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Trigonometry Redefined without Sines And Cosines
Spy der Mann writes "Dr. Norman Wildberger, of the South Wales University, has redefined trigonometry without the use of sines, cosines, or tangents. In his book about Rational Trigonometry (sample PDF chapter), he explains that by replacing distance and angles with new concepts: quadrance, and spread, one can express trigonometric problems with simple algebra and fractional numbers. Is this the beginning of a new era for math?"
I never understood that crap anyway.
I did a stint as a Maths teacher, and it was hard enough trying to convince the kids that it was worth learning Trigonometry then. They'll be even more determined to be ignorant if they hear of this.
Aide-toi, le Ciel t'aidera - Jeanne D'Arc.
This sounds promising, but I have two educational concerns: 1. Is this just a dumbed-down version of trig? ...and on the opposite end of the spectrum...
2. Would this lead to bombarding students with too much math as the requirement shifts from alg/calc/trig to alg/calc/trig/trig2?
(a^2 + b^2)^(1 / 2) = c
to:
a^2 + b^2 = c^2
With a lot of just applying that paradigm to every aspect of trig. Pretty nifty time saver, but I fear the unique insights from this method may be few.
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Erm, I actually read the sample chapter, and one thing I don't get: What did he do that is so revolutionary?
He redefined a side of a triangle with a Quadrance - a square of distance. He claims this removes the square root, but guess what? d^2 has the same effect.
He also defined a spread, which is the relationship between two lines. The catch? It is PROPORTIONALLY EQUAL to an angle. 90 degrees is 1, 0 degrees is 0, 45 degrees is 1/2.
I haven't read the full book, but from what I can tell, all this is doing is redefining the constraints of trigonometry, causing nice even numbers to be used.
ahh Sin= Op/Hyp
Cos = Adj/Hyp
Tan = Op/adjacent.
By as someone who's done some surveying it (which uses angles and distance), its easy to see why people use angles and distance. They are fairly easy to measure, so usefull for plans etc..
Replacing angles and distance with abstract quadrance and spread is exchanging one difficult thing (tan,cos,sin) for another (quad, spread)
Quandrance = distance ^2
Spread hard to see.
You're wrong.
It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).
The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group
theory, is generally not needed." (to use this method)
I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.
Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?
Sigs are for the weak.
The professor seems to have found an interesting way to present trig - however, it should also be noted that he does actually rely on sines as an underlying concept.
I read the article and part of hist concept of spread for an angle is the ratio (in a right triangle) of the side opposite the angle to the hypotenuse of the triangle. As I recall from my high-school geometry classes, though, this is exactly how the sine is calculated. (The cosine is the adjacent side divided by the hypotenuse.)
The interesting thing about his approach, though, is that he defines the concept of spread so that there's no need to refer to the underlying sine concept and the calculations are all (relatively) simple algebra - squaring and addition and subtraction. I would like to read some more and play with it a bit - the fact is I still have bad dreams about those damned trig identity tables, which I never really successfully felt comfortable with!
Interesting.
Sigh. My id isn't prime. 2 2 2 2 2 3 5 313
As a math teacher, it may not be as obvious to you, but as someone who learned trigonometry in school and isn't a math teacher now I can say for certain. When people say they'll never use that in the real world, they're absolutely right.
Now sometimes while slogging through my day, paying bills, shopping, working and things like that I sometimes say "Man I really should calculate the cosine of this electric bill", but as of yet I haven't been harmed by not actually doing that.
What happens when kids get to math subjects where trigonometric functions are used for more than just calculating the dimensions of geometric figures? How does this "spread" thing represent angles greater than 180 degrees without redefining what you are measuring, and how does this really make a persons life any easier unless someone tells you the spread as in the textbook chapter? If his explanation is to be taken as any sort of indication of how to measure the spread, then you might as well just walk off the dimensions of what you're measuring because you'll have to do that anyway to calculate it. How will you integrate problems that call for constant rotation using spread? This seems like trading a little bit of pain now for a lot more down the road, and I pray that it won't catch on in the US. If students are really having this much trouble with this subject then it should be introduced earlier and in smaller portions, not ignored. The last thing we need is for someone to take another slice out of the already anaemic math programs in our primary schools.
`which fortune`
if you've ever progrmmed a for-loop, then you have basically integrated! ;)
If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.
All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.
This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.
Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.
Imagine if we'd been using "quadrance" and "spread" for years - and then some bright spark suggested calculated using sines and cosines. Which would people see as easier?
I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.
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Even kids who go into trades like carpentry would benefit from a knowledge of the fundamentals of trig. Laying out angles for roof rafters, staircase stringers, etc....
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The base 60 number system of the Babylonians was successful enough to have worked its way through time to appear in our present day modern world. We still have 60 minutes in an hour, 60 seconds in a minute, 360 degrees in a circle and 60 minutes in a degree. Even our 24 hour clock is a legacy from the ancient Babylonians.
Well you certainly aren't working in animation or writing simulations, or writing AI programs, or code for robots, or doing any kind of graphics conversion, or audio programming or making any kind of games with your "programming"(I'll stop here, but I could go on and on). I would guess with your attitude toward math you're really not a programmer, you probably just tie stuff together that other people have written with your own code or scripts. You use libraries rather than write them. Not trying to insult what you do, but there's a lot more to programming than that, and it does take math.
And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.
No, Thursday's out. How about never - is never good for you?
It wasn't intended to give rise to unique insights. It was intended to simplify the teaching and calculation of geometry.
My dad is a machinist, and that job is very heavy on trig. If you're trying to figure out how to calculate a correct taper or how to calculate threads, you need to know trig.
I doubt there are any unique insights from his approach: he's basically made angle the fundamental element of trig, rather than the more usual distance.
That said, it might be an interesting way to teach/explore trig. By doing away with the trig functions (which are just the distance->angle mappings,) he gets to solve many simple problems with just algebra and a final square root. Because the sqrt is explicit, this approach might give students a better mental model for trig: as things stand, most students just treat sin, cos, etc, as black boxes, and apply the rules (SOHCAHTOA) by rote.
I'd like to see how his method stacks up when applied to, say, an entire beginner level book or classic text, on geometry/trig.
Here is what he does: He replaces the distances by its square, and calls it a squandrance. He replaces the angle by the square of its sine and calls it a spread.
Ok, the relations between the squandrances and spreads of a rectangular triangle are simpler than those between its sides and its angles -- they are just simply obtained from pythagoras' theorem.
However, two way more fundamental relations suddenly become horribly difficult: Say going from town A to C I go from A to B then from B to C. A to B is 5 squandrance, B to C is 3 squandrance, how much squandrance is it from A to C? No, not 8, but 5 + 3 + 2*sqrt(8). The simple addition of distances becomes a square-root function...
Also, say I turn left by 30 degree, and then do that again. Guess what, I turned left by 2*30 = 60 degrees. However, if you were doing this in spreads, you don't want me to tell you the answer. I think there is a reason why sailors have been using angles and not spreads...
As for engineers: Why do they have to learn trigonometry? Not for geometry, you can just do that with coordinates. But because it turns up in waves everywhere, which in turn turn up everywhere. (Aside: the mathematical reason is that sine and cosine are the solutions of one of the most fundamental differential equations.) For example, the voltage of an AC current will depend proportionally on a sine fucntion. The angle you substitute in sine will be proportional to time. The spread you would have to substitute in a rational geometry function instead is NOT proportional to time, but to the square of the sine of time. Too bad my stopwatch measures time in seconds and not in the square of its sine...
Will his approach lead to faster computations? Of course not. Whereever it would help, people of course already knew last year already (and maybe even, gosh, in 2003) to use some trigonometric substitution....
I don't know quite how to put this, so I am just going to say it.
The degree doesn't make you an engineer. The MATH makes you an engineer. The degree is just your univerity vouching that you have completed your math and other engineering studies competently.
In my opinion, I think the author of this book is a quack and all I had to see was the first paragraph on the first page of his web site where he states that he has dispensed with (geometric) axioms. You cannot do anything in mathematics without axioms. Period. Math is not capable of proving something from nothing.
"Lawyers are for sucks."
- Doug McKenzie
I am a high school mathematics teacher and I train students for mathematics competitions. I think most of you are missing the point of Dr. Wildberger.
Dr. Wildberger is not trying to redefine trigonometry, he is simply trying to give it a new perspective and hopefully, the new perspective would allow new insights into new methods of solving trigonometric problems. Protesting that memorizing the trigonometric functions as side adjacent over side opposite, etc., etc., is very easy and intuitive ignores the fact that in analytic geometry, that is not even how the trigonometric functions are defined!
Yes, really! For example, the sine of theta is defined in analysis as the y component of the radial vector from the origin to a point in a circle of unit radius whose arc distance from the x-axis is theta. The cosine of theta is defined similarly but this time taking the x-component. From this two simple definitions, the entire panoply of the trigonometric identities can be usefully derived!
The analytical definition is certainly not intuitive and not easy to memorize for a high school student! The side opposite, side adjacent trick is just that, a trick that is useful sometimes and certainly useful enough for high school mathematics but it is not a very useful definition as far as analysis is concerned.
For example, computing the derivative of the cosine function is not easy to understand if you restrict the definition of cosine to side adjacent over hypotenuse! Not to mention the fact that most students think there is magic involved in the computation of the trigonometric functions because the method of computation is not in their textbooks. It is only when one studies the calculus that the methods for computing the trigonometric functions are explained!
Dr. Wildberger has an idea that he thinks will make trigonometry more intuitive and I hope he is really onto something here. It would certainly help me with my students. I have read only the downloadable first chapter of the book and the idea is intriguing. Waving off Wildberger's new ideas without reading the entire book and without understanding the mathematics of trigonometry is just tragic.
In times like this I always remember the architect (I forgot the name, help me out here please) who refused to accept an architecture medal because the society that was giving out the medal invited Prince Charles to hand out the medal. That architect said, "I refuse to accept a medal from a person who believes that our grandfathers already know everything there is to know about how to build buildings and that there is nothing we can ever add to that knowledge anymore."
Just my two cents.
confused the area of mathematics with a subset of mathematics called calculus.
Try proving that a sorting algorithm is O(log n) without using any mathematics. Then you'll understand why mathematics is important.
You could apply this to pretty much any field with just as much stretching. The music scale works on logarithmic principles, and don't get started on how complex the calculations can be for what makes a good instrument, or good acoustics. That doesn't mean you average cello player need to know anything about sines and cosines.
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Some of them sucked.. some of them were really good (IMO).
And that is all you can get without using maths to prove that they are good or bad, be it in the average, best or worst case.
In fact, you give away your attitude a bit by mentioning Prince Charles. I'm a republican myself but I think you misrepresent his views. He appears to believe that people should live in human scale buildings designed to give a sense of community, and not live in buildings designed to return the maximum income to the shareholders in developers (who don't have to live in the buildings that bring in that income, and wouldn't want to.)
He also appears to believe that many architects produce stuff intended to glorify themselves and their clients regardless of their fit with the landscape and the environment. This is a neat parallel to the idea that mathematics teaching for the majority should reflect concepts that tie back to the real world so they will be useful, and that people who propose radical changes are often more interested in their own fame and reputation than the real benefits.
Panurge has posted for the last time. Thanks for the positive moderations.
It's hilarious - look at Europe with dozens of little legal systems, languages, political fiefdoms and no constitution and two world wars but goddammit WE ARE BETTER BECAUSE WE ARE METRIC.
I'll get OT and bite -- yes, I'd rather live in a continent whose culture extends more than 300 hundred years back in history, does not have a constitution that ensures the same bad politicians always stay in power and act against the interests of the population and looking into pleasing the big corporations, and does not use freedom as an excuse to bring war to other parts of the world. Any questions?
That said, I don't really understand what one thing has to do with the other. The advantage of the metric system is simply to ensure that everyone uses the same rigorously defined units and the potential for confusion is reduced -- much like your pseudo-argument that the USA are somehow better for having a single language, except that in this case you should look into other factors such as cultural diversity.
Ergo, you must be a troll.
Score: i, Imaginary
What's the point of a standard if it's not pervasive and useful? Most standards are useful, and just by being standards, they are pervasive. But being widely accepted also means incredible costs for invoking changes across most sectors of society on the basis of marginal increases in performance.
... but only having 2 factors makes it insufficient for other uses.
There are proposals to change time, but like changes to the English alphabet, the benefits have to outweigh the almost impossibly large costs of transforming. Look at the English/Metric systems. The USA still has not changed. Metric's benefits still have not outweighed the costs of changing all rulers, indicators, speedometers, odometers, signs, etc. -- and as well, the internal rulers in people's minds that have used feet and miles for centuries.
P.S. The number 60 has these low factors: 2 3 4 5 6. That means that if you use the number 60 for measuring, it's easy to divide whatever you're doing into 2 to 6 parts, and each part is another integer. The number 10 only has factors 2 and 5. Arguably, the number 12 (having factors of 2 3 4 6) is more useful than 10. 10's usefulness is that it matches the base (10) that we use
P.P.S. If there was some real political will behind it, the USA might be able to change all typing keyboards to Dvorak from Qwerty. But that would be a social "Apollo Project" in scale, hence it's never going to be done. Keyboards will probably change format when the entire concept of a keyboard changes, like if pervasive voice recognition or neural connections arise. But then the change will be invoked as the standard becomes nonstandard, and the keyboard fades away.
[You have a stable society when some nut guns down a schoolyard and the law doesn't change.]
Those black boxes are the reason that while I was relatively good at math, I sucked at trig, which screwed me when I got to calculus. I had always thought that when learning math, I could follow the steps to a solution which lead to an understanding as to why it worked. Black boxes, as you described it, do not do this.
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Furthermore, a lot of what this guy did is kind of a trick. Using 'spreads' may work when given an explicit triangle, but the part he's skimming over is that spreads are missing a REALLY nice property of angles. They don't add. Angles are a very nice parameter for rotation because a rotation of 10 followed by a rotation by 20 is the same as a rotation by 30. This property is implicitly used all over the place in graphics. So, in the end you probably have to use some angle-like measure when doing computer graphics (which is all about transformations, not measurements of unknowns). And in doing so, I'm sure you end up computing sines and cosines to do projections based on those rotations.
In the end, you just can't cheat your way out of the fact that a projection based on a rotation is a transcendental operation that numerically requires computing a truncated infinite series.
I wasn't taught trig functions as black boxes. We learned right from the start that they're the ratios of the various sides. Once you understand that, it's easy to know which function to use to find which side or angle, and why. Identities were just s easy: they're just formulas that don't depend on the angle; they're right for any angle, so you can use them to simplify equations. Trig was fun, and I was good at it, but that might be because my teacher understood how to explain it instead of simply demanding rote memorization.
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I wouldn't have any problems with (yet) an other mathematical notation and method. In any case we use different notations for various rules of physics (tensors, vectors, fourier transformations etc.) depending on the aim and whatever method is easier for the problem. The problem would be teaching high-school level pupils because at that age you usually accept anything you are thought as the norm and then get confused when you are in the university and someone shows something completely different (tensors anybody?).
Since the focus of graphics in the last 10-20 years has shifted from getting anything at all in 2D to doing photorealism in 3D, angles are arguably not directly useful to modern algorithms. In fact, in most of the work I do, angles are essentially translated into scaled sines or cosines directly from the relevant vectors (cross products or dot products), and operated from there on as pure trigonometric quantities.