ronthepon Posted February 5, 2007 Report Posted February 5, 2007 I've bought a new scientific calculator last week, and I've been experimenting with it ever since. After hours and hours of mindless entertainment, when I bagan to get bored of this, I decided to check the repeatitive limits of functions. Basically, I experimented as performing one function over and over again on any number to see what number it... kinda like stablises upon. Like experimentally solving [math]f(x) = x[/math] So, here I began with every function on the board. One which interested me particularly was the cosine function. Using radians for the angle, I got a number. But I was somewhat surprised. This did not bleed any of the obviousness as any of the other 'limits' I had seen. So I tried again, this time using the Microsoft calculator. After repeatedly pressing the button, the function settled down to this number: 0.73908513321516064165531208767387 I can really not place it anywhere. Where does this solution to [math]\small Cos\theta = \theta[/math]come from? This is probably an irrational number. Where on earth does it come from? What is it's exact value? What group of other irrationals constitutes this number? Quote
ronthepon Posted February 5, 2007 Author Report Posted February 5, 2007 What I've come across seems to be an example of a fixed point for a function. (See? You can learn from your games as well) Here they mention the fixed point for the cosine function. Apparently, the fixed point for the cosine function is "asymptotically stable" or "superstable", meaning that it moves towards this point upon repeated application of the function. For the sine function this point is zero. The easiest result I got during play time. [math]Sin\theta = \theta[/math] [math]\theta=0[/math] Easy to guess. But how the hell do you come at the fixed point for the cosine function? And what could be the exact value for the number? Quote
Farsight Posted February 5, 2007 Report Posted February 5, 2007 It's just fiddling with functions, ronthepon. Here, have a decko at this: http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/3.5-3.8.pdf Look out for the cobweb. Quote
Pyrotex Posted February 5, 2007 Report Posted February 5, 2007 ...But how the hell do you come at the fixed point for the cosine function? And what could be the exact value for the number?rtp,You are close to discovering Chaos. :) I used to do the same thing back in the 70's with my brand-new TI SR-58, programmable calculator. Then I started using more than one function--I would slap in any number and hit:COS SQRT COS SQRT COS SQRT ... or even,1/X + 2 = LN 1/X + 2 = LN 1/X + 2 = LN 1/X + 2 = LN ... I also noticed that these (and other) function strings would often close in on fixed values. From the book by James Gleick, Chaos, Making a New Science, it was just this idle playing with calculators that led to the discovery of what eventually became called, the first chaos equation. Fixed link!In a previous post, I submitted an Excel spreadsheet which can iterate this loop of calculator button pushing 1,000 times in a fraction of a second, allowing you to actually "see" raw Chaos in action. Quote
Mercedes Benzene Posted February 5, 2007 Report Posted February 5, 2007 Is your calculator on "radians" or "degrees" mode?That could make a difference. Quote
ronthepon Posted February 6, 2007 Author Report Posted February 6, 2007 Here, have a decko at this: Not been able to open up your link, popular; internet connection can't handle all the info at one time I think. What was the thing it was about anyway? You are close to discovering Chaos. :)Chaos huh? Seems interesting. But help me, I've still not found the spreadsheet. And, er, the result showed up both for radians, that's what would make sense. Another number appears, though in the degree system. I will now try to crack this equation. Does anyone know how to solve this? [math]Cos\theta \. = \. \theta[/math] Quote
Farsight Posted February 6, 2007 Report Posted February 6, 2007 Sorry Ronthepon, now it's gone funny on me too. It was calculus, a section on conic sections along with hyperbolas and parabolas. Quote
Qfwfq Posted February 6, 2007 Report Posted February 6, 2007 I can really not place it anywhere. Where does this solution to [math]\small Cos\theta = \theta[/math]come from?If you want a hint, try pondering on the equation: [math]\cos\theta = \sin\theta[/math] This is probably an irrational number.I'd say it's also transcendental Quote
Pyrotex Posted February 6, 2007 Report Posted February 6, 2007 Is your calculator on "radians" or "degrees" mode?That could make a difference.If you were talking to me, then of course it does. like tacking on a multiplicitive factor. The loop will assymptotically approach a different fixed point. If you were talking about cos theta = theta, then I might point out that all trig functions in Calculus and higher math are, by default, expressed with radians, unless you specifically include the radians to degrees conversion. As far as I know. Quote
Pyrotex Posted February 6, 2007 Report Posted February 6, 2007 ...Chaos huh? Seems interesting. But help me, I've still not found the spreadsheet....I fixed the link. Quote
ronthepon Posted February 6, 2007 Author Report Posted February 6, 2007 My, that's a really brilliant idea, Pyro! I've never overused my computer's power more, but this is so cool! Just tell me how to change the limits of the graph... please... I wanna mess with it so badly! If you want a hint, try pondering on the equation: [math]Cos\theta = Sin\theta[/math]That's normally easy, I'd straight out say theta corresponds to [math]\frac{\pi}{4}[/math], but I can guess that's not what I'm supposed to do in order to figure this. I've been tinkering with the infinite series, but I don't seem to be getting any where. Quote
Pyrotex Posted February 6, 2007 Report Posted February 6, 2007 My, that's a really brilliant idea, Pyro! I've never overused my computer's power more, but this is so cool! Just tell me how to change the limits of the graph... please... I wanna mess with it so badly!.... :hyper: :fly: :confused: uuuhh... what idea?what graph?sorry, I'm confused. :( Quote
Pyrotex Posted February 6, 2007 Report Posted February 6, 2007 cos( theta ) = theta I tried to replace cosine with its Taylor's expansion out to the 4th term, but the resulting equation is too difficult for me to solve. Quote
Leo Posted February 7, 2007 Report Posted February 7, 2007 The calculus book that Popular linked to a chapter of which is maybe more accessible there: MIT | Textbook Publications | Calculus by Gilbert Strang Quote
ronthepon Posted February 7, 2007 Author Report Posted February 7, 2007 :confused: :hihi: :confused: :confused: :confused: uuuhh... what idea?what graph?sorry, I'm confused. :(The excel sheet ofcourse! And thanks, Leo, I'll go through it. Quote
Qfwfq Posted February 7, 2007 Report Posted February 7, 2007 That's normally easy, I'd straight out say theta corresponds to [math]\frac{\pi}{4}[/math], but I can guess that's not what I'm supposed to do in order to figure this.Actually, that's a start :) but I guess I made it kind of tricky. What's the sine and cosine of [math]\frac{\pi}{4}[/math]? I'm not talking about an actual calculation, just a rough comparison that requires a bit of imagination.... Quote
ronthepon Posted February 7, 2007 Author Report Posted February 7, 2007 Actually, that's a start :) but I guess I made it kind of tricky. What's the sine and cosine of [math]\frac{\pi}{4}[/math]? I'm not talking about an actual calculation, just a rough comparison that requires a bit of imagination....Uhuh... I think I get what you are driving at. The geometrical significance, right? Like [math]sin\theta = cos\theta[/math]When the non-hypotnuse sides of a right triangle are equal. Hmmm... That way, we could imagine a circle for the [math]\theta[/math]... and assosicate a triangle for the [math]Cos\theta[/math]. Quote
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