[PD] fft beginner question
matju at artengine.ca
Wed Nov 21 17:47:26 CET 2007
On Tue, 20 Nov 2007, Chuckk Hubbard wrote:
> I still don't exactly understand why one couldn't just use (x, y)
> vectors; why the y value has to be multiplied by something imaginary.
The complex product, (a+bi)(c+di) = (ac-bd) + (ad+bc)i, is something that
can be done using vectors, and complex numbers can be thought as an
extension of 2-D vectors by a product like vector*vector=vector. I believe
that complex numbers are easier to use than 2-D vectors for that kind of
thing, but 2-D vectors are still what we used in 12th and 13th grade
physics because they were too chicken to teach us complex numbers.
> I mean, i/j is *defined as* the square root of -1, but it can't really
> *be* the square root of -1... I've accepted it and moved on to more
> practical questions, but that is still mysterious for me.
This is called an algebraic extension. Another example of algebraic
extension is when going from numbers to polynomials. If you take
polynomials in x and force x^2 to be equal to -1 then you end up with a
system of linear equations that work exactly like complex numbers.
"forcing x^2 = -1" is really a modulo, just like forcing 13=0 is called
modulo 13. The complex numbers are polynomials modulo x^2+1.
A more natural way of thinking of complex numbers is that you start with
real numbers and want all square roots to exist. Negative numbers started
as a way to make all subtractions exist. Fractions started as a way to
make all nonzero divisions exist. Why not square root, too?
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