How would you like to trigonometry without sin/cos/tan?

Post #225,258 by Silverlock 9/18/05 9:58:13 AM Reply	How would you like to trigonometry without sin/cos/tan? [link\|http://www.physorg.com/news6555.html\|Algebra-based Trig] ----------------------------------------- George W. Bush and his PNAC handlers sent the US into Iraq with lies. I find myself rethinking my opposition to the death penalty. --Donald Dean Richards Jr.
Post #225,260 by bionerd 9/18/05 10:31:45 AM Reply	That's just wrong.
Post #225,298 by a6l6e6x 9/18/05 1:38:03 PM Reply	Agreed if does away with Euclid. Studying Geometry is the only contact most people have with the notion of a proof. Alex The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt. -- Bertrand Russell
Post #225,394 by ben_tilly 9/18/05 10:57:41 PM Reply	This is a good thing? My memories of having been taught "proofs" in geometry are not good. Actually they are very bad. Besides, people get the notion that proofs are a big, fancy and difficult thing. I'd rather that people think of proofs as straightforward. A highschool teacher did something fun with complex numbers. He introduced an abstract set of operations with pairs of numbers. Proved that algebra worked. And only later mentioned that (x, y) was normally written x + iy. This worked very well for me. Cheers, Ben PS Note: part of my dislike for geometric "proofs" may be that I fall in the 40% of the population that is not visually oriented. I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,403 by static 9/18/05 11:29:23 PM Reply	I also learnt complex numbers via an unconventional route. This strongly suggests that the "standard" ways of teaching mathematics are highly flawed. I believe you already hold this opinion. :-) Wade. d-_-b
Post #225,466 by mmoffitt 9/19/05 1:22:23 PM Reply	Re: think of proofs as straightforward. I would prefer that, too. The problem is that very few people I've met have the reasoning abilities required to follow a modus ponens argument. bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell
Post #225,525 by ben_tilly 9/19/05 6:35:17 PM Reply	I'll have to disagree with you on that I've met a lot of people with the ability but without the patience. I've met far fewer who could not learn it. In any case, real mathematicians almost never write out formal proofs. The exercise of doing so strikes me as nearly useless. Knowing that you can is useful, but doing it is not. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,598 by mmoffitt 9/20/05 11:18:42 AM Reply	Nit. In any case, real mathematicians almost never write out formal proofs. Except, of course, those who must publish to keep their positions. They do it at least once in a while. At any rate, I'm certainly glad Andrew Wiles did. ;0) bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell
Post #225,652 by ben_tilly 9/20/05 7:18:19 PM Reply	Nit denied The proofs that you see published are closer to vague outlines of how a competent person might construct a formal proof than they are to formal proofs. And yes, that includes Andrew Wiles. Whose first "outline" turned out to have a hole that a competent person could not be expected to be able to fill in, though he managed to come up with a workaround. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,741 by mmoffitt 9/21/05 12:37:40 PM 9/21/05 12:39:10 PM Reply	What's your point? That professors of mathematics never write formal proofs? Or that professors are not "professionals"? Because, CLEARLY, professors of mathematics (at least at Purdue and North Carolina) do write formal proofs and submit them for publication. [Edit]: BTW, where'd you get your M.S. in Mathematics? Presumeably, at a place where none of your professors wrote formal proofs, right? bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell Edited by mmoffitt Sept. 21, 2005, 12:39:10 PM EDT Expand All History
Post #225,760 by ben_tilly 9/21/05 2:49:18 PM Reply	My point is that... a formal proof is a very specific thing, and very few proofs that you'll find in the mathematical literature are formal proofs. If you find this strange, then you either have no familiarity with what real math papers look like, or you don't really know what a formal proof looks like. This applies as much at Dartmouth College, where I got my masters, as it does at Purdue and North Carolina. Furthermore if you talked to a logician from any of those three universities, I guarantee that they would agree with me. All of this is well-known within the profession. It even has shown up on the wikipedia, for instance see [link\|http://en.wikipedia.org/wiki/Mathematical_proof\|http://en.wikipedia....athematical_proof]. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,769 by mmoffitt 9/21/05 3:26:10 PM Reply	Ah. Dartmouth. That explains the arrogance. :-p bcnu, Mikem It would seem, therefore, that the three human impulses embodied in religion are fear, conceit, and hatred. The purpose of religion, one might say, is to give an air of respectibility to these passions. -- Bertrand Russell
Post #225,473 by a6l6e6x 9/19/05 1:59:58 PM Reply	It was for me. I ate 'em up. Therefore, QED! :) Alex The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt. -- Bertrand Russell
Post #225,271 by warmachine 9/18/05 10:57:33 AM 9/18/05 10:59:49 AM Reply	What the hell is spread in mathematics? Matthew Greet Choose Life. Choose a job. Choose a career. Choose a family. Choose a fucking big television, choose washing machines, cars, compact disc players and electrical tin openers. Choose good health, low cholesterol, and dental insurance. Choose fixed interest mortgage repayments. Choose a starter home. Choose your friends. Choose leisurewear and matching luggage. Choose DIY and wondering who the fuck you are on a Sunday morning. Choose sitting on that couch watching mind-numbing, spirit-crushing game shows, stuffing fucking junk food into your mouth. Choose rotting away at the end of it all, pishing your last in a miserable home, nothing more than an embarrassment to the selfish, fucked up brats you spawned to replace yourself. Choose your future. Choose life... But why would I want to do a thing like that? I chose not to choose life. I chose somethin' else. And the reasons? There are no reasons. Who needs reasons when you've got heroin? - Mark Renton, Trainspotting. Edited by warmachine Sept. 18, 2005, 10:59:49 AM EDT Expand All History
Post #225,283 by boxley 9/18/05 11:25:45 AM Reply	the value between the over and under, what, you dont bet? "the reason people don't buy conspiracy theories is that they think conspiracy means everyone is on the same program. Thats not how it works. Everybody has a different program. They just all want the same guy dead. Socrates was a gadfly, but I bet he took time out to screw somebodies wife" Gus Vitelli Any opinions expressed by me are mine alone, posted from my home computer, on my own time as a free american and do not reflect the opinions of any person or company that I have had professional relations with in the past 49 years. meep questions, help? [link\|mailto:pappas@catholic.org\|email pappas at catholic.org]
Post #225,354 by Ashton 9/18/05 7:12:46 PM Reply	Fundamental blindness - Most people are more visually/spatially oriented than symbol-manipulation weirdos (like many of us here). To miss That point re pedagogy, re casually eliding this experience, training -- is, is No verbs; I no words.
Post #225,375 by imqwerky 9/18/05 9:48:06 PM Reply	IOW: "Let's dumb down Math for the Masses" "DIVINE PROPORTIONS" - hmmmmmmm Doesn't that smack of ID to yall? Except this wanker is from Down Unda. Coming attraction: Soon to be in a school district near you! Sined, Amy Pray for the survivors of Bush.
Post #225,395 by ben_tilly 9/18/05 11:02:59 PM Reply	Choose which operations are difficult Some of them have to be. Suppose that I go 30 miles, turn 45 degrees and go another 10. Where am I? The answer is an ugly number, and any method of calculation has to agree on that piece of ugliness. With usual trig, the ugliness lies in computing sin and cos. With his method, the ugliness lies in the difficulty of adding vectors. Knowing both approaches can be good because you get to choose the method that is more straightforward for the problem at hand. But neither is going to avoid the intrinsic issue that geometry doesn't always give nice numbers. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,419 by Ashton 9/19/05 4:48:59 AM Reply	I'd have thought most folks can grok Pythagoras, at least - It is rather intuitive to complete the rectangle figure via the complementary triangle. In your sample, the diagram would need only a line and a triangle + a simple squares calculation. Is the answer 'ugly' because it's not an integer? (Maybe your example is just too practical and/or I've seen enough maps to be unable to tune out the 'visual' of what a 45\ufffd angle change means. Which would make me a lousy tutor for the algebra-only crowd, I guess :( Still, I also recall many people who didn't really grok that those trig 'names' are describing merely arithmetic ratios. And a few cases come to mind where the student could manage the flag-pole measurement via shadows from a nearby stick of known height ie A:B::C:D -- They may also have learned that simple manipulation by rote - not associating the idea of 'ratios' there either! as we would Wish they did. {sigh} It's hard to forget what one has gleaned. Yet teachers Must!
Post #225,519 by ben_tilly 9/19/05 6:23:22 PM Reply	Conceptually the problem is easy With standard trig, it is straightforward, you just have to know the sin and cos of 45 degrees. And the number comes out to be somewhat ugly. With the replacement for trig it isn't quite as straightforward. The "simple algebra" doesn't quite stay so simple. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,553 by Silverlock 9/19/05 9:54:34 PM Reply	So this isn't bogus? I'm not enough of a math geek to understand on a cursory view if this is real or a con job. ----------------------------------------- George W. Bush and his PNAC handlers sent the US into Iraq with lies. I find myself rethinking my opposition to the death penalty. --Donald Dean Richards Jr.
Post #225,559 by ben_tilly 9/20/05 12:17:47 AM Reply	Somewhat Often you want to avoid explicit trig calculations. They are complex, slow, and introduce rounding errors. So having a way to think about trig that avoids them is useful. But push come to shove, there is inherent complexity that you aren't going to get around. Cheers, Ben I have come to believe that idealism without discipline is a quick road to disaster, while discipline without idealism is pointless. -- Aaron Ward (my brother)
Post #225,582 by drewk 9/20/05 9:31:36 AM Reply	Seems to me ... Trig is based on measurements of the triangle itself: what is the length of each side, what are the angles, etc. This method seems to enable doing formulas directly on the coordinates. I could be mis-reading -- okay, I didn't read the whole thing -- but that's what it looked like. === Purveyor of Doc Hope's [link\|http://DocHope.com\|fresh-baked dog biscuits and pet treats]. [link\|http://DocHope.com\|http://DocHope.com]
Post #225,402 by tuberculosis 9/18/05 11:27:05 PM Reply	Just another coordinate system If it makes some things easier, great. "Whenever you find you are on the side of the majority, it is time to pause and reflect" --Mark Twain "The significant problems we face cannot be solved at the same level of thinking we were at when we created them." --Albert Einstein "This is still a dangerous world. It's a world of madmen and uncertainty and potential mental losses." --George W. Bush
Post #225,456 by ChrisR 9/19/05 12:43:51 PM Reply	Maths is not singular There are many, many kinds of maths, some that have been abandoned long ago, and many still waiting to be explored. Personally, I found Trig to be the easiest family of Math that I have ever taken. Based on fairly simple principles - if you understand Pythogorean theory, then all else is derived. And even after you forget the theory, sine, cosine, and tangent tables will serve you quite well. The preference of one form of math over another can only be judged by the type of questions being answered. While I was doing measurement in an engineering fashion, Trig was an immensely useful tool. Not that I'm more into mundane aspects of business software, I find it less useful - though not totally useless. The question should be (a) who finds life harder with Trig; and (b) who finds life easier with Trig. Personally, I think some topics should not be addressed by watering down complexity, but rather the complexity should be appreciated from first principles. If someone finds Trig hard, perhaps the problem is not in Trig, but in how the subject is taught. If you're going to be an Engineer of practically any sorts, then you'd be an idiot not to expose yourself to the rich history of triangles - no matter whether you think it easy to grok or not. Anyhow, Trig is but one of the ways to express certain mathematical principles. It's long and useful history makes it likely it will survive and thrive. Not dismiss the alternatives, but they must not only show they are better, but that they are profoundly and pragmatically better on a very wide and far reaching basis. Like it to Programming Languages - under what circumstances would you adopt a new PL if you had hundreds of years invested in an immense library of proven code?

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