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Complex numbers

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Hey again I''ve just started studying at the TU Delft in the Netherlands. We started with math concerning complex numbers. We''ve got transform stuff to polar coords, solve functions that isn''t possible in high-school math. But I was wondering, is there any practical example of complex numbers? My guess is that there are many examples in physics, but I can''t find any information whatsoever. Thanks!

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to clear up slightly what Cedric submitted, Quaternions are related. Real numbers are a subset of complex numbers which are a subset of quaternions, I believe, and the list goes from there.

I''ve just started in complex variables as well, so I don''t know a whole lot of practical uses yet, but one that I''ve seen relates to some older study of physics, specifically special relativity. Using complex numbers artificially, it is possible to transform the non-euclidean spacetime coordinate axes to something that resembles a more classical idea of geometry, although from what I know, this interpretation is not regarded in the highest light. So in conclusion, I have no idea what they''re used for, but I''m learning all about them.

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Complex numbers are often used to simplify calculations in physics. For me at least, it is mus easier to treat an oscillating function by using the complex exponatial than dealing with sin and cos (multiplication becomes easier with complex numbers). In quantum mechanics the wave function is treated as a complex quantity, daming of an harmonic oszillator can be treated by using a complex frequency,...
But always, the quantity that can be measured is a real number.

There are a some more topics in physics, where complex numbers are usefull, too much to remember them all at the moment.

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quote:
Original post by Divide
But I was wondering, is there any practical example of complex numbers? My guess is that there are many examples in physics, but I can''t find any information whatsoever.


This doesn''t seem to be related to game development exactly, but its not homework either. It is an interesting question, and I believe there is some use for complex numbers in game development. So I''ll allow the thread to remain open. In fact, I''ll offer my thoughts in reply.

Complex numbers are often used in control theory, for simulating the motion of robots, airplanes, automobiles, and other objects in motion. So this is one practical use. Typically, you''ll find them used/referenced in classical or linear control theory. Although games often simulate nonlinear dynamics, there are applications of linear dynamics that can be based on linear control, such as some flight simulators (arcadey ones, not hard-core ones), and some vehicle/racing simulators (again, arcadey ones, not hard-core ones). In the engineering world, linear control theory is used to predict instabiliities when operating vehicles near their design condition, for example, determining if an airplane is stable or unstable when it is basically flying straight and level at cruise. A simple analysis can determine whether either of the "Phugoid" or "Short Period" longitudinal/pitch oscillation modes is unstable, for example, due to the center of gravity being too far aft.

Complex numbers also show up in simulating airflow. There are two occurances that I can point out. First, conformal mapping techniques use complex math to map the 2D flow around a circular cylinder to the flow around a complex airfoil shape. Look up "Joukowski airfoil" in google and you''ll find links to this. Second, some boundary layer stability theories use complex math to predict boundary layer separation, which is a key component causing airplanes to stall. (Look up, for example, "parabolized stability theory".)



Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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But I was wondering, is there any practical example of complex numbers? My guess is that there are many examples in physics, but I can''t find any information whatsoever.

Paradoxically life is just a lot simpler when you use complex numbers. I think it was David Hilbert who said "the shortest path between two truths in the reals often leads through the complex."

The whole of quantum mechanics is described in a complex vector space. There is an equivalent formulation in terms of real vector spaces, but it is far more complicated.

And Cauchy''s Residue Theorem which you''ll probably learn in second year makes it possible to do about a billion integrals that would be hideous otherwise.

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quote:
Original post by sQuid
And Cauchy''s Residue Theorem which you''ll probably learn in second year makes it possible to do about a billion integrals that would be hideous otherwise.


An uncountably infinite number I''d imagine . On topic: I did an ocean wave simulation recently, and it used (Fast) Fourier Transforms with complex numbers. It''s described quite well here I think.

Miles

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Complex numbers also arise in the study of electrical circuits and signals. As has been hinted at, many mathematical problems are more succinctly expressed after transforming them into a complex space.

As to whether complex numbers are practical, that depends solely on whether you have a tough mathematical problem to solve that is made easier/solvable by using a complex transformation.

Cheers,

Timkin

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I find them useful for manipulating 2d vectors. Although it isn''t impossible without them - rotation is just so nice and it is good to know that your methods are ''standard''.

Trying is the first step towards failure.

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Not to forget, fractals are done with them. You take the complex coordinates of every pixel, and let it go through an iterating complex function, if after n steps the pixel is outside a certain radius, that pixel belongs to the fractal, otherwise not (or vica versa). Well that''s probably not that clear but I''ve gotta go, anyway here are some fractals I rendered, it''s not that hard (took me only a few hours and some cursing to get them)

zoomed julia:


zoomed mandelbrot:



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