Not that I don't appreciate the snide commentary, but this is why I'm asking. You can't forget the sin*sin part. i stands for "imaginary". (slowly: i is "imaginary"). Turning the product of sines into the negative product of sines is imaginary. There is no number that, squared, equals -1. There is no number that, squared, equals -1. It's not really there.
Date: Mon, 07 Nov 2005 20:58:36 +0100
From: Piotr Majdak piotr@majdak.com Subject: Re: [PD] basic DSP stuff Cc: Pure Data List pd-list@iem.at Message-ID: 436FB1EC.80908@majdak.com Content-Type: text/plain; charset=us-ascii; format=flowed
Chuckk Hubbard wrote:
i*sin(a) and i*sin(b) multiply to -sin(a)sin(b)?
let's forget the sin*sin and look at the i*i part only: since i = sqrt(-1) by definition, i*i is -1 (slowly: i*i= sqrt(-1)*sqrt(-1) = sqrt(-1 * -1) = sqrt((-1)^2) = -1)
Thus, i*sin(a) * i*sin(b) = -1 * sin(a)*sin(b)
br, piotr
Chuckk Hubbard wrote:
Not that I don't appreciate the snide commentary, but this is why I'm asking.
If I offended you, please forgive me. You asked a simple question and I tried to answer, as simple as possible (I don't know your level of education).
Back to your question, you asked:
So why, when you multiply Z1 and Z2, do i*sin(a) and i*sin(b)
multiply to -sin(a)sin(b)?
You see, you wrote an "i" there. If you define "i" by "sqrt(-1)" (I admit I was implying that), then my answer:
i*sin(a) * i*sin(b) = -1 * sin(a)*sin(b)
is correct. And, as you see, "i" is there :-)
But, if you wanted to discuss the fact, that "i" has no physical meaning
- in this case I misunderstood your question. This is complete another
issue, much more philosophical than mathematical and thus outside my education focus.
br, Piotr
On Mon, 7 Nov 2005, Chuckk Hubbard wrote:
Not that I don't appreciate the snide commentary, but this is why I'm asking. You can't forget the sin*sin part. i stands for "imaginary". (slowly: i is "imaginary"). Turning the product of sines into the negative product of sines is imaginary. There is no number that, squared, equals -1. There is no number that, squared, equals -1. It's not really there.
depends on how people define "number". In math, it's usual to use an extended definition which isn't so limited as to forbid sqrt(-1) from existing.
also, to understand the word "imaginary" in math, you have to first understand that its meaning is completely distinct from any meaning of "imaginary" outside of math.
likewise for Real... "Real" is just a name. Small integers have a relatively high tangible quality to them, but it gets mostly downhill from there. "Real" numbers are arguably much less tangible than integer numbers, rational numbers, algebraic numbers, and then some. The numbers PI and E are a lot more tangible than almost all (100%) of the Real numbers.
Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju Freelance Digital Arts Engineer, Montréal QC Canada
What I'm getting at is that expressing rotation as complex numbers is no different than using cartesian coordinates. Why, when you multiply two points, would one of the multiples turn negative? I see no reason you couldn't say i = 1/0. Then 4*0*i=4. That makes as much sense.
On 11/10/05, Mathieu Bouchard matju@artengine.ca wrote:
On Mon, 7 Nov 2005, Chuckk Hubbard wrote:
Not that I don't appreciate the snide commentary, but this is why I'm asking. You can't forget the sin*sin part. i stands for "imaginary". (slowly: i is "imaginary"). Turning the product of sines into the negative product of sines is imaginary. There is no number that, squared, equals -1. There is no number that, squared, equals -1. It's not really there.
depends on how people define "number". In math, it's usual to use an extended definition which isn't so limited as to forbid sqrt(-1) from existing.
also, to understand the word "imaginary" in math, you have to first understand that its meaning is completely distinct from any meaning of "imaginary" outside of math.
likewise for Real... "Real" is just a name. Small integers have a relatively high tangible quality to them, but it gets mostly downhill from there. "Real" numbers are arguably much less tangible than integer numbers, rational numbers, algebraic numbers, and then some. The numbers PI and E are a lot more tangible than almost all (100%) of the Real numbers.
Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju Freelance Digital Arts Engineer, Montréal QC Canada
-- "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"
Chuckk Hubbard wrote:
What I'm getting at is that expressing rotation as complex numbers is no different than using cartesian coordinates. Why, when you multiply two points, would one of the multiples turn negative? I see no reason you couldn't say i = 1/0. Then 4*0*i=4. That makes as much sense.
<OT lecture_and_rant> Not really. Any number multiplied by zero becomes zero. You can't break the existing rules. There is a philosophical need for i because intuitively every number ought to have a square root, even negative numbers. The other philosophical need comes from the idea that the energy of an oscillator should be constant if the oscillator is at a constant amplitude and frequency, but the observable output of the oscillator passes through zero, which implies there is no energy there. Where is the energy? In an imaginary dimension...imaginary just means unobservable. This is basically how in quantum mechanics one decides what is actually observable versus just virtual, so there is some connection with reality here, but it's probably hard to grasp since humans have never used such concepts until about a hundred years ago simply because there was no practical use for them. It also took a long time for people in Europe to accept the concept of zero after it had been invented in North Africa and Central America and elsewhere...in Roman numbers (I, II, III, IV...) there is no zero but in Arabic numbers (0,1,2,3,) there is. The Maya used a shell glyph for zero and dots and bars for the other numbers. I wouldn't get hung up on the use of i, it's like using an empty shell for zero, a kind of handle on a concept. Some people think that concepts cannot be expressed in a language that doesn't already have words for them, but they are wrong. The concepts are just expressed in a crooked, crooked way. </OT lecture_and_rant>
Martin
By the same token, any number times itself is non-negative. i breaks existing rules too. (lol)
You know, I could just have said, oh, since you're multiplying i times i the result is negative, by definition, and accepted that it works. I could just be the kid who writes down what the teacher says and memorizes it for the test, instead of asking questions. I suspect there is something else at work for the result to be correct, because the only reason given for using i was to represent the vertical axis. At no point was there actually a quantity of -1, for which we found the root before multiplying by the second part of the complex number. -Chuckk On 11/11/05, Martin Peach martinrp@vax2.concordia.ca wrote:
Chuckk Hubbard wrote:
What I'm getting at is that expressing rotation as complex numbers is no different than using cartesian coordinates. Why, when you multiply two points, would one of the multiples turn negative? I see no reason you couldn't say i = 1/0. Then 4*0*i=4. That makes as much sense.
<OT lecture_and_rant> Not really. Any number multiplied by zero becomes zero. You can't break the existing rules. There is a philosophical need for i because intuitively every number ought to have a square root, even negative numbers. The other philosophical need comes from the idea that the energy of an oscillator should be constant if the oscillator is at a constant amplitude and frequency, but the observable output of the oscillator passes through zero, which implies there is no energy there. Where is the energy? In an imaginary dimension...imaginary just means unobservable. This is basically how in quantum mechanics one decides what is actually observable versus just virtual, so there is some connection with reality here, but it's probably hard to grasp since humans have never used such concepts until about a hundred years ago simply because there was no practical use for them. It also took a long time for people in Europe to accept the concept of zero after it had been invented in North Africa and Central America and elsewhere...in Roman numbers (I, II, III, IV...) there is no zero but in Arabic numbers (0,1,2,3,) there is. The Maya used a shell glyph for zero and dots and bars for the other numbers. I wouldn't get hung up on the use of i, it's like using an empty shell for zero, a kind of handle on a concept. Some people think that concepts cannot be expressed in a language that doesn't already have words for them, but they are wrong. The concepts are just expressed in a crooked, crooked way. </OT lecture_and_rant>
Martin
-- "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"
man,
admitting there are so-called imaginery numbers and that it makes sense because it has a reflexion ( a projection ) back in reality, it means the 'reality' is actually a projection by itself of something that has more than 3 dimensions.
.it looks like believing in magic and the paranormal, that's why i liked maths and physics for a while.
then, they asked to build motors, and i just left )
saludos, sevy
Chuckk Hubbard wrote:
By the same token, any number times itself is non-negative. i breaks existing rules too. (lol)
You know, I could just have said, oh, since you're multiplying i times i the result is negative, by definition, and accepted that it works.
I could just be the kid who writes down what the teacher says and memorizes it for the test, instead of asking questions. I suspect there is something else at work for the result to be correct, because the only reason given for using i was to represent the vertical axis.
At no point was there actually a quantity of -1, for which we found the root before multiplying by the second part of the complex number.-Chuckk
"The other philosophical need comes from the idea that the energy of an oscillator should be constant if the oscillator is at a constant amplitude and frequency, but the observable output of the oscillator passes through zero, which implies there is no energy there. Where is the energy?"
The fact that an oscillator passes through zero is just admitted to trace on X Y a signal where Y axis is the edge between positive and negative solutions, but if we make a translation of Y axis to only have positive solutions for X, we wouldn't have to use imaginary numbers?
Patco
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