From: | J Y S Czhang <czhang23@...> |
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Date: | Wednesday, April 23, 2003, 4:10 |

>Christophe Grandsire wrote:En réponse à Tristan McLeay : > >>> And what, pray tell, is the purpose of these beasts? What problem can't >>> be solved with real numbers that it requires us to take the sqrt of >-1? > >> A lot! And modern physics just can't be correctly presented without using >> complex numbers. From electricity-magnetism to Quantum Mechanics >> passing by system technology, you just can't solve any problem withoutresorting>>to complex numbers. They are vital for anything that has to do with waves.Like in acoustics (Fourier, Gudermannian inharmonioc series, etc.)>> They are also extremely practical to solve trigonometric problems, and >> simplify greatly calculations involving sines and cosines (both things >> are related of course). And for things like optics (and dynamics too, >> although there people tend to avoid using them ;)) ), you even have to gofurther>> and resort to Quaternions, which resort not to one but three roots of >-1, all perpendicular to each other, and get rid of the commutative >> assumption of multiplication.Or in the "fractal/chaos"-inspired music of the _non-Just-Intonated, non-12 Equal Tempered, non-octave_ kind ("nj n12TET n8tve"), there is a lot of use of transcendental numbers [so-call "irrational numbers"] in the creation of new musical scales. Examples: - tunings derived from iterated systems. Iterated scales can be self-similar or non-self-similar. i.e. square root of 3, 25th root of 5, iterated square root of 3, iterated pi, iterated phi, iterated 7.1235, etc. also scales based on the inharmonic series - modes of vibration of a mathematically ideal sphere, modes of vibration of a mathematically ideal cube, modes of vibration of a mathematically ideal tube, modes of vibration of a mathematically ideal plate, rod, disc, membrane, cable [which is a rather head-ache-inducing complex "cross mode" system constantly shifting somewhere between a membrane, a rod and a tube with material factors further complicating matters], icosahedron, trunacated icosidodecahedron, Buckyball, ad infinitum... ... or based on fractal geometry like: z = recip(fn(z) x pixel + square root of 5, z = z to the 2nd + c [z being a changing iteration of some transcendental number], etc. etc. http://classes.yale.edu/fractals/Panorama/Music/Mus/Music.html In a message dated 2003:04:22 08:17:15 AM, h s teoh writes:>Without geometry, life would be pointless. -- VS:) I love that quote. Who's "VS"? In a message dated 2003:04:22 07:40:10 AM, David Starner (dvdeug@ISPWEST.COM) writes:>Ignoring the physics questions (which is probably why I went from >math/cs to straight math) complex numbers are an extension of the real >numbers closed under the square root operation. They also come in very >handing in analytical geometry: if you represent the plane by complex >numbers, a rotation is equivalent to multiplying by a complex number >whose complex absolute value equals 1 - e.g. multiplying a point by i is >like rotating it by 90 degrees. > >If you don't like complex numbers, then you really don't like >quaternions, do you? (Quaterions, if you haven't run into them, are like >complex numbers, with j and k such that j^2=-1, k^2=-1, ij=-ji, jk=-kj, >and ijk=-1. Turns out they're sort of like a souped up form of vectors >no one uses because it's overkill.)Brilliant LMAO. BTW what's "wrong" with just a lil overkill 0_o? *snarfle!* --- Hanuman Zhang "In the beginning was noise - raw sound, the seed sound, the One, _Nada Brahma_, the Big Bang. And noise begat rhythm. And rhythm begat everything else. And thus the Dance began. Rhythm and noise. There is terror in noise, and in that terror there is also power." - adapted from writings by Mickey Hart "I have the feeling that the English word 'noise' has more negative connotations than our German word 'Gerausch'. We would describe the sound of wind blowing as Gerausch, to imply that it's a beautiful and natural sound. It's so stupid when people say that instead of making beautiful sounds, I make noise...I like these sounds and this has nothing to do with 'anti-beauty'" - Helmut Lachenmann "We cannot doubt that animals both love and practice music. That is evident. But it seems their musical system differs from ours. It is another school...We are not familiar with their didactic works. Perhaps they don't have any." - Erik Satie What strange risk of hearing can bring sound to music - a hearing whose obligation awakens a sensibility so new that it is forever a unique, new-born, anti-death surprise, created now and now and now. .. a hearing whose moment in time is always daybreak. - Lucia Dlugoszewski improvvisazione liquida, sospesa temporalmente e profondamente "aliena"