[PD] fft beginner question

[Mathieu Bouchard]

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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| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada