[cloud8]

You can get rid of all the sines and cosines, etc., and replace them with "spread" and "quadrance," or "Jennifer" and "Pamela," and I still won't understand trigonometry.

[InterceptPoint]

Take a look at Chapter 1 of the book. What I see is a complex replacement of sin/cos/tan with what amounts to the Pythagorean Theorem. You have to give up the concept of angle and distance and replace it with "quadrance" (distance squared) and "spread" (don't ask) and you still (obviously) have to be able to do square roots to get answers. So he makes you give up the intuitive idea of distance and angle just so you don't have to push the SIN, COS or TAN buttons on your calculator. Instead you have to push the SQR button several times.

This is progress?

[mtbopfuyn]

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

And yet he's left me still scratching my head since he failed to explain the five main rules.

[Texas Eagle]

Mathematics students have cause to celebrate.

Wow. That oughta be a party.

[Final Authority]

for later read

[JasonC]

What horsefeathers. You need to understand sines etc to understand any periodic phenomenon, it is hardly just about triangles. Then you need it to understand families of orthogonal functions, expansions, frequency spectra, light, fields, yada yada yada. This is just dumbing math down for nabobs who'll never learn enough of it to tie their shoes.

[fzx12345]

Hooray for new math,
New-hoo-hoo-math,
It won't do you a bit of good to read math.
It's so simple,
So very simple,
That only a child can do it!

---Tom Lehrer

[dr_who_2]

"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."

Verily I say unto you, you will become like a rock, knowing and unknowing, aging and ageless, dense and very porous, dumb and yet so friggin' clever.

[dighton]

REPENT OF YOUR SINES

[TAdams8591]

Trigonometry was easy, especially compared to calculus. I see no reason or purpose in making it even easier.

[Excuse_My_Bellicosity]

This author sounds like he's full of crap to me. I'm not going to spend $80 on his book to find out, but I find some things on his book jacket (http://wildegg.com/papers/DivineProp_LR.pdf) very suspect.

Rational trigonometry makes it all much simpler, by replacing transcendental functions like cos, sin and tan with arithmetic and high school algebra.

I learned trig in the 8th and 9th grade using basic arithmetic and high school algebra. Tangent = opposite length/adjacent length, Cosine = opposite length / hypotenuse length. I don't know how you can get simpler than that.

For sines, cosines, and tangents of angles, a $10 calculator eliminates look-up tables, gives you trig values for any odd angle you can punch in there (so that you don't have to use linear interpolation for angles like 28.7893 degrees), and they generally give you results better than 6 decimal places. How is this guy's results more accurate than that?

I also noticed that when you Google this guy, you get nothing but the link to his book at wildegg.com. If this guy is a brilliant mathematician who is rewriting Euclidian mathematics and devising systems better than Newtonian coordinate systems, why doesn't he have write-ups in publications like Scientific American and mathematical periodicals? This whole thing is looking more and more like a money-making gimmick to me. I would think engineering firms would be beating this guy's door down if he were for real.

[Calvin Locke]

"Sined...Seeled...Delivered"

[Fzob]

I found this "Why rational trigonometry?"

http://merganser.math.gvsu.edu/david/reed05/projects/halserogers/html/why.html

[No Truce With Kings]

I have to admit to being skeptical about his claims. A bald assertion such as the one the author is making is not necessarily the naked truth. I will have to get his book and see if it really *is* a breakthrough.

That said, people will have to check his claims before dismissing them. In the late 1700s a sailing captain with little formal education wrote a new approach to spherical trigonometry that allowed badly-educated sailors to understand it and use it for navigation. The man was Nathaniel Bowditch, and his book is *STILL* in print, recently having had its 200th anniversary of continuous publication.

[megatherium]

My assessment:

Wildberger happens to be an "abstract harmonic analyst," which is an area of mathematics devoted to the development of analogues and generalizations to harmonic analysis (what mathematicians call the area of math that is based on trigonometry, and which includes such important mathematical ideas as the Fourier transform or Fourier series). This is my own specialty as a mathematician. Wildberger is extremely good at abstract harmonic analysis. (Much better than me!)

This book, which I have only read the first part of, seems pretty solid to me. It might be oversold -- certainly it will not replace angles and ordinary trigonometric functions, especially in physics and engineering. But based on my initial impression, it is a very pretty piece of work. (Certainly, there are no obvious mistakes; Wildberger at the least is fully competent.) What the book is likely to do is keep a fairly small group of mathematicians rather busy for a few years, as they work out generalizations in Wildberger's framework of modern harmonic analysis. A worthwhile enterprise (exactly what we do in my biz) but again, not terribly likely to change the mathematical landscape from the point-of-view of your average engineer or physicist (and certainly not your average 11th grade trig student).

One detail of the trig system he proposes strikes me as simple yet clever. This is using the square of the distance instead of the distance between two points (his "quadrance"). This reminds me of the crucial insight by Fisher in statistics to use the "variance" instead of the standard deviation. The variance is simply the square of the standard deviation -- but what this accomplishes is that when one adds independent random variables (normally distributed), the variance of the sum is just the sum of the variances. (Standard deviations do not add across that way, you must use square roots.)

[snarks_when_bored]

I've glanced briefly at Wildberger's first chapter. 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).

In fact, his two fundamental concepts, Quadrance and Spread, are straightforward ways of hiding some of the computational complexity underlying the ordinary trig functions, which functions derive ultimately from the infinite series for the exponential function. The 'Quadrance' of two points is just the square of the distance between the points. And the 'Spread' between two (not necessarily distinct) lines is just a ratio of two Quadrances (which ratio ends up being the square of the sine of the angle between the lines—yes, such an angle is not unique, but that's okay).

So ordinary trigonometry is lurking just beneath the surface of Wildberger's 'rational trigonometry'. For example, we find that what he calls his 'Spread law' for triangle A₁A₂A₃ is just the square of the customary Law of Sines. Similarly, his 'Cross law' is just the square of the Law of Cosines (what he calls the 'cross' is just the square of a cosine).

His approach has advantages—perhaps it's less computationally challenging to beginners; it generalizes easily to different kinds of fields, including finite fields; etc. But his approach also has disadvantages—angles are additive, spreads are not; the ordinary trig functions will still have to be learned at some stage by those intending to take higher math courses since they occur in vital places in calculus and have wide applications throughout physics, chemistry and the various engineering disciplines, including computer programming; etc.

From a slightly more philosophical perspective, what Wildberger says on p. 20 about the 'vagueness' in the foundations of modern mathematics and the 'logical deficiencies' of mathematical analysis is just wrong. And his claim that no axioms are required to do his rational trigonometry is also mistaken. He's assuming as given the field of real numbers, but this field is defined by a collection of axioms (almost every type of object in modern mathematics is so defined). Surely he knows this, so why he says otherwise is a mystery.

Enough.

[HoustonCurmudgeon]

Trig is the only course I took in High School or College that I truly hated, and still do decades later.

[LibWhacker]

Oh, yeah, I went to one of his lectures once.

[Condor51]

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

Huh?
Oh well I guess this means I'll never have to; sine another contract, cosine for a loan, or .... wait I'm going off on a tangent; what do I do with my

Post VersaTRIG Slide Rule????

[paulat]

...off on another tangent, I see....

[June Cleaver]

Ping!

let me know what you think about this