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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?"
Perhaps Dr. Wildberger is trying to take geometry off on a weird tangent.
I never understood that crap anyway.
If only he could redefine Calculus to use simple algebraic expressions.
is 42
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.
He does this the year after I take Algebra II/Trig. Bastard.
Note to mods: I'm probably being sarcastic.
It's the era where I don't fail Calculus IV because I never went to math in 8th grade.
I'd be sceptical if it will take off - 'Wildbergers' just isn't as catchy as 'degrees' or 'radians'....
When does a rectangle become a line?
The "New" refers to South Wales, rather than the uni. Might wanna read a bit closer next time.
I'm a failing engineering student at Iowa State University- or at least I was.. My failing point: math. I can out code some of the instructors in my classes, but I can't do math or memorize functions and derivitives of trigonometric functions.
If this book pans out, it would ultimately change Calculus for the better and might allow me to pass my classes- I wonder if I figured out how to apply this book to my classes and came up with the correct answers despite having worked the problems with his methods, if I'd still pass the class?
As per the article .. Dr Wildberger is from UNSW, the University of New South Wales .. in friggin' Australia. South Wales is somewhere all together different. But as always people don't RTFA
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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?
are trigonometric functions REALLY that hard to learn?
is trigonometry one of the root causes of the layman's hatred for math?
that's doubtful and even if it was true, his version of trigonometry still requires algebra which has a far greater hatred among joe sixpack.
-- Believe your Justice!
Just wait until someone reimplements many software functions that rely on sine, cosine, and tangent, using these new concepts, and get a patent on them.
Inconceivable!
DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry by N J Wildberger is a link to the advertisment for his upcoming book, which also has a PDF dowbload of the first chapter.
I am Slashdot. Are you Slashdot as well?
just a new way of coming to the same solutions. that's like saying the transportation industry has been revolutionized because an alternate route has been found between your house and where you work...
: )
(actually...i would probably say that if there WERE an alternate route...but...eh...)
It takes just a moment and an action to destroy. It takes some time and thought to create.
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.
My high school math teacher used to march around class chanting SOH CAH TOA, SOH CAH TOA, SOH CAH TOA!
(and now, thirty years later I still remember)
Sine = Opposite over Hypotenuse (SOH)
Cosine = Adjacent over Hypotenuse (CAH)
Tangent = Opposite over Adjacent (TOA)
(when dealing with right-angle triangles)
TDz.
Uh... that's not just redefining trig, that's totally redefining mathematics and logic. I find that hard to believe. Is it just marketing talk? Or did this guy revolutionize the axiomatic system upon which we built all human knowledge? I find the latter doubtful.
And it shows how to apply this new theory to a wide range of practical problems from engineering, physics, surveying and calculus. Wait... This is math. There are no theories. It's either proven or unproven. There might be conjectures waiting to be proven but I've never heard of theories being used in mathematic. Then again, I am not a mathematician.
Maybe someone much more knowledgable can explain this for me.
EvilCON - Made Famous by
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.
its like saying the transportation industry has been revolutionised because you can now cycle to work instead in walking. Oh wait, it has.
No idea if that book is junk or not though, but you'd be wiser to take a course in rhetoric (or failing that English composition) instead anyhow.
I am wondering if this could be used to make faster calculations
in raytracers and 3D engines by using integer numbers.
Isn't that what sines and cosines are in the first place?
Remember them and trigonometry is a doddle.
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.
This is horrible for ray tracing. The angles are non-linear. In computer graphics, it is easy to add anagles 45deg+45deg=90deg. That is the beauty of regular angles.
With his method you can't just add angles line that. You have to do an elaborate calculation.
tan(88.6361deg) = 42.
Is that what you wanted?
sin 90 ...is 90 wrong things u did against will of god
42 is wrong answer why?....its cos 90
tan 90 ...is the new skin care product
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`
This has already been done by a man named Karl Weierstrass who came up with a way to express continuity in algebraic terms. You know, the "epsilon-delta" definition you learn in your first week of Calculus. In a nutshell, before this definition, everyone knew that Calculus worked, but no one was sure *why*.
The World is Yours.
Does this make things easier for computers?
Would be very nice to have a performance boost at the math level for 3D calculations.
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 never really got a good answer to that one from my maths teachers.
Deleted
Ok, so using squares of distances instead of plain distances, and relations between lines instead of angles makes calculations easier. But isn't that shifting the problem? Now measuring becomes more complex, as do calculations based on angular velocity. Still, it's good that someone is trying to provide a new perspective; back when I was doing trigonometry I always thought there must be something simpler underneath.
Please correct me if I got my facts wrong.
No.
but does it have a simple mnemonic like:
Orange
Hippos
Always
Have
Orange
Angles
which yields...
Opposite
Hypotenuse = sin theta
Adjacent
Hypotenuse = cos theta
Opposite
Adjacent = tan theta
Just use quaternions and be done with it.
It really seems to me that his concept of "spread" to measure the orientation of two lines is much less intuitive than angle. The concept of angle is just not hard to grasp compared to this weird construction of dropping perpendicular lines.
And it isn't true that you need calculus to understand cosines and sines, you just need some simple plane geometry (right angle triangles inscribed in circles and so on). You can even plot the cosine and sine functions without calculus.
The world is everything that is the case
Now THESE are some divine proportions.
A man's reach must exceed his grasp, or what's an erection for?
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....
Remember "News for Nerds, Stuff that Matters"? Help make it a reality again! http://soylentnews.org
reinventing the wheel...
http://kelvin.quee.org
It looks like all that is being done is removing squareroots and negative numbers.
quadrance is the square of the distance
spread is the square of the sin angle
If you want to teach trig this way, it will certainly be easier to learn. But then, your students will not appreciate or understand square roots and negative numbers.
It is highly unlikely to lead to any revolutionary new algorithms. It appears that you are changing an angle to something rational, because all most of the popular sine values have square roots in them. However do not be fooled by this, it is just as hard to calculate spread as it is to calculate the sine of an angle. (except in a few cases).
If it was going to take off as a standard approach to trigonometry the there's a lot of follow up stuff in engineering and physics where the traditions would have to be redefined and textbooks re-written. eg angular momentum, theory underlying fourier transforms, forces etc.
It may be better but the world has been using this system for a while and its pointless teaching a new format of trig if people who *are* going to have to use it only have to relearn the current conventional system because thats how its always been done in the field of application.
Professor Karmadillo Songs of Science
In the same way, as I understand it based on the sites referenced in the story, this guy makes use of an angle replacement that is not useful to most people. If I want to divide a circular cake into n portions, using spread will not be helpful. Although degrees are a unit with no rational basis(there are not 360 days in a year, and 480 might be better-divisible by 5,3 and powers of 2) they are relatively easy to use for circle division, which is all the circle math that most people need to do most of the time.
Unfortunately, mathematics lectures at Cambridge left me with the permanent belief that mathematicians' ideas of what is simple and what is complex merely illustrate the fact that physics and engineering use math, but they use a useful subset (which changes with time) and do not necessarily buy into ideas which mathematicians regard as self-evident.
Panurge has posted for the last time. Thanks for the positive moderations.
I've always assumed that angles are so fancy that they need to have transcendental numbers to actually use them - even if everything else is rational. I'm glad this guy has challenged that assumption and showed that you can do cool stuff while staying in a smaller, closed system.
He may be just formalizing stuff serious graphics/geometry programmers already know: you can do many interesting geometric computations without taking sines, cosines, or square roots (or at least deferring them until you actually need the distance or angle, which is rarer than you might think). He may have done better using dot products and cross products, the standard tools of the trade, but that wouldn't have let him claim new math.
here's something to twist your nose
Scientology maintains that misunderstanding or not understanding something not only interferes with your ability to use it, it also leads to people developing and justifying a dislike of the thing later on. Never mind other stuff.
How many times have you heard folks say "math is so stupid, y'know. Hey! let's pick on the geeks!"
Hey, that stuff can't be very important. George Bush got to be president without knowing any of it!
The one where the solution involves sqrt(7). The fact is, you don't need trig to solve that problem and people shouldn't be using trig to do so. His approach isn't new, it's what a mathematician should do anyway. If there's one thing that is taught wrong it's a tendency to use trig when pythagoras's theorem and similar triangles will do the job anyway. But this guy isn't doing anything new.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Trigonometry, even spherical trigonometry, is an elementary subject, thousands of years old; No XXIth century active mathematician engaged in real mathematical research is interested in trigonometry. This is a subject suitable for high school teachers, not for real mathematicians.
Even arithmetic is more interesting than trigonometry, because it is vaguely related to a very important branch of pure mathematics, number theory.
I still don't have a good reason we continue to use it though. Other than, "it's always been that way".
Deleted
My question is, does it do away with the unit cirle and being required to no how many radians is in say 30 degrees (I believe that was PI/6 Rads). That thing was a pain in the ass to memorize. However it was neccesary to use radians. I wonder how his book will confront the use of radians with just simple algebra.
Take off every sig!
I was taking a real analysis class in my first semester of grad school. I did a good job on the first question, figured everything out, and got that the answer was the integral of 1/(1 + x^2).
I got no points at all for the question - despite solving the parts relevant to the class - because I didn't know off the top of my head that the integral of that is the arctan function.
I love abstract math but I hate trig.
42.0008?
ok, tan(88.63607247deg)=42...ya happy now, dr precision?
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.
slashdot still uses the 2+2=5 chalkboard icon.
.. eventhough I never remembered what was what with sinus and tangens. The simple trick is to allways have the Unit circle (http://en.wikipedia.org/wiki/Unit_Circle) aka 'Einheitskreis' (german, http://de.wikipedia.org/wiki/Einheitskreis) handy before your "inner eye". With that finding out which tangens and sinus is which and which needs to be applied in a certain situation is a piece of cake.
Draw yourself a nice unit circle, memorize it and you'll allways know which relation to use.
Trig math can be real fun. I actually considered becoming a surveyor after our surveying project in 10th grade. A friend and I went to the local surveyors office and even did a little on site trip with the surveying team. That was real cool - and we both got an A for the project in school.
Little portable computers (Sharp PC 1402 back then, mid 80s), lot's of intelligent stuff to do, pratical math and your outside at the fresh air all day. Very cool job for a geek actually...
We suffer more in our imagination than in reality. - Seneca
I can see the utility of this from the point of view of trigonometry. At one point in the chapter he gives an example of a simple problem that is needlessly complex to do using tan (if you dont have a calculator.) As such a knowledge of this would surely be useful.
However it is not clear to me how you could use spread to map a continuiously increasing quantity (such as time) on to a periodic variable (such as displacement.) Surely to do this using his simple ratio of quadrances would be more complex than using sin? Then what about things like Fourier series? This would surely be very clunky in this framework.
This stuff must still be equivalent to classical trig. Thus it cant possibly be 'revolutionary'. You still need to start with the same axioms.
You've hit on a major downside to this new method. How do you measure quadrance and spread in the real world? Until Home Depot starts selling quadrance tape measures and "spread" mitre saws, you're not gonna see this idea really in use at all outside of a classroom environment.
It's an interesting read, but I think that the "rational geometry" isn't as fundamental a change as the author claims.
I would say that it is a lot like switching between polar and rectangular coordinates. Some things are easier to do in one system, some things are easier to do in the other.
The author makes an excellent point that when working with quadrance and spread, trigonometry and spread become easy. So if I am working with a triangle, I can do everything with rational numbers. However, it often happens to me that I want to do things like add two angles. In the new parametrization, this often useful operation becomes a lot less trivial. I can't just add spreads. Also, if I want to know the distance between two points A and B on a line, and I happen to know the distances from both A and B to an intermediate point C, then my life will be simpler if I use distances.
I'm happy for the insight that working with distance squared and spread (which is just sine squared) can make some computations simpler. But I wouldn't see this as more than a change of coordinate systems. You win something and you loose something. In any given problem, you still should pick the coordinate system that is most suitable.
But if you think simplification is a Good Thing, let's see the US *finally* go metric.
mark "the inch is what, the length of the
King's thumb, from knuckle to tip?"
The "New Kind of Math" introduced in the book is an interesting perspective on high school trig, but the author has centuries of precedent to overcome in getting his methods applied there. In particular, conventional trig concepts like angle, sine/cosine, etc., are so pervasive throughout actual science and engineering that it would require reinventing higher science and engineering. Not a pretty thought, as it would mean a divide between the scientists and engineers learning this new method and the ones who learned it the normal way - a divide in a language that's supposed to be universal.
On the other hand, who would it benefit? A bunch of high school students taught by an instructor who hates calculators? Trig is cake compared to some of the things you have the opportunity to wrap your head around in college (calculus, for starters).
If you implement a geometrical function whose input is a bunch of lengths and their ratios and whose output is a bunch of lengths and their ratios then you frequently shouldn't need any trig in your function, instead you should be using algebraic functions (+, -, *, /, nth root etc.). But many people find it easy to solve their problem by converting a bunch of stuff to angles using trig and then converting back. In fact, I optimised someone's rendering code recently by literally looking for every line of code where he'd used trig and replacing it, where possible, with the algebraic version. It worked well and speeded up the code massively. In addition I uncovered what you might call bugs. For example at one point this guy did a 180 degree rotation by converting to polar coordinates, adding M_PI to the angle and converting back. Rotating (x,y) by 180 degrees gives (-x,-y). The guy who'd written it was 'trig happy' and should have stopped to thing about the geometrical significance of what he was doing rather than just doing the obvious thing. If this book encourages people to use trig less it might be a good thing. But you don't need to talk about 'quadrance' and 'spread' to use the principles I'm talking about.
But of course there are times when you need trig. I'd hate to see the guy differentiate rotations without trig.
Doesn't it make you feel good to know that our freedoms are protected by politicans, lawyers and journalists.
Would anyone who understands this stuff care to speculate on how the use of Rational Trigonometry might affect the mathematics of Complex Numbers?
It seems to me that there might be a simplification, given that we work with the squares of things, we might be able to avoid the square root of -1, and just work with -1. Also, Rational Trigonometry already has the concept of perpendicularity, in the simple value (spread) of 1.
But I don't remember enough of my university math to speculate further.
Can anyone explain it in simple terms?
how does one espress pi algebraically?
I went to high school some 15 years ago, and our teacher derived all the trigonometry from square equations of triangles+circles using cartesian geometry - very simple and very similar to this "new stuff". Also, doing a lot of work in graphics applying all that linear algebra, one quickly notices dot products and cross products, and rather known equations with known properties (well, at least if developing 3D graphics is your job). It's nice to rewrite basic trigonometry, but I'll have to read the other chapters to see if there's anything actually new.
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 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.
will this be an introduction or a replacement to the standard trig ?
...as one goes more in depth traditional trig becomes invaluble, even in high school. So would they now have to reteach trig for physics ?
whithout sine, cos etc... it would be really hard to resolve forces and other quantities.... and thats just in the very basics
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.
I Am A Math Student Teacher, and one of the classes I'm covering is trig analysis honors. They're just getting the hang of sines and cosines now, and learning about function graphs on the way toward inverse trig functions in the next week.
Glancing at the sample chapter, I see that this professor is "eliminating" sines and cosines by replacing them with "spreads", ratios inherent in the triangle. This is easy to do with triangles, but since geometric sines and cosines derive from ratios anyway, he's essentially just given them a new name.
He's eliminated the unit circle as an essential part of his trigonometry, but I can't think of any post-trig class that doesn't rely on the unit circle and trig functions in one way or another. In other words, rational trigonometry looks like a good way to learn trig and only trig without sines, cosines and angles -- but since you need those functions and angles in later courses anyway, why would you want to?
I use it all the time, though I rarely go through the machinations of calculation. There's a certain amount of usefulness in the concept of "mod 2Pi" (if you will, or "mod 360" if you won't) that I find extremely useful.
Of course, it rarely applies to a sine, cosine, or tangent specifically, but any set of random cyclic phenomena I care to compare.
in radians please
The world is everything that is the case
Try the wikipedia on 360 (number).
360 goes way back thousands of years before the Greeks.
Infuriate left and right
The notion of spread removes the dependence upon circles
Well, I've got the sample PDF and I'm looking at the definition of spread in 1.2. I still see a circle there. And it's the classic trig circle with the inscribed right triangle.
I applaud the guy for trying to make things simpler for students to start with, but ultimately I feel this won't help students in the long run by renaming everything and making their studies incompatible with the rest of the world.
Weaselmancer
rediculous.
I'm not sure what to think after reading through the chapter available. I'm thinking I should print it off and wave it in the face of my maths teacher, making the last semester of study completely redundant...
.____.
Too bad it won't help me with trig identities though...
eg. prove
cos^2(x-y)-cos^2(x+y)=sin(2x)sin(2y)
However, I don't find trig particularly hard, so I don't see what the problem is
Let the commencement BEGINULATE!
You're making an ass of yourself.
It is not only silly, it breaks common sense and physical meaning. It maybe useful for properies of a triangle but nothing more. Now if you you have two intevals on a straight line, the total distance is a sum of two distances. It is no longer true with quadrance. Imagine a ruler with quadrance! The same with spead instead of angle. Spread are no longer additive.
It makes the things that are easy now difficult.
Save the bandwidth. Don't use sigs!
Of course, you could always interpret the sig as suggesting a democratic republic is essentially the same as an aristocratic fascism, and your choice of OS is similarly irrelevant. That's clearly not the implicit intent, though.
Math is not those things. Math is a collection of ways of thinking about things. And if you are using those ways of thinking, or any of the rules that hide behind them, you're thinking mathematically.
Perhaps the most important part of mathematics is the proof. Proofs are ways to convince yourself (and others) that you thought about the problem in the right way. And programming is all about proofs. Every time you write a loop and convince yourself it works, you're doing an inductive proof.
I like the characterization by Juris Hartmanis - offered, he says, half in jest, that "computer science is the engineering of mathematics".
Perhaps you'd like a more practical and less abstract notion though....
Essentially all programming is just practical algebra - it just uses slightly different rules. It all hinges on the notion, that you can abstract a thing to a name ("x") and manipulate that name in ways that are common to all of the things of the same kind.
That computer science in many, many fields (other than simple web servers) depends on mathematics in (sometimes very) deep ways, is also pretty clear to those who look at the field in any deeper way than "I never use maths" - category theory is important in the idea of types, calculus in anything involving moving objects (like many computer games), linear algebra in graphics, automata theory in just how computers work, and the list goes on and on and on....
And to top it off, even if you never do any of this other stuff (or prefer to believe that you do not), graph theory permeates the field, and even if you don't do Hamiltonian circuits, you're using and manipulating graphs every time you build a tree (like that file system you're using).
Yea, that is cool, but isn't it like something you would come up with later when you already know "tan" is the answer?
I'm still trying to figure out what people mean by 'social skills' here.
You have to be pretty dense to call any branch of math useless in such an computer/science focused board like slashdot, and I suspect you must be a pretty shitty math teacher if you don't have future engineers/physicists in mind. I suppose you could just be a troll as well. Either way, I wouldn't want your slacker ass teaching my kid.
Ignorance kills, complacency kills, hatred kills, but usually not the ones guilty of them.
You need sines and cosines for differential equations, time series analysis and spectral analysis, not to mention orbital mechanics. If people are taught to think geometrically, you don't need redifine trig. No revolution here, definitely no insights that haven't been part of every scientists tool kit for the last few generations.
On page 16 in the preview chapter (when presenting a "rational" solution to a triangle) author has to solve a quadratic equation, obtaining the square roots already in "quadrances".
Square roots (Sqrt(7), in particular) are not more rational then Pi. Thus, calculations in this "rational" formulation of geometry still involve irrational numbers even for simple problems. What's the point then ?
K.L.M.
Some People Have Curly Brown Hair Till Painted Black.
Some People Have
Sin = Perpendicular / Hypotenuse
Curly Brown Hair
Cos = Base / Hypotenuse
Till Painted Black.
Tan = Perpendicular / Base
Mumia Abu-Jamal is *laughably guilty*. Check the evidence.
My guess is it's intended so that those who understand it laugh at those who don't. It won't change anybody's mind, because the information, like you said, is implicit.
I noticed this subject describes how to measure angles without using irrational numbers, but to illustrate it the slashdot used a Pi icon. :-)
Would not it be unappropriated?
STOP
CLOSING
WITH
FUCKING
IDIOTIC
QUESTIONS
This isn't a goddamn teaser for the 6pm news!
I'm going to try to implement the Fourier Transform using this ...
Wildberger has put a cleaner theory underneath the kind of geometry game engine developers use. This may turn out to be useful.
Lately I've been doing robot motion planning, which has too much unnecessary trig in it. With enough work, it's often possible to derive a trig-free solution to some of the key problems. Better ways to think about trig-free solutions will help.
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.
This isn't the first time someone has tried to get rid of sines and cosines, and use squared distances to avoid a square root later. In my day, we ALL did that.
I'm talking about the world of game programming up until about 1995 (and maybe beyond). When you're drawing a dungeon on a 386, or steering a sprite to point at a target on the Amiga, or, lord help you, writing Corncob 3D, you don't have time to find sines and do square roots. Sure, you can build tables in advance, but when you only have 640k main memory, that's a last resort.
Hence the need for a whole way of calculation that I suppose isn't used now...
Whence? Hence. Whither? Thither.
Robotics is not the best example to use if you are trying to say programmers need math. Most if not all the calculation probably would be better suited to an electrical or mechanical engineer.
Ooo man the floppy drive is broken. No wait. The computer is just upside down.
That's right! And I'm typing this out on a morse code key right now. I can't believe that anyone would suggest that a keyboard is a revolutionary improvement in communications!
The CB App. What's your 20?
What did I miss?
You forgot the algebra.
I'm just about to RTFA, but I usually start by reading a few comments first.
From my point of view, as someone that has studied math, and considers himself a mathematician, this appealing. While nothing appears to be technically new, it shifts the perception of what trigonometry fundamentally is and how it fits into the framework of geometry.
By abolishing square roots with the quadrature, the real numbers can be done away with where they aren't really needed. It should be noted that mathematicians disagree about the standing of real numbers in some ontological senses. Removing them from the discussion can help to better align trig with more modern ideas of geometry which include non-Euclidean spaces.
The spread also has some intuitive appeal. We won't be doing away with sine or cosine, as these are concrete functions that give precise values and are especially useful on continuous domains for applications like signal processing. Angles greater than 90 degrees are really just a cyclic extension to the values between 0 and 90 degrees, and it wouldn't be too hard to generate a similar cyclic structure for use with spreads. In fact, I like the idea of spread better, because it doesn't embed the angle measure into the 2-dimensional space; in effect, by limiting you to the 0 to right angle bounds, it limits you by the geometry itself, making it more intuitive how to apply the concepts to more dimensions.
I expect the immediate impact to be minimal. Long term, this may take hold in education, but I don't expect any groundbreaking increases in understanding or simplicity.
This goes for algorithms as well. This will just make algorithms easier to represent in the short term. To those that research this kind of thing, this is basically a notational footnote, expected to be picked up by any competent practitioner. Notational conventions can be powerful, however, as they often expose simple truths that can be lost in the jumble otherwise. For this reason, I expect the long-term impact to be a low-investment incremental increase in the conceptual reach of mathematicians.
For me, it provides a framework for doing some calculations I've been looking at trying to frame for a while. I'll use it and see where it leads. I get to dig a little further, but I'm not expecting miracles.
Hope this clears things up for a lot of people!
Now I'm off to RTFA...
How does he excuse using the Pythagorean theorem as a basic concept? As I remember, the Pythagorean theorem has to be proven first ... and it uses ratios of distances, not quadrances.
Most programming I see and do appears to be a form of compression- maximise compression of decisions[1] (minimal code) to provide a solution whilst trying to maintain flexibility to cope with _likely_ changes in the problem with minimal change in the code (minimal extra work - work/time compression).
:) ). While I've seen some programmers do stuff like this, that's not really programming eh?
Whilst the trig and sorting stuff seems quite well dealt with by the Math people, I'm not so aware of more formal ways to deal with the decision compression stuff- yeah there's Boolean logic and all that, but it's not helpful to type the entire uncompressed decision tree out, and then only let a computer compress it.
So sure, maybe programming is still maths, but I don't see that much help from the Maths people. They can talk about lisp and set theory till they turn blue, but we could do with a bit more help if possible...
I suppose maybe we're in the sine and cosine stage of Computer Science, and nobody has done the quadrant and spread thing yet? Or is it worse - we're in the roman numeral stage, and nobody has discovered the number zero and magnitude by numeric position?
[1] The most naive uncompressed program is probably a whole bunch of IF statements with all the possible input states followed by each of their corresponding THEN statements which produce the output states, (and GOTOS I guess
Some of this is meant to be memorized (ie basic differnations and integrals) while some of it is meant to be remembered from a book. Im doing my homework right now as I type. Will I ever remember it (Runge-Kutta approximations). No. Even my math teacher says no one memorizes it.
Ooo man the floppy drive is broken. No wait. The computer is just upside down.
This was the first thing I noticed. I am used to working geometry problems by adding and subtrating angles. if a, b, c=a+b are angles with corresponding spreads A, B, c then
C = sin^2(asin(A)+asin(B))
I haven't read the book, but it sure looks like this makes some calculations more complicated.
But this complaint is really beside the point, and may just show my bias.
Here's a more cutting criticism. What happens when students taught this form of trig study calculus? Looks to me like they are at a distinct disadvantage when it comes to the topic of trig substitutions in integrals. Requiring remedial trig in calculus class to introduce the concept of radians and trig functions is not time well spent.
More importantly, even if you can consider it as math, how can it help you?
;) ).
Is there a function or math method that will help create the best procedure/algorithm for you to get your groceries?
If there isn't then who cares if it's math or not? (OK so I'm an engineer...
Get back to me when some math guy can use math to help me refactor or debug some broken code.
Maybe we're using the wrong programming languages I guess (no, I doubt Lisp is it).
Hopefully someone can come up with Computer Programming/Science with "quadrants" and "spread".
Right now I think we're still stuck in the "roman numerals" stage of Computer Programming. Look at C for instance... Sure we can solve problems with it. But hey the Romans built roads, aqueducts etc too.
As a software engineer(yes engineer) who develops algorithms dealing mostly with geometry, I love these ideas. I've already sent an email to my boss asking for a copy of the book. Anytime you can simplify geometric math, you not only make your own life easier, but you can make the computer games of the world faster....and that's a good thing.
That's how I used to do it too... it was hard to convince people why it was a good idea, though,.
And it acheives the same result as this book (polynomials for most analysis) although through different means.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
what about Pi being basically irrational? does his ideas have any affect of figuring the dimension of curved objects, or just polygons?
you still get a degree!
Bert
I know this is a lowblow, but I smell PHB.
Ignorance kills, complacency kills, hatred kills, but usually not the ones guilty of them.
To be even more pedantic, what we call Maxwell's equations, in their short and memorable form, were written by Oliver Heaviside.
--grendel drago
Laws do not persuade just because they threaten. --Seneca
So it is basically a square of a sine. So it is useless in measurement. You rotate something by 60 degress in two parts 30 degrees each. With spread they are not equal. Explain it to a car mechanic :)
Come on. Distance and angle are intuitive and physical. Quadrance and spead are not.
Save the bandwidth. Don't use sigs!
Trig should be about a 1 to 2 week topic in school. If instead of having students memorize endless identities you simply teach them 1 (Eulers equation) and show them how to easily derive the rest then it becomes pretty trivial.
Euler's equation:
e^(i*x) = cos(x) + i*sin(x)
Need a double angle formula? No problem.
e^(i*2*x) = cos(2*x) + i*sin(2*x)
e^(i*2*x) = (e^(i*x))*(e^(i*x))
= (cos(x) + i*sin(x))*(cos(x) + i*sin(x))
= (cos(x))^2 - (sin(x))^2 + i*2*cos(x)*sin(x)
So you can clearly see that
cos(2*x) = (cos(x))^2 - (sin(x))^2
sin(2*x) = 2*sin(x)*cos(x)
All of the trig identities fall out as simply through simple algebraic games when you start with Euler's equation. It also makes looking at the calculus of trig pretty trivial, greatly simplifies the study of waves, etc. I cannot for the life of me understand why everyone doesn't teach it this way.
If you read the first page of his site, you probably noticed that he put the word axioms in quotes.
Math is all about discarding old "axioms" and coming up with new axioms. You just have to realize that as axioms age, they often become "axioms". Get it?
-------
Incite and flee.
If the computations are 2-d then complex algebra will work very well, and often be quite rapid, and the formulae clean and compact.
And you don't have to worry about domain problems (-pi, to pi) or (0 to 2pi) or whatever.
You vote, right?
Well, having a sense of persepctive and knowing "Hmmm... this political situation seems familiar" is something that you gain invaluable insight too through history.
Everyone should be familiar with history and geography. We live in a connected, opinionated world.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
This means that all numbers are represented with rational numbers. This will do wonder in eliminating round-off errors in computerized calculations.
If you can't do basic analysis (which might require some rudamentary calc) then you won't be able to model (and thus justify) any algorithmic choices you make as a project manager.
Well, if performance isn't your main objective then it might not matter so much... you're probably one of those get-it-to-work-then-shove-it-out-the-door kinda project managers.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
Anytime you say: take the limit (as some parameter goes to 0, goes to inf.) you need calc.
L'Hopital's rules ARE calc.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
The ultimate test for the new rational approach to trigonomety is the success in many different classrooms and with many different teachers. If it is successful for one teacher, it does not mean that it will work for others. There are many examples of elegant new ways to approach things which failed because the teachers could not adapt or because the elegance only seems so for a narrow circle of people. An example is nonstandard analysis.
I personally doubt that you can avoid trigonometric functions, because they are eventually used, for example when dealing with differential equations. Still, it is amazing to see a rare case of somebody thinking about such a basic mathematical concept in a new way.
No. Why do you think this article is about Wonderful New Math being released in a book? This isn't the 1600s; we have systems in place to accept and critically review claims such as this. There are math journals and societies and the like. I don't think we can outright dismiss anything that hasn't travelled the "proper" channels, but this guy is an academic by profession. He actively chose to avoid his peers and that says something.
If aspiration is a virtue, achievement cannot be a vice.
It is not revolutionary, nor redefined. What it does it to rewrite trigonometry in similar thing as the good old "SOH-CAH-TOA" method taught in American high schools.
Wildberger's sole insight are the following:
- Instead of using the linear norm, he chooses to use the equivalent quadratic norm for distances, thus removing the squares from Pythagorean theorem. (So, for a right triangle, his version would be BASE + HEIGHT = HYPOTENUSE).
- Instead of using angles and calculating sines and cosines from it, he uses the concept of Spread, which is essentially just the sine of the angle squared!!
Well, one immediately sees a problem with the second point when trying to do something more than traditional planar Euclidean geometry: an obtuse angle will have the same spread as one other acute angle, and they share spreads with two other angles greater than pi radians!His argument is foundamentally flawed even in the first chapter when he "explains" the traditional method of measuring angles. He claims it as the following (paraphrasing):
so far so good, but he goes on to argue that That is complete bullshit. When I was in middle school in Taiwan, we were taught that 1) we start with two rays instead of two lines. The angle between rays A and B would be given by taking the arc length of on the unit circle by going counter clockwise from A to B. So, semantically, the angle between A and B, and the angle between B and A, would sum up to a nice 2pi. While colloquially, we often prefer to deal with the smaller of the two measurements.There are obvious problems with the Quadrance and Spread concept when applied to real life. For one thing, you can't just add Quadrances: say I know that points A B and C are on a line, in that order. Say I also know the distance AB and BC, then the distance AC = AB + BC. But if I measure using the Quadrance, sqrt(AC) = sqrt(AB) + sqrt(BC) would hold. He is just passing the buck in his "redefinition". While making things simpler for trigonometry, he is making things more difficult for other bits of Euclidean geometry.
Similar, while traditionally angles can just sum, his definition using a spread cannot sum angles.
In summary, his "getting rid of sines" is just replacing it with its valuation, which is hardly novel: it is something taught standard in American classrooms (remember SAHCOHTOA?).
Engineers also speak PDE, only in a different dialect.
is to define the triangle as consisting of three sides: A (Adjacent), O (Opposite) and H (Hypotenuse. Then :
Sine S = O / HCosine C = A / H Tangent T = O / A
Now these are easy to remember, since they echo the name of the famous American Indian Chief SOHCATOA. [Big "Aaaargh!" from trig class occurs here, but too late, the mnemonic invariably sticks, even in the dullest mind].
The inverses of the above three are in order: Secant, Cosecant and Cotangent.
Now you have memorized all six basic trig functions. All trig manipulations and formulae are results of shuffling the above ratios to your heart's content.
Upon seeing the above, I ceased memorization of all but the simplest trig formulae, since any complex result can be derived from the above and some algebraic manipulation. My trig teacher was surprised but accepted my use of this methodology on both tests and homework.
Engineers use Laplace transforms in order to turn ordinary differential equations into algebraic equations all the time. There are many transform systems that do this for various kinds of ODEs and PDEs. They all start with the observation that the eigenfunctions of analytical equations are the differential and integral operators, which are duals.
The principal benefit is not so much in simplifying calculations, but rather in that spread may be specifiable in contexts where angle is not.
and figure out the derivatives that way.
As a linear algebra professor of mine put it "Don't memorize these--by the time you're my age you'll be senile and won't remember them anyway. Instead, learn how to do them so that you can always work them out in the future."
Integrate Keynote and LaTeX
You don't actually want your code to be doing that many difficult calculations per second, you just want a big table that tells you what the cosine of 23.2 is in a single memory operation.
The ______ Agenda
Beat the horse some more, will you? This joke is not funny anymore, and when it's not even vaguely related to the subject at hand, it's even worse. So utterly, completely, devastatingly boring.
What makes it so sad is that people actually modded you up.
While I don't see why it wouldn't be possible to rewrite the fourier sine series and such in this notational system, I don't see how it would make it any easier to work with.
Integrate Keynote and LaTeX
tan(1.5469913 rad) = 41.9999989. Close enough?
From TFA:
Divine Proportions: Rational Trigonometry to Universal Geometry opens up new areas of research not only in Euclidean geometry, but also in algebraic geometry, number theory, combinatorics, sprecial functions, Lie theory and non-Euclidean geometries.
This guy is clearly nuts if he thinks that! not only are sin and cos as natural as can be (cos(t) is just the real part of exp(it)), most of those fields would not be affected, except maybe "sprecial functions", which sounds made-up.
The toad can't burp - and for some reason can't fart either, so it swells up and eventually explodes. --Anonymous Coward
Well, I took that example from a Paul Graham essay, so blame him. ;)
Please, for the good of Humanity, vote Obama.
sin(x) = [e^(ix) - e^(-ix)]/(2i)
cos(x) = [e^(-ix) + e^(ix)]/2
Not using transcendental functions is another matter... Interesting looking book.
Excuse me, but that's a load of stinking brown matter.
Some people, myself included can not memorize things to save our lives. Period.
I spent my entire school career being labeled Lazy and disinterested, simply because I have a memory that approaches the consistency of swiss cheese and forgot all kinds of important things like assignments and subjects we just studied.
Your assertion of disinterest is a rude and close- minded reaction to something you obviously don't understand, and as such I would suggest you stop labeling people according to your ignorance.
There is a very real thing people suffer from called a learning disability - perhaps you should go memorize some facts about it.
As someone who graduated from an Engineering University...
Sines and cosines are everywhere. If you don't teach it to them in such a way that they understand it, you will be doing them a *horrible* disservice even if they only go so far as to take Calculus I. I can't tell you the number of times I've had to do "remedial trig" for someone in an introductory physics class because it wasn't taught properly the first time they studied it.
That hurts just in Physics I. It hurts worse in something like Surveying, Material Science, or--the gods forbid--electrical engineering.
Integrate Keynote and LaTeX
I'll be making a visit to your homepage soon. This is really useful information for someone like me who is going to start Calculus for the first time next week.
Yup, contractors have tried that with me, hoping I wouldn't figure out that even though I'd paid for 1000 sq ft, they only delivered 700. I needed to know trig formulas to get it right.
Okay, maybe there's some use for this, but the author has a really screwed up idea of what's allowed to be "fundamental". He says that angles are no good because they require calculus to define precisely. Putting aside the fact that you could come up with a working approximation (to any precision) of the standard definition of "angle" using standard tools like the diameter of a circle and angle bisectors, does that mean that we can't teach "speed" because it requires differential calc, and we can't teach "volume" of anything that's not a rectangle, because you can't prove it without integral calc?
Really, this has the same tone as hundreds of years of mathematical quackery: I will now revolutionize mathematics by employing THE GOLDEN ANGLE of the COSINE BETWEEN THE PLANETS as revealed to me by JESUS 2500 YEARS AGO.
I think it was Ann Landers who wrote once that smart people talk about ideas, average people talk about events, and dumb people talk about other people.
What the hell is that? I started reading the first chapter. OK, maybe there's something mildly interesting here; some calculations could be simpler expressed in these terms. But alarm bells went off when I read "decimal number plane" (let alone everything about how this will revolutionize mathematics).
He seems to mean real plane, but with a pejorative connotation of needing decimal numbers to do ordinary trigonometry.
Google the phrase (in quotes); you get exactly one hit - this book.
Sure, so anyone can derive a bubble sort but they generally use this kind of bubble sort.
for x = 0 to 10
for y = 0 to 10
(Mathematically the sort is of the order x^2)
But with a bit of match knowledge you can do this kind of bubble sort
for x = 0 to 10
for y = x to 10
Which is of order (x * (x + 1)) / 2
almost twice as quick, just with a little maths.
Now how many programmers who don't know maths would be able to derive a quick/heap/merge sort from a binary search using first principles? A quick sort has an order around log2(x) * x which is usually many orders of magnitude quicker than a bubble sort.
And that's just something simple like a sort, for instance could you write a good hashing algorithm for a hash table?
Programmers who know maths are more often then not going to write faster, less buggy code and are capable of proofing why there code is faster, and less buggy.
thank God the internet isn't a human right.
I'm not a mathematician and I didn't read his book, though I read the 1st chapter: interesting, but...
I think it is obvious why the Ancients cared a lot about distances and why distances are more intuitive. If I want to drive back to Kansas, or if Achilles wants to go to Thebes, he cares about distances, even if there might be a detour to Troy (NY that is... hence a triangle). Ok, I can perhaps swallow that quandrances are a computational convenience at times -- I don't like square roots either, but...
Most of his thesis actually rests on the notion that angles are not fundamental. The spin (hah!) that he uses involves the Greeks loving circles, but circles aren't the most basic object. Maybe not geometrically, but... there's that important practical notion of TIME... which then ties into MOTION...
I would argue that what is fundamental is CIRCULAR MOTION. It is intuitive that many things in our world rotate and pivot: wheels, planets, joints, eyeballs, AND... VECTORS, from which one gets the use of sines in describing waves, etc.
I simply don't see us replacing angles with spreads for many uses, because of ubiquitiousness of rotational motion.
I've used match to prove that a fault wasn't just down to noise or running 12 AM - 12 AM instead of 12 AM to 11:59PM. It didn't help fix the fault, but it did prove that the fault was real. I also used math to show that the error was less than 5% so the results were still statistically significant even if they weren't 100% accurate.
thank God the internet isn't a human right.
Your hinting at Euler's Identity...
exp(i*x)=cos(x)+i*sin(x)
if x = pi
exp(i*pi)=cos(pi)+i*sin(pi)
exp(i*pi)=-1+0
exp(i*pi)=-1
Amazing.
This has the downside that it can't distinguish the sign of an angle, but you get a lot from the convenient identity that sin^2(x) + cos^2(x) = 1.
Thus, the normal double angle formula, sin(2x) = 2 sin(x) cos(x) turns into spread(2x) = sin^2(2x) = (2 sin(x) cos(x))^2 = 4 sin^2(x) cos^2(x) = 4 spread(x) * (1-spread(x)).
Really, it's all exactly the same math, just squared.
Personally, I prefer to do everything in terms of complex numbers cis(x) = cos(x) + i * sin(x), which is the antilogarithm of the angle. Then everything becomes beautifully simple. cis(a+b) = cis(a)*cis(b).
Very odd, last night (after watching the Threshold premiere) I had a conversation with my friends.
Essentially, my point was that in all of these sci-fi shows, "Math is the universal language" seems to be taken for granted. My thought was: why?
Granted that any spacefaring civilization would have to have certain capabilities when it comes to math (plotting intercepts with planets, especially in other systems requires certain prerequisites), I said it seems reasonable to assume that any advanced civilization would have to have some knowledge of trig. But -- why would they necessarily have developed the concept of sine, coosine, etc? We don't know how math could develop when 'people' take a radically different approach to it (i.e., by races not 'standing on the shoulders' of our mathematical predecessors).
I proposed that a race could probably create a trig system that didn't have a concept of sine, cosine or tangent, by working with / directly manipulating angle values, etc.
And then the next day this article goes up. Odd.
But I think it does prove my point that Math As The Universal Language might have to be revisited. =)
Just when I've been getting set to help my daughter out with "The New Math," along comes "The New New Math."
Gods don't kill people, people with gods kill people.
I also tend to use an approach that's light in trigonometry to many of the problems that others solve with trig. You can do nearly everything you need to with dot products, cross products, ratios, and algebra.
That said, I do use trigonometric functions quite a bit in generating unit vectors.
I also do think that the relationship between polar and rectangular coordinates is one of the most beautiful things in simple math. It's stuff that everyone takes for granted, but sines and cosines are just beatuiful functions. They seem to say something fundamental about the way space is. And that the same function that relates these two coordinate systems arises from exceedingly-simple differential equations - that describe simple systems of springs and masses or subatomic particles - is just spooky to me. My favorite (albiet not-very-efficient) algorithm for drawing circles remains to use the simple parametric form, and to compute the sine and cosine for each point not by using Taylor expansions (as standard library functions do), but by Euler-integrating from the previous values...
Does no one ever read far enough to learn that the real, CORRECT answer is 54? /F
Stupidity... has a habit of getting its way.
Scientific types aren't the only ones who use trig; a couple weeks ago a friend of mine wanted to draw a perfect pentagram on his guitar. We didn't have a a compass or protractor, just rulers and a calculator. It was fun determining the points on the diagram; we hadn't used trig in years, but it really wasn't hard, just basic sines and cosines.
This also reminds me of M.C. Escher; his repeating tiled diagrams (like the reptiles one) were all based on tiling the plane with geometric figures. I imagine he used trig and other math quite extensively in his work.
If you wanted to teach trigonometry using Euler's equation you would have to teach complex numbers ahead of time so that pupils would be comfortable with them.
:-)
That is probably a good idea any way. For example, turn 2+i into a right-angled triangle, two wide and one high, draw on squared paper, cut out with scissors. Do the same for 3+2i and 4+7i.
Note that (2+i)x(3+2i)=2x3+3i+4i-2=4+7i.
Place the two smaller triangles side by side to add the angles. Place on top of the big triangle and see that the angles are equal.
I think that children could grasp that there is a funny way of multiplying pairs of numbers that makes angles add up. Whether their teachers could cope with complex numbers is another matter
Time to teach them some quantum-physics' math!
|| Geshem ||
-- Premise : school in Italy starts when you're six, you take five years of primary school, three of middle school, and three or five of high school. You must have done five years of high school to go to university, and usually a course is 3 (~Bachelor's Degree) + 2(~Master Degree) years long. Doctorate takes another 3yrs at least.
In Italy trigonometry is more or less "compulsory" at the 3rd class of every high school conceivable (private schools aren't a lot, and statal school are required to teach it), so _every_ student has to face it sooner or later (since you _must_ go to school until you're at least 16 in Italy, soon it'll be 'til 18).
By the way, I've always found trigonometry a bit sterile. We are used to do geometrical and algebraic (simple) theorem demonstrations at high schools here, now _that_ opens your mind, imho.
You usually ends the fifth year working with integration and differentials.
Note that here in Italy you cannot choose what subjects you want, you know all the subjects you're gonna attend in the 3/5-years-long program when you start high school.
At least Math, Italian, English and History are always there.
Chemistry, Physics, Latin, Phylosophy, Law & Economy, Biology, French and/or Deutsch, Greek and Geography are often there too (not one or two of those at once, the _most_ of those), plus of course PE.
I never understood the American model much. We're brough up to try and follow the biggest number of subjects while we're young, so we can better decide what to do of our lives, and we can adapt to different situation without being tied to a specific work / situation.
How you can do without a strong Math course at high school, is beyond me and most Europeans.
42.
Parent may be "4, Interesting", but nonetheless is factually incorrect.
He didn't replace distance with angles, nor is one of the two "the fundamental element of trig" - they are together the fundemental elements of (standard) trigonometry.
What he actually did was that he replaced distance with distance squared and angles with sine squared.
Not that there's anything particularly mentally challenging in forcing people to rote-memorize a set of equations and other numbers.
That's just the way trig is taught... but you can learn trig in a historical context starting with chords of a circle, then the idea of a unit circle, and then a half chord (i.e. sine), and then the cosine.... and it all makes logical sense, because that was the order in which it was developed.
And instead of making students memorize the trig identites, they should be taught how to derive them. Most of the derivations are pretty neat.
Most kids hate math, because most of the math teachers hate math and don't really understand the underlying beauty of it... and they just teach it as memorizing a set of formulae and equations to get high standarized test scores.
There are 10 types of people in this world, those who can count in binary and those who can't.
ok...His method is good, and his work is good( Me, degree in Math )...
But a few things:
Trig without sines is impossible. even he does not avoid it. Sine is simply the function created by the ratio of the length of adjacent line segment, to the hypotenuse in a unit triangle as the vertex traverses a unit circle. It is possible to define some special values as rational, but in general, the function is not continuous through the rational numbers, therefore it is NOT a rational function.
It DOES remain in the range of transendental, so it is a transdentential function.
The Square root of 2 is also a transendential number, and the square function is also a transendental number, hence any system of mathematics based upon sines can be called a transendental system, and any system of mathematics based upon square roots is also a transendental system. You may gain simplicity by removing sines from your system of analysis, but you will not make it rational, until you also eliminate square roots. Hence its name 'Rational Trig' is not rational. ( Rational in the sense of using only ratios ).
BUT: The methods of HIDING the functional definitions of sine, cosine, forces the student to learn the essential concepts behind their definition. ( similar to the definition of multiplication as repeated addition, and powers as repeated multiplication )
Degrees is only a measure of convenience, and no serious treatment of proofs involves it.
All this leads to a rather novel treatment of the study of angles and their related line segments.
"If in order to make progress we must leave reality, by all means, lets leave reality!" - Steven Brooks.
Mathematics, being the purest of sciences, easily lends itself to new and innovative treatments. Its still the only thing you can study for 12+ years, and still NOT cover ANYTHING done since the seventeenth century.
Yours,
Arthur
But why o why doesn't logic get taught in school? The concepts of propositional logic, and even predicate calculus are sufficiently simple to even learn in the very first years of high school. You can apply this to language and even teach those 'alpha' kids to recognize invalid reasoning. No, what we get at first is 'math', primarily aimed at calculations on the real axes, not the root of math, valid and exact reasoning. Math is necessary, but without logic it lives in a void.
By not teaching logic as a proper subject in school, the politicians have already won!
gah, should have previewed the subject, corrected above.
Telling a bunch of nerds how to remember some trivial math actually concepts qualifies for a +5 interesting?
Mod me flamebait but that must be some mind bogglingly fucked education.
The method doesn't matter, as long as the answer correct in base 13.
-
- - You can't take something off the Internet! That's like trying to take pee out of a swimming pool.
So you're smarter than the prof, and you figured out how to play the game.
But, you settled for lower marks because you didn't want to play anymore? And you blame them?
Let me tell you a little secret from someone who's out of school (a good school) and been in the real world for a while.
It's the same fucking game everywhere, don't give them what they ask for, give them what they actually want. Being right or wrong is mostly irrelevant, fufilling your "customer needs" ie prof, boss, customer, wife. is what brings sucess.
Is basically: sin^2(A), where A is the angle between the lines. Basic trig identities then says that this is (1-cos(2A))/2. His argument that circles and angles are non-intuitive, though, is a bit of a stretch. I think we intuitively understand rotational symmetry. His argument that you can apply this theory to other fields is true, except for the fact that the spread assumes that a^2+b^2 != 0 when a or b !=0. But that doesn't hold in lots of fields. What sense does it make to have a geometry that applies to Z/3 but not Z/5? I'm really not sure. It can't be used for complex numbers. What he's basically saying is that you can do a lot of this stuff without ever taking square roots. I guess that's true, and it is quite a nice alternate view on the trig world, but I don't think it is a good way to teach the subject.
In Sydney, Australia.
I have done programming involving coordinates and trig from time to time - originally, stuff like finding where a line is clipped by a polygon. I can remember and do SohCahToa, but that was close to the limit of my knowledge of trig.
The big problems that I found, while trying to write the code, were positive versus negative angles, infinite-angle of vertical lines, and having to calculate a lot of square-roots.
I found that two principles were a great help...
- Like the man says/implies, if distance-squared works as well as distance, use it; you avoid a square-root calculation.
- Express angles as a pair of numbers dx/ds and dy/ds (change in x and y as you move along the line).
The second point eliminated a lot of if-statements and similar but not quite identical code (if both angles are positive..., if angle A is positive and angle B is negative..., etc.)"When the going gets weird, the weird turn pro" -- HST
on page 14, example 1.5 The classical approach "does" provide the choice of solving the example problem algebraically and accurately. For example, x is the length of A1B. Using the cosine law once for the triangle A1A3A2 and twice for the triangle A1A2B: 4^2=5^2+6^2-60 cos(alpha) -----(1) d^2=5^2+x^2-10 x cos(alpha)-----(2) x^2 = 5^2 +d ^2 -10 d cos(45 degree)-----(3) we know cos(45 degree)=sqrt(2)/2, from equation(1) cos(alpha)=3/4 equation (2)+(3)=> 3x=20 + 2 sqrt(2) d -----(4) plugging equation (4) into equation (3) will give similar algebraic equation of the "rational solution"
Even if it has low relevance to what most people do with their time, trig has great value as a quick proxy of whether someone paid attention in school and has a few brain cells. Ask somebody 3 quick and easy trig/calculus/stats questions in a job interview for any position in any company -- somebody who gets it might wonder why you asked such a trivial question, but I assure you from personal experience that you'll weed out a lot of no-talent ass-clown candidates. Seriously, do you want someone to be an employee in your company who can't bring even simple analytical tools to the table?
why
Once I showed people the nature of their statement/position, I said, bring all the lawyers you want, my friends are engineers...
End of discussion and bs.
He uses a right angle to define spread.
The complaint about redundant information with the suplemental angle is nonsense.
And he limits himself to triangle geometry.
I really doesn't buy that argument about it beeing more natural. Why should I square the pages of a triangle? And he talks about special cases only shortly after complaioing about suplemental angles, and he also adds ac/ob.
And how should young kids learn geometry?
How does he intend to handle circular things?
What the finite fields means is a mystery to me.
The example on page 14,15,16 clearly shows how simple the proposed method is.(irony).
Inaccurate with the classical way? If he did it the classical way without calculating any of the approximate values, like the alpha angle, he would not only find the "mysterical" sqrt(7). He would also find sqrt(2).
Nice though to get the second solution.
Finaly. He's just making a fool out of himself. He complains about the flaws in the classical way, but gladly uses perpendicular things, and other intuitive things.
It might be of interest as a curiosity, nothing else. There is no value in this for calculations unless it handles all the fields where trig things are used.
Evolution of Language Through The Ages: 6000 BC : ungh, grrf, booga 2000 AD : grep, awk, sed
Sine = opposite / hypotenuse
Cosine = adjacent / hypotenuse
Tangent = opposite / adjacent
We were taught to memorize this mnemonic.
So, a good test of does this lead to new insights would be him showing us how to do digital signal processing or the like with the new system, and see if anything improves, right? Pretty hard to compute spectra without trig functions, although there are other orthagonal basis that do other interesting things. I don't see how this improves on the other and more useful usages, other than making it a bit easier to pass some test in high school.
I just bought a calculus 100 textbook. it was the cheaper softcover version. only $116 dollars!!!
Jesus.
Am I the only one that figured this out in high school?
~/ssh slashdot.org ssh: connect to host slashdot.org port 22: too many beers
Ask any vector graphics program (Adobe Illustrator, Corel Draw, etc, etc) to generate an outline around some text and you will rapidly see the limitations of conventional trigonometry. Increase the width of the outline and/or the complexity of the text and sooner or later the maths will blow up. .....!
A few years ago my software house needed a subprogram to create paths offset any chosen distance from another 2D path. (Necessary for machining in the sign-making industry.) I fondly imagined this was half a day's work for a clever visiting student.
Alas, no, it turned out to be a 3-month coding nightmare. Finding the precise intersection of two nearly parallel vectors (expressed as lines, circle arcs, or Bezier curves) is surprisingly difficult, within the limits of precision and time set by computers. You end up dealing with special case after special case.
In ignorantly fumbling towards a better way of expressing the calculations, I got as far abandoning angles and using quadratures. If only Rational Trigonometry had been around at the time
but i think sin and cos aren't done in hardware on even modern processors. The difference is that the floating point operations required to do them to the accuracy required are much faster nowadays.
note: i'm known as plugwash most places but i screwd up registering that here somehow in the past and now can't register
Wrong. There are things that cannot be proved true or false in any formal system sufficient for arithmatic.
"It is not how things are in the world that is mystical, but that it exists." -Ludwig Wittgenstein
Eh, I sign my shit manually out of habit. Sometimes I manage not to, but I'm so damned used to it that it's automatic by now.
I hadn't really thought of the meta-moderation bit. Honest.
Laws do not persuade just because they threaten. --Seneca
The angles given for the 4,5,7 triangle on page 1 of chapter 1 are incorrect to the number of digits specified. These values are quoted as
:-) j/k
theta_1 ~= 33.92 deg
theta_2 ~= 102.44 deg
theta_3 ~= 43.64 deg
But the accurate values (according to my trusty HP48SX) are
theta_1 ~= 34.05 deg
theta_2 ~= 101.54 deg
theta_3 ~= 44.42 deg
Maybe the author explored alternatives to traditional trigonometry, because he was not good at traditional trigonometry?
damm I just finish calculus1 and 2... too bad I didn't knew this... I dunno if my teacher would of agreed the use of i though... oh well I,l use it in my physics class
Here's the part I don't get: the apparent idea is that we keep everything simple by keeping everything well within the realm of rational numbers. That's great, but how do we pull this off? Many things have an irrational number somewhere. Does this version of trig just let us somehow mask over the irrational numbers in an intermediate step so we don't have to look at them?
... poof! infinite accuracy and marshmallow goodness :-) )
It's great, I guess, but from what I'm reading, I somehow doubt this is any kind of a real solution. I was taught right-triangle trig first, then circular trig later. I liked circular trig better, but at the same time, I'm not sure the average Joe on the street should be forced to go through all that. Oh, but in any case, circular trig, and all that early Calculus stuff would have gone along *much* more smoothly if only the teachers had, had some frikkin' animations actually *showing* how the size/shape of the right triangle inscribed within the Unit circle changed as we moved along the circle. Animations would have also been helpful for concepts like "as x approaches y" and Reimann sums (see how the number of rectangles gets ever larger as their width gets every smaller and we have increasing accuracy? now, we wave our magic wand and "take the limit"
Calculus concepts that took me months and *months* to get my head around could've been "gotten" in mere days or weeks if the concepts had been demonstrated with little videos.
Furry cows moo and decompress.
This dude is a professor at the New South Wales University - in Sydney, Australia.
South Wales is somewhere in the UK, and I don't know if they even have a university there.
It's only one word missing but it makes half a world of difference =)
Specialist Mac support for creative pros, Melbourne
There's a big difference between the two. Wildberger wants to make it easier for the average student to learn basic trig (which is the study of triangles, not circles, afterall).
;)
:) He'd probably tell you that he invented numbers and that they're alive and only he understands them properly or something to that effect 8-)
Consider his analogy of his system being like switching from Roman to Arabic numerals. You can get right answers doing addition/subtraction/multiplication/division with Roman numerals, its just a pain in the ass
Wolfram... well, he's just nuts
...you're probably not doing your job as well as you could.
Seriously. I work in IT and study graduate CS, all of it in discrete, not continuous math. I don't have to do any continuous math.
That said, applying principles of continuous math routinely saves me time, and familiarity with it routinely helps me notice things that I'd otherwise miss entirely.
I could get by without trig, differential calculus, and even linear algebra. I'm sure many of my classmates and coworkers do. That's why I'll be getting promoted and getting my degree faster than they will.
There's no failure quite as dissatisfying as a complete and total solution to the wrong problem.
The only way this has a chance of taking off is if it is legislated as mandatory that all newly manufactured equipment and devices that perform measurements of this sort (including any that are imported for domestic sales), must conform to this system (which isn't going to happen, of course).
File under 'M' for 'Manic ranting'
C'mon, it'll be easy as pi! (That was my math joke. Is pi taken, yet?)
Old Hippies Are High On Acid...a much more memorable way for me...haha...a very good method for memorizing certain formulas is making a rhyme
twinkle twinkle little star...power equals I^2 R!!
I personally found a way to display all elipses and circles 2d/3d in my game withou any high level math functions (like cos/sin). But I use geometrical way to discover these algorithms not a pure math way. And these are based on subdivisions and creasings, very nice and ultra quick, with only multiplications used. Is is how nature itself do this, no calculus or trigonometry LOL. Same goes for physics..
Nature itself do not use calculus or differential equations to solve those problems.
But this is pretty sweet. All he's doing is saying instead of talking about distances and angles, let's talk about distance ^ 2 (quadrance) and sin^2 (spread). Then just find the old usual trig equations in terms of these new parateters, but they're greatly simplified, and lots of the old ambiguities are gone. The equations now generalize to all other fields. It's awesome and it's intuitive.
The guy who'd written it was 'trig happy'
Actually, he was simply incompetent and didn't pay attention in math class; this stuff is being taught.
If this book encourages people to use trig less it might be a good thing.
This isn't a question of "encouraging" or "less". Any computer programmer dealing with geometry should know how to use the minimum number of trigonometric expressions, error and roundoff properties, argument ranges, etc.
The problem he gives as an example in the sample chapter can be solved without resorting to any trigonometric functions at al. Just use Pythagora's theorem a couple of times plus the fact that the sides of a isoceles triangle are equal. You then solve a 2nd degree equation and voila. This wouldn't have worked if the angle wasn't 45o, but then his method wouldn't have yielded a neat result either. My guess is that any problem that can be solved in closed form by his formalism can be as easily solved using run of the mill geometry.
Of course, his method may turn out to be more intuitive, but I'll reserve my judgment on that.
(I am sleepy so I might have missed it...but did he mention that spread=sin^2 or no?)
o s^2(a)=4sin^2(a)(1-sin^2(a))=4spread(a)(1-spread(a ))
From the definition on pg 6...if you look at the example, the sin of BAC is |BC|/|BA|, so if you square that value, you get the rule for the spread.
The example on page 8 uses this 4-5-7 triangle and calculates the spreads s1=384/1225, s2=24/25, s3=24/49
The law of sines states that the ratios sin(angle_i)/oppSideLength_i are all equal...so we have sin1/L1=sin2/L2=sin3/L3, and if we square these values, we get
(sin1/L1)^2=(sin2/L2)^2=(sin3/L3)^2
and if we assume that spread=sin^2, (and, of course Q=L^2 by definition...) then the formula turns in to
Spread1/Q1=Spread2/Q2=Spread3/Q3
which is the spread law defined on page 10.
Again on page 10 that example of the "triple spread" formula has S2(s) = 4s(1-s) which is interesting in that it's the logistic map, but recall the double angle formula from regular trig:
sin(2a)=2sin(a)cos(a).
If we set spread(a) = sin^2(a), then
spread(2a)=sin^2(2a)=(2sin(a)cos(a))^2=4sin^2(a)c
The example saying that spread(60)=3/4, spread(45)=1/2, spread(30)=1/4...it's well known that the identities for the sin's of those angles are sqrt(3)/2, sqrt(2)/2, 1/2 resp...so if you square those sin values, you get 3/4, 1/2, 1/4 which are the spreads.
It appears to be redefining trig in terms of the squares of the trig functions instead of the trig functions themselves. That might make it easier to do certain things and may make it easier to learn the material. But, I'm not sure...since this gets to the problem of "It's easy to calculate variance but standard deviation is the thing that's useful in the real world..." However, maybe there are times and places where it would be easier to convert to this stuff to make calculations easier before going back to Euclidean space.
Best. Comment. Ever. Enjoy!
Is this a common practice?
Publisher:
http://wildegg.com/about.htm
From his personal site:
"Wild Egg is a new, small publisher of high quality mathematical texts. I am the director of this fledgling outfit, and hope to establish in the years to come a spare but illustrious line of mathematical texts that break out of the usual mold. The first offering will be Divine Proportions: Rational Trigonometry to Universal Geometry. hopefully due out in September 2005, and available over the internet at http://wildegg.com./"
Douglas Calvert
It sounds like you missed the Alan Sokal affair a few years back.
He basically pulled together a bunch of philosophical jargon, made some stuff up relating it to quantum mechanics, loaded it with red flags for anybody with a minimal knowledge of physics and had no trouble getting it published in the journal Social Text. He even wrote a book critical of philosophers misusing physics, and did it in french because he thought the worst culprits were francophone. You can read all about it here: Sokal affair
Of course, there are also scientists who could use a little refresher in math, too. One of my favorite papers is some psychiatrists who were inadvertantly testing the equivalence principle in a study on clozapine and weight gain. It was reported that clozapine causes weight gain, and they proposed that it might also cause an increase in body mass index (BMI). BMI is defined as: m/h^2, where m is weight in kilograms and h is height in meters. If you read the paper they weren't suggesting that clozapine affects your height. Abstract available here: Clozapine and Body Mass Change.
The amazing thing is that the reviewers didn't at least make them change the first two lines of the abstract.
My first thought from the parent post (the part you quoted) wasn't numerical solvers (approximators if you like), but programs like Maple and Mathematica, which can symbolically produce the same solutions that one normally associates with pencil and paper calculus. There have even been pocket calculators capable of some of this since at least the late 80s.
No floating point involved at all unless you want it to plug in some values at the end and pop out a number.
Say I turn left twice with each turn having a spread of 1/4: then we can use a 'double-angle formula' to calculate the total spread:
sin 2x = 2 sin x cos x
so total spread = (sin 2x) squared
= 4 * 1/4 * (1 - 1/4)
= 3/4.
You'd get the same answer by thinking about angles, of course.
Would high level math on chip make programs and computer do more thing in less cycles or more accuracy or efficiency compare to add/sub multi/divide and simple algebra?
Is it faster for a computer to do short cuts in calculations instead of pure Mhz in number crunching.
would it make my games more interesting?
It would not have been hard for me to draw (or find a drawing on the Net) an x/y axis cross, a circle and a triangle, and put the appropriate labels on them. Starting from there the explanation is one short paragraph.
However, I can't be bothered. There are plenty on the Net already. Both ASCII art, pretty drawings and interactive demos (Java applet). You may have to search a little longer for a truly minimalistic and concise answer to your question, but noone is stopping you.
We have to remember that the author is a mathematician!
He doesn't like the real numbers (i.e. including transcendental numbers, crucial for sin/cos) because they're hard to define formally -- which in his mind leads to confusion. So he wants to avoid this by keeping everything confined to algebraic numbers. This lets you avoid defining uncountable sets, etc.
This is a shame for CS folks, since uncountable sets, and the proof that there are more real numbers than natural numbers, are so important to computer science and undecidability!
(Sadly most high schoolers never really see the proof that there are more real numbers than natural numbers... but that's another problem...)
hooray for math! wheeeeeee!
I studied mathematics with Dr. Wildberger at the University of New South Wales (not South Wales University).
His classes were well thought out and engagingly presented. Although the link to his book is slashdotted at the moment, I'm sure that his take on trigonometry is both elegant and interesting.
"Where is the wisdom we have lost in knowledge, and where is the knowledge we have lost in information?"-T.S.Eliot
Actually, every calculator I've ever had didn't use base 2, but base 10. And there are vast swaths of numbers that simply aren't available.
Finally a conversation where my numerical analysis class paid off!
A friend of mine's children were having trouble with their math homework, and she, not being a math whiz, was having trouble with this. She called me up, and asked me to help. At first, when I heard the problems, I was like "WTH?" - because her son was in the the 4th grade (albeit in some advanced math program), and this stuff looked like some test in a Mensa book. Once I saw what was going on, I saw what was being done - what I don't understand is why don't they just use what is standard...
You see, the math being taught to her son was the equivalent to simple algebra - but it didn't look like it at all. Take this "fictional" problem (the problems looked like this, but this isn't an actual problem - although it might be able to be worked out - it is here for illustrative purposes only): Circle plus square equals triangle, triangle minus square equals circle, circle times three equals triangle (imagine the appropriate symbols and shapes on a page). Find the values for circle, square and triangle (for the purposes of the problem, circle, square, and triangle must be numbers between 0 and 9).
Now, at first this baffled me - but then I saw that they could have (and more importantly, should have) used standard symbols like x, y, and z to represent the geometric shapes instead. Furthermore, they could have taught the rules of algebra in how to manipulate and solve for x, y, and z - like you learn in algebra. Strangely, I tried to show this to her son, and he couldn't get it - he was telling me that all they (his teachers) showed was plugging in numbers until they fit. Huh?
Here they were, teaching something akin to algebra, using trial and error to fit numbers in. It makes me wonder, given the complexity of the problem being shown - if the actual learning exercise did involve algebraic formula to solve, and was supposed to be taught this way - but the teacher either couldn't get her fourth grade class to understand this, or she didn't understand it herself! I will probably never know the answer, but it would seem to make sense to me to teach the kids to use the standard symbols (of letters and such) that they will be using once they move on to real algebraic notation in middle and high school - rather than substituting in symbols that are more "fluffy" for kids.
In fact, in the latter half of the 19th century and the early part of the 20th, this is exactly what they did - I have an old math "textbook" written for grade schoolers (and not "advanced placement" kids, either) which teaches not only algebra, but geometry and a bit of calculus as well. It doesn't take much to realize that today's children (in America, at least) are getting the short end of the proverbial stick when it comes to their education...
Reason is the Path to God - Anon
Oscar / Has = Sin (opposite over hyp)
A / Heap = Cos (Adjacent over Hyp)
Of / Apples = Tan (Opposite over adjacent)
He uses these formulas and simply doesn't ack the angle. Big deal. If all you have is an angle and a side, trig is still the best way in town to deal with it. Other than that, he seems to use standard tricks of dealing mathematically in Eucledian geometry that we have always used. *Yawn* Wake me back up if I missed something.
Are you sure you took that seriuosly enough?
It's about time someone proposed a way to teach maths so that by the time we teach trig we can teach it like they teach poetry, where as currently we teach it more like grammar.
Trig and exponential are beautiful, but are seems a hideous due to the way they are taught.
Teach this as early as can be understood and trig as early as it can be understood. They are definitely not the same audience.
for a second there I actually thought about what you were saying and it's screwing up my nicely memorized trig equations.
GENERATION 26: The first time you see this, copy it into your sig on any forum and add 1 to the generation.
How does simply giving people what they really want make you unhappy?
Personally I am quite happy knowing that people are satisfied with my work, I would actually be unhappy if they were constantly dissappointed.
The fact that they might not express exactly what they want is a reality of the world we're in. Pretending communication problems don't exist won't help you and won't help them.
differentiation is easy because there are simple rule for differentiating functions of functions and products of two functions so if you know how to differentiate the individual functions you can differentiate basically any combination of them.
with integration there are no such rules so integration consists of guessing what methods to use and hoping you get to an answer which may not even exist. or you know the answer from past differentiation of something else.
are there integrations that have been done by a human but can't be done by programs like mathematica. I'd guess the answer is probablly a yes though i don't know for sure as i've never used mathematica or studied really advanced maths.
note: i'm known as plugwash most places but i screwd up registering that here somehow in the past and now can't register
While we're at it, we should refer to all distances in base pi. That would make things a LOT simpler. For example, the area of the unit circle would be 10, and its circumferene would be 20. So many things could be simplified this way. I think I'll suggest this idea to the author, and maybe I can share in his Nobel Prize!
Any sufficiently simple magic can be passed off as mere advanced technology.
yes, that is a pretty constructive method.
I'm still trying to figure out what people mean by 'social skills' here.
.... is not part of living in the real world, yeah, you are absolutely right.
Many people will experience the thrill of problem solving for a very few precious years while they are students. After that life will be a routinary exercise in which the knowledge of the ancient is not used at all.
My goodness, trigonometry is deeply embeded in the history of civilizations and culture. No wonder we repeat history's mistakes: we are always finding excuses to avoid learning something tha "will not be useful".
What a shame that there are people so materialistic. While I go through my bills I may not need to use trigonometry, but heck, I remember fondly how well it felt to solve thos problems. It gave me a sense of achievement, a sense that I was cleverer than I thought.
But nooooo, it is not useful. But repeating "do you want ketchup with that" in a neutral English accent is I suppose.
IANAL but write like a drunk one.
There are descriptions of the functions, but (if memory serves, someone please correct me if I'm wrong) you'd be hard pressed to find a closed-form solution. (I think they are described as infinite series that converge.)
As you say, they are "the ratios of the various sides". More specifically, for example, given a right triangle, the cosine of an angle theta is the ratio of the adjacent side over the hypotenuse, or
cos(theta) = A/H
but still you never really know how the cosine function works. (i.e. Given an arbitrary angle, you would be hard-pressed to be able to calculate its cosine by hand.) The explanation of the relationship the classical trig functions describe gives you something of an intuitive feel, but it's kind of a pseudoexplanation--it tells you what it does, not how it works.
Black box.
Thanks to mensan for posting here.
I've taken a first pass at designing protractors to measure spread:
http://www.ossmann.com/protractor/
The only reason angles are easier to measure is that we have lots of angle measurement tools. All we have to do is build spread measurement tools to make spread just as easy. From playing with the protractors, it seems to me that it would be just as easy to learn to eyeball spread as angle, with the possible exception of lines that are very close to parallel or perpendicular.
No, the correct answer is 10000000. Each term has only one significant figure, so after truncating to the correct precision you get the calculator's displayed answer.
Well, you're both right actually, depending on how you arrived at those numbers. The correct answer is 10000000.00000001 if they are exact numbers. It is 10000000 if they are measured quantities. The concept of "significant figures" is a way of accounting for measurement error, which obviously only applies when things are being measured in an imprecise manner. (If you don't believe that, I encourage you to try telling your bank that 5% of $251 is $10 instead of $12.55.) Since the original poster didn't use scientific notation, it's most likely that he meant the numbers as exact. But then again, I imagine you knew that and were just trying to be clever. Better luck next time.