On Sep 1, 3:54 am, Grzegorz S odkowicz <[EMAIL PROTECTED]> wrote: > > You're mixing definition with application. You didn't say a word about > what complex numbers are, not a word about the imaginary unit, where > does it come from, why is it 'imaginary' etc. > ... > I'd also like to see a three-dimensional vector > represented by a complex number.
Well, maybe you'd like to learn something about Geometric Algebra. :) I am a bit surprised that today, September 2007, in a thread about complex numbers, no one has mentioned about geometric algebra. There is an older way of looking at complex numbers: the imaginary unit as square root of -1. And then there is a new way of looking at complex numbers: as the multi-vector space associated to the two-dimensional vector space. So, yes, complex numbers are a bit like vectors, but more precisely, they are "multi-vectors", where the first component (the real part) is a "scalar", and the second part (the imaginary part) is an "area". This may all be just paraphrasing. But it gets more interesting when you go to higher dimensions. You'd like to know whether there are extension of complex numbers when you go to three dimensional space, and the answer is definitely YES! But the new multivectors live in 8 dimensional space. Geometric product not only make this possible, but this product is invertible. Moreover, complicated equations in electromagnatism in physics (Maxwell's equations) can be written in a single line when you use geometric algebra. When you see some of the features of geometric algebra, you will realize that complex number are but a small part of it. (There is a paper with the title "Imaginary numbers are not real...", I guess the title says it all.) Anyway, there are always many ways of looking at the same thing. Geometric algebra is one. Who knows what tomorrow brings? But as of today, I'd say that it's better to teach school children about geometric algebra, instead of the present way of introducing imaginary unit. Just my opinion. -- http://mail.python.org/mailman/listinfo/python-list
