name_xox wrote:Most 3D programmers or 3D computer Artist will use the 4 Dimensional Co-ordinate of W, X, Y, Z lines or points
The W used in 3D graphics is kind of an artifact of how projections and transformations are done. See here.
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Zagadka wrote:Would it be correct to say that a nominalist viewpoint would be that a number is "of something"... saying 1,748,392 means nothing, but saying 1,748,392 kg means something.
Similarly, pi (or tau if you prefer) is only a thing when it is applied to a circle. Without the circle, it isn't anything.
I can certainly say that there are zero bananas on my desk, and I can understand a negative number in a more abstract way... I can't say there are -1 bananas on my desk (don't ask me why I'm fixated on bananas), but I can comprehend there being one less banana on my desk than could potentially be there. I usually just think of such things on a graph.
I also don't have a problem with sqrt(-1) or other imaginary numbers, since I take the operative word "imaginary" literally. Maybe fictionalism makes more sense there, since it literally doesn't fucking exist ever anywhere, and that may work for more abstract concepts.
Dividing by zero I can imagine since I picture it on a graph
I'm not sure if that is technically nominalism or platonism or fictionalism though.
But I don't think that "1" exists. "1" is specifically 1 of something. You don't have to necessarily mention or define that something, but 1 unit divided by 2 units is half of 1 unit.
Thoughts?
Rancid wrote:An imaginary number is a means to help deal with transitions (or rotations) between positive and negative sides of the number line. We're just adding a dimension to the number line. I'm not so sure that imaginary really mess with other math concepts. The hell would I know though. It sure as hell makes sense to me from an engineering sense.
Smertios wrote:I have been thinking a lot about this lately, and I think this approach to complex numbers is not the most didactic one. Don't get me wrong, you are 100% correct in all your assertions. I just think that explaining how i = srqt(-1) is a better alternative.
Smertios wrote:
I have been thinking a lot about this lately, and I think this approach to complex numbers is not the most didactic one. Don't get me wrong, you are 100% correct in all your assertions. I just think that explaining how i = srqt(-1) is a better alternative.
While it is true that from going to R to C, you are adding one dimension, and from going back from C to R, you are subtracting one dimension; that is counter-intuitive to someone who is learning about number systems. And that's mainly because it gives the false impression that, to go from one set to the other, all you need to do is add or subtract dimensions, and that you can add and subtract dimensions freely.
You can't do that.
In the sequence N - Z - Q - R - C - H - O, the only situation in which you are adding/subtracting a dimension is when you go from R to C (and vice-versa). When you go from R to Q, you are not subtracting a dimension (the rationals don't have 0 dimensions. Similarly, Hamilton showed that you can't have a number with 3 dimensions. The next step after the complex numbers are quaternions, which have 4 dimensions — e.g., (4 + 2i + 7j + 2k) ∈ H.
Also, it's important to notice that C is different from R². In R², you have points formed by a pair of coordinates (x,y) = xî + yĵ, and x and y, or between î and ĵ. In C, you have a number (a,b) = a + bi formed by two coordinates a and b. But there is a clear relationship between i and 1. That is, i² = 1.
lucky wrote:I much prefer the modern treatment, with proper definitions.
Somebody saying "imagine sqrt(-1) exists even though you already know it doesn't" would just cause a protest and a rejection of the whole idea as bogus from my younger self.
Kind of how I rejected "pretend an infinitely small real number dx exists, even though we all know it doesn't" as dumb when my high school physics teacher tried to do calculus that way. It wasn't until college that I really appreciated the beauty of mathematical analysis because until then it was tainted to me with nonsense notation like that, so I was repelled from the whole subject. Even though I did already know proper definitions of derivatives and such. It is unfortunate that people still teach calculus like that. The whole "dy/dx" notation should be expelled from school books because it only causes a lot of confusion about what it really means.
Harmattan wrote:Hence why "imaginary" is a very fitting name: it has no physical importance.
Harmattan wrote:And I do not see what else you would like to use instead for multivariate functions.
In physics the imaginary part is of no interest in the end result! Never. Hence why "imaginary" is a very fitting name: it has no physical importance.
lucky wrote:Actually, complex numbers are central to quantum mechanics. The Schrödinger's equation is a differential equation over complex functions, there is a factor of i in it.
I'd be more accepting of the df/dx notation if everybody was clear that it's just a funny notation and not an actual ratio of two objects. Unfortunately, they aren't. For instance, they start writing shit like df = x dx. This immediately leads to errors, for instance, it's not true that df/dt = df/dx * dx/dt when f is a function of two variables x and y, which are both in turn a function of t. Thoroughly confusing and unnecessary nonsense.
Rancid wrote:In signal processing, the imaginary component of say an IIR filter is very important and most certainly has physical importance.....
Harmattan wrote:And it could be rewritten to not use complex numbers. For example the Schrodinger equation could be split into two real equations a wave function should obey.
Harmattan wrote:But those are ratios and what you did is actually legit.
lucky wrote:Ha. Of course. Instead of complex numbers, we could be using pairs of real numbers.
when in fact:
df/dt=120 at t=2.
Besides: even if you were getting the right answers by doing this, it's still pointless. Define what you mean by "dt" before you start talking of ratios.
Harmattan wrote:Instead of a complex wave you could use a single real wave. But it should obey two equations instead of one.
Harmattan wrote:Ok, the problem is different from what you have stated but I get your point: there are students who think that f'(t = 2) can be computed by doing (f(2) - f(0)) / (2 - 0).
Harmattan wrote:I really fail to see why there would be a problem to define dt, why it would involve trust into authority or magical rules. It is just a fucking division [...] dt is infinitesimal.
lucky wrote:Show me.
So you're saying dt is a real number and dt is infinitesimal? Define infinitesimal.
Harmattan wrote:The invert of infinite.
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