigonometric tables or calculators, yet often with greater accuracy.

Established by the ancient Greeks and Romans, trigonometry is used in surveying, navigation, engineering, construction and the sciences to calculate the relationships between the sides and vertices of triangles.

"Generations of students have struggled with classical trigonometry because the framework is wrong," says Wildberger, whose book is titled Divine Proportions: Rational Trigonometry to Universal Geometry (Wild Egg books).

Dr Wildberger has replaced traditional ideas of angles and distance with new concepts called "spread" and "quadrance".

These new concepts mean that trigonometric problems can be done with algebra," says Wildberger, an associate professor of mathematics at UNSW.

"Rational trigonometry replaces sines, cosines, tangents and a host of other trigonometric functions with elementary arithmetic."

"For the past two thousand years we have relied on the false assumptions that distance is the best way to measure the separation of two points, and that angle is the best way to measure the separation of two lines.

"So teachers have resigned themselves to teaching students about circles and pi and complicated trigonometric functions that relate circular arc lengths to x and y projections – all in order to analyse triangles. No wonder students are left scratching their heads," he says.

"But with no alternative to the classical framework, each year millions of students memorise the formulas, pass or fail the tests, and then promptly forget the unpleasant experience.

"And we mathematicians wonder why so many people view our beautiful subject with distaste bordering on hostility.

"Now there is a better way. Once you learn the five main rules of rational trigonometry and how to simply apply them, you realise that classical trigonometry represents a misunderstanding of geometry."

Wild Egg books: http://wildegg.com/ Divine Proportions: web.maths.unsw.edu.au/~norman/book.htm
Source: University of New South Wales

[Paladin2]

Somewhere around here I have a DOS?/Win3.1? graphics file/program that lets you have a working slide rule right on your screen.

Amaze your friends. There was a Scientific American article (Amateur Scientist column IIRC) a few years back that went through the various, still useful, things a slide rule can do.

[Excuse_My_Bellicosity]

Okay, I read Chapter 1. I'll take an angle and a distance over quadrance and spread any time, and I swear I am trying to read this all with an open mind. The author claims that he's simplifying trigonometry. So far, I'm not seeing the simplification here.

My problems with what I'm seeing so far are:

• Regarding the distance between 2 points, he states "quadrance is the more fundamental quantity, since it does not involve the square root function." That is not a true statement. Try explaining both concepts to a 5 year old and then tell me which is simpler and more fundamental. When riding a bike down the old dirt road, determining gas mileage, or doing anything else that involves a one-dimensional distance, I care about the distance a hell of a lot more than the "quadrance". When Dr. Wildberger takes away the square root symbol from a Cartesian distance calculation and calls it something else ("quadrance"), that doesn't make the problem simpler, it only means that he has to incorporate the square root in subsequent calculations (unless it simplifies out, which doesn't happen in Dr. Wilberger's calculations; at the end of his example, he performs D = SquareRoot(Q)). I don't view this as simplification.
• The equation: s(l₁,l₂) = Q(B,C) / Q(A,B) = (a₁b₂-a₂b₁)² / (a₁²+b₁²)(a₂²+b₂²) is a simpler concept than an angle????
• "The problem is that defining an angle correctly requires calculus." That's a true statement, but really defining Dr. Wildberger's coordinate systems and quantities also requires calculus, a fact that he glosses over. When I took AP calculus in high school college calculus courses, I had to use calculus to really prove that the circumference of a circle is C = pi * D = 2 * pi * R and various other fundamental mathematical principles but that does not mean that I had to do so before learning basic geometry, trigonometry, and analytical geometry. The good doctor is purposely introducing a moot point in an attempt to describe why his system is supposedly better.
• "Without tables, a calculator or calculus, a student has difficulty in answering this question, because the usual definition of an angle (page 11) is not precise enough to show how to calculate it. But how can one claim understanding of a mathematical concept without being able to compute it in simple situations?" Uh huh. But a student is supposed to able to compute s(l₁,l₂) = Q(B,C) / Q(A,B) = (a₁b₂-a₂b₁)² / (a₁²+b₁²)(a₂²+b₂²) without tables or a calculator?
• "If the notion of an angle cannot be made completely clear from the beginning, it cannot be fundamental." That is not a true statement. Mathematical proof can be difficult but an angle is easily visualized. It's not a difficult concept. And the author does not show that his concept of "spread" is any more simple or fundamental than the angle.
• Section 1.5 shows a comparison of solutions for a problem using conventional trigonometry and Dr. Wildberger's method. Classically, I can solve this using the Law of Sines such that d = (5 * sin 41.4096 deg) / sin 93.5904 deg = 3. 3137. Using the other approach, I have to find the root of the equation: (7/16 + 1/2 + r)² = 2(49/256 + 1/4 +r²) + 4 * 7/16 * 1/2 * r. Call me crazy, I'd rather punch up a couple of sines than find that root!
• "Clearly the solution using rational trigonometry is more accurate." You might want to expand on that thought, doc. You haven't shown me that it is. In a mathematical expression, writing the solution with the square root of 7 intact is more accurate than the decimal expression of the anwswer, but I can make the classical answer the same accuracy as Dr. Wilberger's by not simplifying for the sine or cosine. And it should be noted that the numerical methods that a calculator uses to derive a decimal for the square root of 7 is no more accurate than it is for sin, cos, or tan of angle XX.XXXX....
• "Defining arc lengths of curves other than line segments is quite sophisticated, even for arcs of circles. So it does not make mathematical sense to treat circles on a par with lines, or to attempt to use circles to define the basic measurement between lines." A circular arc length, L = (angle in degrees) / 360 * pi * diameter, is a difficult concept?

I'd like to critique the rest of his book but I'm not about to fork over the $80 to do it.

[maro]

I hope you are joking. Pi is an irrational number that cannot be expressed as a finite decimal.

[kitkat]

***Cast out that apostrophe first.***

You're mistaken. I did not write the sentence you referred to. Pardon me, "to which you referred."

Perhaps you would be interested in the following site:

http://www.apostrophe.fsnet.co.uk/

[ScreamingFist]

Amaze your friends. There was a Scientific American article (Amateur Scientist column IIRC) a few years back that went through the various, still useful, things a slide rule can do.

My brother, the big brain of the family, whipped out a slide rule in a Starbucks recently. He was considered a god by the new purveyors of "math by calculator". He laughed all the way home......

[Excuse_My_Bellicosity]

Since ordinary trigonometry is correct, he's not by any means replacing something incorrect with something correct. Rather, he's attempting to reduce the computational difficulty of some aspects of ordinary trigonometry (although, in the end, his answers do still often require the extraction of square roots, which aren't so easy to do by hand).

True, but I think he's failing to make his case. Maybe it becomes more apparent if you read his whole book. I'm sure not impressed with his comparison of classical trig and his rational method is section 1.5. Classical trig solves the problem such that d = (5 * sin 41.4096 deg) / sin 93.5904 deg = 3. 3137. The other method requires applying the Quadratic equation to find r in the equation: (7/16 + 1/2 + r)2 = 2(49/256 + 1/4 +r2) + 4 * 7/16 * 1/2 * r.

He's not showing me that his method is easier or more straightforward!

[Paladin2]

I still want to find one of those huge slide rules that used to hang at the blackboard in some math/science classrooms.

[ScreamingFist]

I still want to find one of those huge slide rules that used to hang at the blackboard in some math/science classrooms

Pickett Model N803, google my FRiend and you still can get one "relatively" cheap.......

[dr_who_2]

Same here.

[metmom]

I think I like trig better.

[MarkL]

New math was great (for those continuing in math and science).

Ever try casting out 9's?

Well, once in prehistoric times, I was taking a computer science class in assembly language, and we worked almost exlusively in octal... Before I realized it, I had began balancing my checkbook in octal, and it caused me some problems... The funny thing was that there was about 2 month period when none of the checks I wrote had 8s or 9s, so it just "happened," and I didn't notice!

Mark

[Professional Engineer]

Crap, now I'm gonna have to replace my sliderule.

[InterceptPoint]

My problems with what I'm seeing so far are ...

Well done Excuse_My_Bellicosity.

I'm going to keep my $80 also. Maybe I'll use it to buy 16 credit card sized pocket calculators that can do SIN, COS and TAN.

[Neanderthal]

"Trig methods will still appeal more to the spatially-oriented than algebraic solutions."

I agree. Trig is beautiful as it is.

[Paladin2]

Well the "new" math used base 7, so it was pretty much useless except for teaching. Using octal to demonstrate number systems would have been more useful to many (hex might have been too tough for many kids to wrap their minds around). Unfortunately around that time (60s), I had to live a deprived life as my parents wouldn't buy me an IBM 360 for my bedroom.

[R. Scott]

Darn! I kind of liked Geometry and Trig.

[RightWhale]

$200 give or take. I would rather have a '55 Chevy Bel-Air in my living room.

[spokeshave]

""I wonder when this will make it into my Calculus book..."

Forget it....how you expect to understand the basis of calculus (rates of change - tangents) without trig.

And as for hyperbolic functions....eh what?

I guess we can depend on immigrants for all our higher maths.

And don't epect to go back to the moon, much less mars without classical trigonometry.

[Paladin2]

Well for certain geometry/trig situations (Orienteering, radar target tracking) polar coordinates are the way to go (heading & range). There's no way I'm going to be doing squaring and unsquaring as I'm running through the woods looking for a check point. It's hard enough to run in a straight line while doing dead reckoning on distance.

[cloud8]

> Cast out that apostrophe first.

LOL
No apostrophes in plurals or possessive pronouns!!!
When I replied to Paladin2's post, however, 9s and 6s didn't look well on screen. I resorted to sixes.