That's just it, it's convenient. Suddenly you have the rule for
cosine of a sum. But you get it by arbitrarily inserting a
convention that, when you multiply the x values for two complex
numbers, they turn negative. I mean, maybe it's not arbitrary,
but I don't understand how it isn't. If a complex number is just
a way to express two numbers without relating them to each other, then
the decision that they should relate when multiplied seems kind of
weird, giving cos(a)cos(b) - sin(a)sin(b). Seems like
cos(a)cos(b) + i*sin(a)sin(b) would be more accurate.<br>
<br>
I think sine and cosine are a little more real than i. Maybe you
don't actually have a triangle, but it's not like saying 2+2=5.
Or i = 1/0.<br>
<br>
As regards e, wouldn't it make as much sensel to use a and b as
non-mixing terms, but each equivalent to units? Instead of
e^4+9i, have e^4a+9b?<br>
<br>I can accept that this i*i stuff works- I'm not saying I can't follow that- but I'm not ready to stop asking about it.<br>
<br>
-Chuckk<br>
<br><div><span class="gmail_quote">On 11/7/05, <b class="gmail_sendername">Martin Peach</b> <<a href="mailto:martinrp@vax2.concordia.ca">martinrp@vax2.concordia.ca</a>> wrote:</span><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
Chuckk Hubbard wrote:<br><br>> I understand complex numbers used to represent rotation, so as to<br>> encode frequency in an easy to manipulate form, and I understand that<br>> the imaginary part can be disposed of when converting back to real
<br>> signals and nothing is changed, but there's one bit that is hanging me<br>> up:<br>><br>> Using i is just a convention, a way to keep from mixing the two<br>> numbers. The square root of -1 is not really involved in any of it,
<br>> because it doesn't exist. So why, when you multiply Z1 and Z2, do<br>> i*sin(a) and i*sin(b) multiply to -sin(a)sin(b)?<br>><br><br>The square root of minus one is a number that, when multiplied by<br>itself, gives minus one:
<br>i*i = -1<br>so i*sin(a)*i*sin(b) = -1*sin(a)sin(b)<br>The terms containing just one i cancel out: that's very convenient. You<br>never have to deal with 'naked' i so you don't have to worry if it<br>exists or not.<br>
If you think of a wave that is momentarily passing through zero with<br>momentarily no acceleration and then ask yourself 'where is the energy<br>of the wave?' you may see why i is useful: the energy is all in the<br>imaginary dimension for that one instant. Otherwise the law of
<br>conservation of energy would be violated.<br><br>><br>> Also, the decision (this is coming from Miller's "Theory and<br>> Techniques") to multiply the complex constant A by the unit-value<br>> complex number Zn; this is convenient, but it seems rigged. Instead
<br>> of letting Z have whatever amplitude it really has, in which case<br>> multiplying it (which already seems forced) by itself changes its<br>> amplitude, you force it to be 1 and add on the amplitude later.<br>
> Nothing violated, but it seems artificial, like we fudge the numbers<br>> to make it come out right. If it is artificial, and just a way of<br>> simplifying sinusoid manipulation, then why even use<br>> artificial mathematical operations to explain it?
<br><br>Well actually sin and cos are hacks as well, being simply the result of<br>drawing giant circles and measuring the lengths of perpendiculars to the<br>radius at many points. It was later discovered that the exponential
<br>function exp() is more natural and that sin and cos can be expressed in<br>terms of e if the number i is introduced. This is a more sophisticated<br>hack in that the numbers produced by powers of e can be discovered to
<br>any desired precision by doing a long series of multiplications without<br>having to 'exit' mathematics to measure the length of a line.<br><br>><br>> I already tried just moving ahead accepting these, but eventually
<br>> something always throws me.<br><br>Wolfram in his "A New Kind of Science" claims that there are any number<br>of mathematical systems possible and humans have just chosen the ones<br>that work for them in this universe. Because really the universe is not
<br>understood by humans, they are just good at manipulating the symbols<br>they use to model it, and these symbols arose from empirical interaction<br>with the universe: they work. And I'm sure Nietszche would agree, even
<br>if we did understand it, who would we tell?<br><br>><br>><br>> One more question... regarding how filters work, is there no intuitive<br>> way to express it? No shortcut so that rpole~ and rzero~ will at<br>
> least make sense in theory before I push through all the math?<br><br>Roughly that poles are resonant and zeros are damping at a particular<br>frequency.<br><br>><br>> -Chuckk<br>><br>><br>> --<br>> "It is not when truth is dirty, but when it is shallow, that the lover
<br>> of knowledge is reluctant to step into its waters."<br>> -Friedrich Nietzsche, "Thus Spoke Zarathustra"<br>><br>>------------------------------------------------------------------------<br>
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http://lists.puredata.info/listinfo/pd-list</a><br>><br>><br><br></blockquote></div><br><br clear="all"><br>-- <br>"It is not when truth is dirty, but when it is shallow, that the lover of knowledge is reluctant to step into its waters."
<br>-Friedrich Nietzsche, "Thus Spoke Zarathustra"