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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?"

Actually, it does look like just a tangent of traditional trigonometry. After reading the first chapter, most of his math seems to be the switching forms of the Pythagorean theorem from:

(a^2 + b^2)^(1 / 2) = c

to:

a^2 + b^2 = c^2

With a lot of just applying that paradigm to every aspect of trig. Pretty nifty time saver, but I fear the unique insights from this method may be few.

Re:No sines and cosines? (0)

Anonymous Coward | about 9 years ago | (#13584193)

Sort of like the "few unique insights" of Einstein looking at Maxwell's equations in a "simplified" manner.

Re:No sines and cosines? (0)

Anonymous Coward | about 9 years ago | (#13584232)

Whenever you needed to get an answer in trig class, could you give it in the form of "c^2" or did you have to take the square root? My teachers always wanted the non-squared version... thus, the practical form of the Pythagorean Theorem that I actually used was "(a^2 + b^2)^(1 / 2) = c." I really only meant that as a base comparison between this and traditional trig.

Although, you could argue that changing the form thusly leads to the creation of the convoluted mathematics behind sine and cosine, et al. You know, I think that's Dr. Wildberger's point. Heh =]

Cool (0, Insightful)

Anonymous Coward | about 9 years ago | (#13584114)

Actually, i think this is a perfect point. My major problem when learning integral calculus was my utter loathing for trigonometric identities (and my inability to remember them) required for solving all sorts of weird integrals. I'm curious how this stuff plays with calculus.
If it does, maybe this'll make my life easier if i ever go back and attempt calculus again.
anyway, reading TFA, hopefully it says something regarding this:)

This has already been done by a man named Karl Weierstrass [wikipedia.org] 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*.

Re:Now ... (-1, Flamebait)

Anonymous Coward | about 9 years ago | (#13584293)

Been there, done that - it's called Laplacian Transforms.

Mark Edwards -- Proof of Sanity Forged Upon Request

The method doesn't matter, as long as the answer (3, Funny)

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.

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.

Re:Don't worry... (0)

Anonymous Coward | about 9 years ago | (#13584256)

I dunno... I recently wanted to replace my four-by-three panel with a widescreen one of the same height. Given I was going to order online, so couldn't just look at them and get a feel for what was right, trig came in quite handy.

I think it kind of depends on what you do for a living. If you work at Wal-Mart, you probably don't need it. If you are and Physicist or Engineer (like my wife and I), you probably do. And thats hard to know in High School

Not just physicists or engineers use trig.... (4, Insightful)

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....

Everyone complains how trig is not useful, but perhaps its useful because it is hard and is one of the few things left in schools today that actually mentally challenges students.

I take it you taught math in elementary school (K to 5th) then, as your point is completely wrong. For a physicist or computer scientist, the principles of trigonometry are invaluable. All those games you might play. All those electronic devices (cell phones, tv, etc) you use on a daily basis. Much of the theory used to devise how they could possibly work was done with *gasp* trigonometry and to a greater extent, calculus.

Simply because you choose a profession does not use it, does not mean it doesn't have value.

Things that you don't actually need in the real world:

Trigonometry Calculus Physics Biology History Geography Square roots Imaginary numbers Graphs IT Foreign languages Algebra

Things like that should be taught at university to people who actually want to learn them, not school. I've never used any of those things outside of school, ever.

If you'd R'd TFC then you would know that spread is a unitless quantity. It is a ratio between two quadrances (lengths squared), and as long as the quadrances are homogenous with respect to their units then they cancel out.

I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.

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?

Anonymous Coward | about 9 years ago | (#13584177)

Listen brother if you are failing math then engineering is not your thing. Switch to a less math intensive dicipline, like culinary arts.

Re:Wow (-1, Troll)

Anonymous Coward | about 9 years ago | (#13584226)

When you say you can "out code some of the instructors", I'm assuming this is essentially a competition that involves you and your instructor, but your instructor hasn't been informed that he's participating?

I think this is a classic case of beginner programmer. I can hack some c code, so I must be brilliant.

If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.

All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.

This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.

Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.

That's an old idea, largely discredited... from the days when the math teachers where the computer science teachers.

Programming isn't maths.. maybe simple algebra, but it's a lot more about creativity and logically solving problems. I've been programming professionally now for 15 years and never needed more than rudimentary maths knowledge - nor can I imagine any situation when I would actually need it.

Algorithms are worth learning, but algorithms aint maths either.. they're just the 'known best' way to solve problems.

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.

Oooo come on. This will not make things any more especially when you can't memorize with subjects like differntial equations. Those equations you have to memorize what to do on top of figuring out what type of equation to use.

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

How about "People don't read" ?? Though this does assume that editors are people.

Maybe "People can't think" is better. But I am leaning more to "People can't critically think". If they did ..editors would edit.

Re:UNSW .. not South Wales (0)

Anonymous Coward | about 9 years ago | (#13584269)

While your statement should be correct (IMHO), you don't really need to critically think to observe that blatant error. A simple "diff" would show the obvious mistake, and it doesn't take more than the mind of a trained monkey to do this.

Re:UNSW .. not South Wales (0)

Anonymous Coward | about 9 years ago | (#13584275)

The editor and submitter probably just found it easy to believe that nothing of intellectual consequence comes out of Australia... as do most people, to be honest.

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?

Re:Hopefully (0)

Anonymous Coward | about 9 years ago | (#13584208)

1. Yes - sine and cosine are just expressions of the ratio of x&y for the unit circle. Redefine your angle as a vector from the origin of length 1 in a certain direction, pick out your X and y and divide... Duh.

2. huh? I got trig before calculus. It's not that hard. Hard is applying it to probability, physics, engineering dynamics, chemistry, etc. The language of math and solving stuff is way easier than figuring out what is the right math to solve.

I just read the sample chapter. It is not
dumbed down. Its concept of "spread" is closely tied to how angles are viewed by mathematicians on abstract manifolds. I.e., the only property of the angle that really matters is its cosine. Essentially the angle is "named" for its cosine. This works both in simple Euclidean space, multi-dimensional Euclidean space, AND in spaces with non-euclidean metrics. So learning trigonometry in this way will make understanding of subjects like Tensor Calculus much more natural. It's a good attempt. But, of course, it must be carefully ironed out before we start comitting generations of children to this as the world view.

I can do C, C++, and Java, and have written a few MUDS with minimum functionality (They were all in C -- part of the reason I switched to C++ and then to Java), but I never got past Algebra in High School. I can't do Trig. Calc makes me want to die.

I guess I could just say, "I fucking hate integration!" Who's with me?

You are right, it is either sines and cosines or square roots and fractions (I am guessing here, it may be something else)...in any case a lot of algebraic computation is rather unpopular among pupils:(

It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).

The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group theory, is generally not needed." (to use this method)

I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.

Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?

I just read the first chapter of his book, and while I agree that he does away with calculus for dealing with angles (no more sines and cosines) one of his sample solutions ends up with the square root of 7. Now how do you evaluate that with out a recourse to numerical method?

Which is why you break down complex angles into half-angle and double-angle identities, which lets you get the precise answer (with lots of addition of irrational radicals, but whatever)... Take the funky angle sin 22.5 for a simple example. Your calculator will tell you that the answer is 0.38blah. However, since you know that cos45 = sqrt(2)/2, and you know the half-angle identity for sin is sin(x/2) = sqrt((1-cosx)/2), then you know that sin(22.5) = sqrt((1-sqrt(2)/2)/2) = sqrt(2-sqrt(2))/2. Using repetition of this method, you can find out the exact values of very small angles, which you can then add together to find precise values of larger angles. Of course, it takes a long time, but how is that worse than finding the "spread" of an angle, particularly in curves, where you need the angle itself to calculate arclengths?

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.

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...)

Lousy analogy (0)

Anonymous Coward | about 9 years ago | (#13584166)

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.

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.

If you were a programmer that relied on an implimentation that used traditional trig, you would understand why 'redefining' the route to the correct answer to use simple algebraic expressions would be a good thing...precision.
I am a computer graphics enthusiast and I dwell in alot of 3d math that involves calculus (mainly all sorts of complex curves). Guess what, that crap all likes trig! If I can define the formula for a three dimensional sphere without trig, thank you, thank you, thank you.
I'm gonna go read this book when it comes out.

Absolutely. Just think of all the 3d math that goes on behind the scenes of games today. All those lookup tables can be kissed goodbye.
I wonder if this method would actually be faster (even if it were easier to implement) than traditional lookups? Has Carmack read this yet?:-)

Quaternions (0)

Anonymous Coward | about 9 years ago | (#13584294)

Spread is not proportional to angle at all. If you read the pdf summary, he states that.5 is 45deg while.5 and.75 are equal to 30 and 60deg respectively. The relatiosnship is not linear, but would plot as a curve with an inflexion at 45 degrees.

There are 360 degrees in a circle because there are 365 (point whatever) days in a year.
The ancient Greeks were more primitive than we are today; lacking computers, they couldn't manage a simple off-by-one error, and had to fall back on the less sophisticated off-by-five-and-a-long-decimal error.

anybody remember the chant: SOH CAH TOA (1, Funny)

Anonymous Coward | about 9 years ago | (#13584155)

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)

It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'.

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.

I am hoping that he is refering to a Theorem: which is a statement that can be proven via logic. An Axiom otoh is something that is simply generally accepted to be true.

Look up the definition [answers.com] of axiom. One of the basic axiom's we use everday is that our memory of the past is accurate, and that events we remember will influence the present and the future.

SOHCAHTOA and abstract survery results (4, Insightful)

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)

I am wondering if this could be used to make faster calculations in raytracers and 3D engines by using integer numbers.

Non-Linear Angles (1, Interesting)

Anonymous Coward | about 9 years ago | (#13584203)

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.

Re:Faster calculations ?? (3, Interesting)

Anonymous Coward | about 9 years ago | (#13584228)

AFAIK most 3D engines already use tables with values for different angles and extrapolate for faster trigonometric calculations since you don't need that much precision in a game anyway

SOH CAH TOA. SOH (Sine(x) = Opposite / Hypotenuse, Cos(x) = Adjacent / Hypotenuse, Tan(x) = Opposite / Adjacent where x is an angle of a right angled triangle)

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!

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.

The math is correct and the expressions he is using are really just a rational parametrization of the circle - you can find two rational expressions x=R(t) and y=Q(t) that will trace out a circle as t goes along real line.

However, people use angles for a reason - the angle variable just measures the arc cut out on a circle of unit radius. So, it would be rather hard to compute angular momentum in terms of spread, or say, define a periodic monochromatic signal (for which you are much better off using sin() and cos()

The most important point is that the worse thing a teacher can do is dumb down the material - the students feel it and are a lot less interested. If someone has trouble accepting trigonometric functions as is, teach them algebra of complex numbers (useful also as miniature version of vector algebra) and then explain Euler's formula.

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?

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.

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.

## No sines and cosines? (5, Funny)

## Joey Patterson (547891) | about 9 years ago | (#13584108)

## Re:No sines and cosines? (3, Insightful)

## biryokumaru (822262) | about 9 years ago | (#13584149)

(a^2 + b^2)^(1 / 2) = c

to:

a^2 + b^2 = c^2

With a lot of just applying that paradigm to every aspect of trig. Pretty nifty time saver, but I fear the unique insights from this method may be few.

## Re:No sines and cosines? (0)

## Anonymous Coward | about 9 years ago | (#13584193)

## Re:No sines and cosines? (0)

## Anonymous Coward | about 9 years ago | (#13584232)

## Re:No sines and cosines? (3, Interesting)

## SilverspurG (844751) | about 9 years ago | (#13584243)

What did I miss?

## Re:No sines and cosines? (1)

## biryokumaru (822262) | about 9 years ago | (#13584309)

Although, you could argue that changing the form thusly leads to the creation of the convoluted mathematics behind sine and cosine, et al. You know, I think that's Dr. Wildberger's point. Heh =]

## Cool (0, Insightful)

## Anonymous Coward | about 9 years ago | (#13584114)

## Now ... (3, Funny)

## LordKaT (619540) | about 9 years ago | (#13584115)

## Re:Now ... (0)

## Anonymous Coward | about 9 years ago | (#13584128)

## Re:Now ... (4, Interesting)

## NoTheory (580275) | about 9 years ago | (#13584161)

## Already done (0)

## Moderator (189749) | about 9 years ago | (#13584239)

## Re:Now ... (-1, Flamebait)

## Anonymous Coward | about 9 years ago | (#13584293)

Mark Edwards

--

Proof of Sanity Forged Upon Request

## The method doesn't matter, as long as the answer (3, Funny)

## PtrToNull (742886) | about 9 years ago | (#13584117)

## Re:The method doesn't matter, as long as the answe (1)

## helioquake (841463) | about 9 years ago | (#13584220)

Is that what you wanted?

## Wonderful! (5, Insightful)

## h4rm0ny (722443) | about 9 years ago | (#13584118)

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.

## Don't worry... (3, Insightful)

## tgd (2822) | about 9 years ago | (#13584196)

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.

## Re:Don't worry... (0)

## Anonymous Coward | about 9 years ago | (#13584256)

## Re:Don't worry... (4, Insightful)

## anderm7 (68050) | about 9 years ago | (#13584271)

## Not just physicists or engineers use trig.... (4, Insightful)

## Ellis D. Tripp (755736) | about 9 years ago | (#13584312)

## Re:Don't worry... (5, Insightful)

## Dr_LHA (30754) | about 9 years ago | (#13584281)

## Re:Don't worry... (4, Insightful)

## PocketPick (798123) | about 9 years ago | (#13584296)

Simply because you choose a profession does not use it, does not mean it doesn't have value.

## Re:Wonderful! (-1, Flamebait)

## drsquare (530038) | about 9 years ago | (#13584311)

Trigonometry

Calculus

Physics

Biology

History

Geography

Square roots

Imaginary numbers

Graphs

IT

Foreign languages

Algebra

Things like that should be taught at university to people who actually want to learn them, not school.

I've never used any of those things outside of school, ever.

## Figures. (5, Funny)

## Musteval (817324) | about 9 years ago | (#13584119)

## Yeah (1)

## Evanisincontrol (830057) | about 9 years ago | (#13584120)

## Units? (1)

## mrhale (872484) | about 9 years ago | (#13584121)

## Re:Units? (4, Insightful)

## TheRaven64 (641858) | about 9 years ago | (#13584292)

I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.

## Re:Units? (1)

## tantrum (261762) | about 9 years ago | (#13584304)

deg, grad, rad and WILD guess what i'd like to use on my calculator

## The "New" has an initial capital for a reason (3, Informative)

## Bewbewbew (871127) | about 9 years ago | (#13584123)

## Wow (3, Interesting)

## Loconut1389 (455297) | about 9 years ago | (#13584124)

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?

## Re:Wow (1)

## DarkPixel (570153) | about 9 years ago | (#13584170)

## Re:Wow (0)

## Anonymous Coward | about 9 years ago | (#13584177)

## Re:Wow (-1, Troll)

## Anonymous Coward | about 9 years ago | (#13584226)

I think this is a classic case of beginner programmer. I can hack some c code, so I must be brilliant.

## Re:Wow (5, Insightful)

## lobsterGun (415085) | about 9 years ago | (#13584230)

If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.

All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.

This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.

Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.

## Re:Wow (0)

## Tony Hoyle (11698) | about 9 years ago | (#13584283)

Programming isn't maths.. maybe simple algebra, but it's a lot more about creativity and logically solving problems. I've been programming professionally now for 15 years and never needed more than rudimentary maths knowledge - nor can I imagine any situation when I would actually need it.

Algorithms are worth learning, but algorithms aint maths either.. they're just the 'known best' way to solve problems.

## Re:Wow (1)

## technoextreme (885694) | about 9 years ago | (#13584274)

Oooo come on. This will not make things any more especially when you can't memorize with subjects like differntial equations. Those equations you have to memorize what to do on top of figuring out what type of equation to use.

## UNSW .. not South Wales (4, Informative)

## OzPeter (195038) | about 9 years ago | (#13584126)

## Re:UNSW .. not South Wales (1)

## Gothmolly (148874) | about 9 years ago | (#13584136)

## Re:UNSW .. not South Wales (1)

## OzPeter (195038) | about 9 years ago | (#13584206)

Maybe "People can't think" is better. But I am leaning more to "People can't critically think". If they did .

## Re:UNSW .. not South Wales (0)

## Anonymous Coward | about 9 years ago | (#13584269)

## Re:UNSW .. not South Wales (0)

## Anonymous Coward | about 9 years ago | (#13584275)

## Hopefully (3, Insightful)

## JasonEngel (757582) | about 9 years ago | (#13584127)

## Re:Hopefully (0)

## Anonymous Coward | about 9 years ago | (#13584208)

2. huh? I got trig before calculus. It's not that hard. Hard is applying it to probability, physics, engineering dynamics, chemistry, etc. The language of math and solving stuff is way easier than figuring out what is the right math to solve.

## Re:Hopefully (1)

## superwiz (655733) | about 9 years ago | (#13584222)

is notdumbed down. Its concept of "spread" is closely tied to how angles are viewed by mathematicians on abstract manifolds. I.e., the only property of the angle that really matters is its cosine. Essentially the angle is "named" for its cosine. This works both in simple Euclidean space, multi-dimensional Euclidean space, AND in spaces with non-euclidean metrics. So learning trigonometry in this way will make understanding of subjects like Tensor Calculus much more natural. It's a good attempt. But, of course, it must be carefully ironed out before we start comitting generations of children to this as the world view.## huh? (1)

## vapor22 (410851) | about 9 years ago | (#13584129)

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.

## Re:huh? (1)

## PakProtector (115173) | about 9 years ago | (#13584168)

I can do C, C++, and Java, and have written a few MUDS with minimum functionality (They were all in C -- part of the reason I switched to C++ and then to Java), but I never got past Algebra in High School. I can't do Trig. Calc makes me want to die.

I guess I could just say, "I fucking hate integration!" Who's with me?

## Re:huh? (1)

## labyrinth (65992) | about 9 years ago | (#13584195)

## Re:huh? (1, Insightful)

## siplus (796514) | about 9 years ago | (#13584229)

## Re:huh? (0)

## Anonymous Coward | about 9 years ago | (#13584246)

I guess I could just say, "I fucking hate integration!" Who's with me?The Ku Klux Klan. I've heard they prefer segregation to integration.

## Re:huh? (1)

## promatrax161 (913597) | about 9 years ago | (#13584180)

## Re:huh? (4, Insightful)

## HateBreeder (656491) | about 9 years ago | (#13584189)

It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).

The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group

theory, is generally not needed." (to use this method)

I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.

Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?

## Re:huh? (1)

## OzPeter (195038) | about 9 years ago | (#13584278)

## Re:huh? (1)

## Roguelazer (606927) | about 9 years ago | (#13584291)

## Just Wait... (4, Interesting)

## DataPath (1111) | about 9 years ago | (#13584133)

## Re:Just Wait... (1)

## Loconut1389 (455297) | about 9 years ago | (#13584153)

*fires up vi*

## Better LInk (1, Redundant)

## OzPeter (195038) | about 9 years ago | (#13584145)

## Re:Better LInk (1)

## OzPeter (195038) | about 9 years ago | (#13584154)

## new era? nah.... (1)

## cryptocom (833376) | about 9 years ago | (#13584146)

: )

(actually...i would probably say that if there WERE an alternate route...but...eh...)

## Lousy analogy (0)

## Anonymous Coward | about 9 years ago | (#13584166)

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.

## Redefinition? (3, Insightful)

## AndreiK (908718) | about 9 years ago | (#13584150)

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.

## Re:Redefinition? (2, Insightful)

## DarkPixel (570153) | about 9 years ago | (#13584205)

## Re:Redefinition? (1)

## AndreiK (908718) | about 9 years ago | (#13584215)

## Re:Redefinition? (1)

## ab8ten (551673) | about 9 years ago | (#13584255)

I wonder if this method would actually be faster (even if it were easier to implement) than traditional lookups? Has Carmack read this yet?

## Quaternions (0)

## Anonymous Coward | about 9 years ago | (#13584294)

## Re:Redefinition? (5, Insightful)

## sameerd (445449) | about 9 years ago | (#13584213)

spread is the square of the sine of an angle.

## Re:Redefinition? (1)

## ab8ten (551673) | about 9 years ago | (#13584235)

## Re:Redefinition? (1)

## AndreiK (908718) | about 9 years ago | (#13584308)

## Why are there 360 degrees? (1)

## Colin Smith (2679) | about 9 years ago | (#13584259)

## Re:Why are there 360 degrees? (2, Funny)

## Skirwan (244615) | about 9 years ago | (#13584285)

## Re:Redefinition? (1)

## thrashbasket (880168) | about 9 years ago | (#13584303)

## anybody remember the chant: SOH CAH TOA (1, Funny)

## Anonymous Coward | about 9 years ago | (#13584155)

(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.

## Re:anybody remember the chant: SOH CAH TOA (1)

## AndreiK (908718) | about 9 years ago | (#13584167)

## Re:anybody remember the chant: SOH CAH TOA (1)

## eyebits (649032) | about 9 years ago | (#13584181)

## Re:anybody remember the chant: SOH CAH TOA (1)

## freewaybear (906222) | about 9 years ago | (#13584251)

## Re:anybody remember the chant: SOH CAH TOA (1)

## mikael (484) | about 9 years ago | (#13584187)

## Re:anybody remember the chant: SOH CAH TOA (1)

## drooling-dog (189103) | about 9 years ago | (#13584266)

## Uh... (1)

## Comatose51 (687974) | about 9 years ago | (#13584158)

It develops a complete theory of planar Euclidean geometry over a general field without any reliance on `axioms'.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 newWait... 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.theoryto a wide range of practical problems from engineering, physics, surveying and calculus.Maybe someone much more knowledgable can explain this for me.

## Re:Uh... (1)

## maddhatt (149310) | about 9 years ago | (#13584183)

## Re:Uh... (1)

## gonerill (139660) | about 9 years ago | (#13584201)

the axiomatic system upon which we built all human knowledgeYou think your knowlege of where you parked your car is built on an axiomatic system?## Re:Uh... (1)

## starwed (735423) | about 9 years ago | (#13584301)

Yes; of course it is. ^_^

Look up the definition [answers.com] of axiom. One of the basic axiom's we use everday is that our memory of the past is accurate, and that events we remember will influence the present and the future.

## SOHCAHTOA and abstract survery results (4, Insightful)

## acomj (20611) | about 9 years ago | (#13584164)

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.

## Re:SOHCAHTOA and abstract survery results (1)

## AndreiK (908718) | about 9 years ago | (#13584190)

## Faster calculations ?? (5, Interesting)

## AeiwiMaster (20560) | about 9 years ago | (#13584176)

in raytracers and 3D engines by using integer numbers.

## Non-Linear Angles (1, Interesting)

## Anonymous Coward | about 9 years ago | (#13584203)

With his method you can't just add angles line that. You have to do an elaborate calculation.

## Re:Faster calculations ?? (3, Interesting)

## Anonymous Coward | about 9 years ago | (#13584228)

## fractional numbers? (1)

## snoig (535665) | about 9 years ago | (#13584178)

## Three words (0, Redundant)

## DrXym (126579) | about 9 years ago | (#13584179)

Remember them and trigonometry is a doddle.

## Re:Three words (1)

## DarkPixel (570153) | about 9 years ago | (#13584247)

## Interesting - but not entirlely new (4, Insightful)

## caffeined (150240) | about 9 years ago | (#13584192)

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.

## New meanings from now on.. (1)

## mayhemt (915489) | about 9 years ago | (#13584223)

## Re:New meanings from now on.. (1, Funny)

## Anonymous Coward | about 9 years ago | (#13584280)

## Great for eighth grade, but ... (5, Insightful)

## levin (170168) | about 9 years ago | (#13584225)

## easier for computers? (1)

## El_Muerte_TDS (592157) | about 9 years ago | (#13584244)

Would be very nice to have a performance boost at the math level for 3D calculations.

## Several points: (1)

## Ruie (30480) | about 9 years ago | (#13584253)

## I don't see how this is "easier" (4, Insightful)

## Curmudgeonlyoldbloke (850482) | about 9 years ago | (#13584254)

## Re:I don't see how this is "easier" (1)

## freewaybear (906222) | about 9 years ago | (#13584287)

## Great, but (1)

## RAMMS+EIN (578166) | about 9 years ago | (#13584265)

## New beginning for maths? (1)

## pedicabo (753738) | about 9 years ago | (#13584273)

## Yes, (1)

## lobsterGun (415085) | about 9 years ago | (#13584290)

OrangeHipposAlwaysHaveOrangeAngleswhich yields...

OppositeHypotenuse = sin thetaAdjacentHypotenuse = cos thetaOppositeAdjacent = tan theta## A bit crackpotty? (1)

## geordieboy (515166) | about 9 years ago | (#13584295)

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.

## Bah!! (2, Funny)

## doi (584455) | about 9 years ago | (#13584310)