On 11/7/05, Martin Peach <martinrp@vax2.concordia.ca> wrote:

Chuckk Hubbard wrote:

> I understand complex numbers used to represent rotation, so as to
> encode frequency in an easy to manipulate form, and I understand that
> the imaginary part can be disposed of when converting back to real
> signals and nothing is changed, but there's one bit that is hanging me
> up:
>
> Using i is just a convention, a way to keep from mixing the two
> numbers.  The square root of -1 is not really involved in any of it,
> because it doesn't exist.  So why, when you multiply Z1 and Z2, do
> i*sin(a) and i*sin(b) multiply to -sin(a)sin(b)?
>

The square root of minus one is a number that, when multiplied by
itself, gives minus one:
i*i = -1
so i*sin(a)*i*sin(b) = -1*sin(a)sin(b)
The terms containing just one i cancel out: that's very convenient. You
never have to deal with 'naked' i so you don't have to worry if it
exists or not.
If you think of a wave that is momentarily passing through zero with
momentarily no acceleration and then ask yourself 'where is the energy
of the wave?' you may see why i is useful: the energy is all in the
imaginary dimension for that one instant. Otherwise the law of
conservation of energy would be violated.

>
> Also, the decision (this is coming from Miller's "Theory and
> Techniques") to multiply the complex constant A by the unit-value
> complex number Zn; this is convenient, but it seems rigged.  Instead
> of letting Z have whatever amplitude it really has, in which case
> multiplying it (which already seems forced) by itself changes its
> amplitude, you force it to be 1 and add on the amplitude later.
> Nothing violated, but it seems artificial, like we fudge the numbers
> to make it come out right.  If it is artificial, and just a way of
> simplifying sinusoid manipulation, then why even use
> artificial mathematical operations to explain it?

Well actually sin and cos are hacks as well, being simply the result of
drawing giant circles and measuring the lengths of perpendiculars to the
radius at many points. It was later discovered that the exponential
function exp() is more natural and that sin and cos can be expressed in
terms of e if the number i is introduced. This is a more sophisticated
hack in that the numbers produced by powers of e can be discovered to
any desired precision by doing a long  series of multiplications without
having to 'exit' mathematics to measure the length of a line.

>
> I already tried just moving ahead accepting these, but eventually
> something always throws me.

Wolfram in his "A New Kind of Science" claims that there are any number
of mathematical systems possible and humans have just chosen the ones
that work for them in this universe. Because really the universe is not
understood by humans, they are just good at manipulating the symbols
they use to model it, and these symbols arose from empirical interaction
with the universe: they work. And I'm sure Nietszche would agree, even
if we did understand it, who would we tell?

>
>
> One more question... regarding how filters work, is there no intuitive
> way to express it?  No shortcut so that rpole~ and rzero~ will at
> least make sense in theory before I push through all the math?

Roughly that poles are resonant and zeros are damping at a particular
frequency.

>
> -Chuckk
>
>
> --
> "It is not when truth is dirty, but when it is shallow, that the lover
> of knowledge is reluctant to step into its waters."
> -Friedrich Nietzsche, "Thus Spoke Zarathustra"
>
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