Posted on 09/18/2005 8:41:47 AM PDT by 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
And I like tangents. Especially the seedless ones.
Damn you!
Yet Wildberger is a very good mathematician. I suspect there might be something interesting going on there if one were patient enough to read more of his book. He refers to some sophisticated mathematical ideas, such as orthogonal polynomials, that might be impacted by his ideas. But perhaps it's just as likely he's let himself be captured by an idea that seems very pretty, but is actually very thin. I know the feeling myself, as a scientist; one should always be skeptical, most of all of one's own work.
Are you wishing me well or ill? :-)
> SMSG made prefect sense and allowed me to sit in the back of the class reading Heinlein and biographies of U.S. Frontier Explorers.
LOL! That's what I used to do in English.
After college I substituted in Boston area high schools. Once I had to fill in for math. The advanced group was using the SMSG text! So I explained that I couldn't be of much help. Instead, I said, we'll go around the room, each of you giving the answer to a homework problem. If someone disagrees or has a question, you explain how you got your answer. They all participated and it was a very productive class. I haven't "taught" math since.
They are lucky they weren't going to Mars using Pi as 3.14.
You can't even use 3.14159 for those kind of calcs.
Thanks for the correction, I was scratching my head big time on that one......inductive and capacitive reactance/reluctance, who leads who(?) LOL!
H***, even I know that Pi is 3.14159. No wonder it was all screwed up. (and haven't had any math courses in 10 years.)
ELI the ICE man?
Ping!
let me know what you think about this
:)
Sorry for the unintentional mistake.
During high school and college I developed the opinion that many math teachers purposely made the subject difficult to enhance their status. They liked being experts in a very difficult subject. Like you, when I later learned the stuff on my own I realized it wasn't as difficult as I had thought.
hmmm. I think I'll read the book, I read the first chapter and it reminds me of work I've done to speed up numerics code loaded(larded) with transcedental functions (written mostly by PhD engineers ;).
I was impressed you could still remember the equation....LOL! I couldn't......
I had to use something similar to it very recently at one of the top two semiconductor manufacturers (sorry i cannot take either of their names publicly because I risk being thrown in jail).
yea, true though...it was like one of those topics coverd in "Engineering circuit analysis". But again, being that it is first order analytical expression for ckt behaviour and these days I am doing a lot of modeling stuff....i had to remember it :)
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