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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Division of two complex numbers
Date: Mon, 20 Jan 2025 13:45:35 -0800
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On 1/20/2025 1:35 PM, Python wrote:
> Le 20/01/2025 à 22:28, "Chris M. Thomasson" a écrit :
>> On 1/20/2025 1:09 PM, Python wrote:
>>> Le 20/01/2025 à 22:06, "Chris M. Thomasson" a écrit :
>>>> On 1/20/2025 1:04 PM, Python wrote:
>>>>> Le 20/01/2025 à 21:59, "Chris M. Thomasson" a écrit :
>>>>>> On 1/20/2025 12:51 PM, Python wrote:
>>>>>>> Le 20/01/2025 à 21:44, "Chris M. Thomasson" a écrit :
>>>>>>>> On 1/20/2025 12:20 PM, Python wrote:
>>>>>>>>> Le 20/01/2025 à 21:09, Tom Bola a écrit :
>>>>>>>>>> Am 20.01.2025 20:33:12 Moebius schrieb:
>>>>>>>>>>
>>>>>>>>>>> Am 20.01.2025 um 19:27 schrieb Python:
>>>>>>>>>>>> Le 20/01/2025 à 19:23, Richard Hachel  a écrit :
>>>>>>>>>>>>> Le 20/01/2025 à 19:10, Python a écrit :
>>>>>>>>>>>>>> Le 20/01/2025 à 18:58, Richard Hachel  a écrit :
>>>>>>>>>>>>>>>>> Mathematicians give:
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> z1/z2=[(aa'+bb')/(a'²+b'²)]+i[(ba'-ab')/(a'²+b'²)]
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> It was necessary to write:
>>>>>>>>>>>>>>>>> z1/z2=[(aa'-bb')/(a'²-b'²)]+i[(ba'-ab')/(a'²-b'²)]
>>>>>>>>>>>>>
>>>>>>>>>>>>>> I've explained how i is defined in a positive way in 
>>>>>>>>>>>>>> modern algebra. i^2 = -1 is not a definition. It is a 
>>>>>>>>>>>>>> *property* that can be deduced from a definition of i.
>>>>>>>>>>>>>
>>>>>>>>>>>>>  That is what I saw.
>>>>>>>>>>>>>
>>>>>>>>>>>>>  Is not a definition.
>>>>>>>>>>>>>  It doesn't explain why.
>>>>>>>>>>>>>
>>>>>>>>>>>>> We have the same thing with Einstein and relativity.
>>>>>>>>>>>>>
>>>>>>>>>>>>> [snip unrelated nonsense about your idiotic views on 
>>>>>>>>>>>>> Relativity]
>>>>>>>>>>>>
>>>>>>>>>>>>> It is clear that i²=-1, but we don't say WHY. It is clear 
>>>>>>>>>>>>> however that if i is both 1 and -1 (which gives two 
>>>>>>>>>>>>> possible solutions) we can consider its square as the 
>>>>>>>>>>>>> product of itself by its opposite, and vice versa.
>>>>>>>>>>>>
>>>>>>>>>>>> I've posted a definition of i (which is NOT i^2 = -1) 
>>>>>>>>>>>> numerous times. A "positive" definition as you asked for.
>>>>>>>>>>>
>>>>>>>>>>> I've already told this idiot:
>>>>>>>>>>>
>>>>>>>>>>> Complex numbers can be defined as (ordered) pairs of real 
>>>>>>>>>>> numbers.
>>>>>>>>>>>
>>>>>>>>>>> Then we may define (in this context):
>>>>>>>>>>>
>>>>>>>>>>>           i := (0, 1) .
>>>>>>>>>>>
>>>>>>>>>>>  From this we get: i^2 = -1.
>>>>>>>>>>
>>>>>>>>>> For R.H.
>>>>>>>>>>   By the binominal formulas we have: (a, b)^2 = a^2 + 2ab + b^2
>>>>>>>>>
>>>>>>>>> Huh? This is not the binomial formula which is (a + b)^2 = a^2 
>>>>>>>>> + 2ab + b^2
>>>>>>>>>
>>>>>>>>> (a, b)^2 does not mean anything without any additional 
>>>>>>>>> definition/ context.
>>>>>>>>>
>>>>>>>>>>   So we get: (0, 1)^2 ) 0^2 + 2*(0 - 1) + 1 = 0 + (-2) + 1 = -1 
>>>>>>>>>
>>>>>>>>> you meant  (0, 1)^2 = 0^2 + 2*(0 - 1) + 1 = 0 + (-2) + 1 = -1 ?
>>>>>>>>>
>>>>>>>>> This does not make sense without additional context.
>>>>>>>>>
>>>>>>>>> In R(epsilon) = R[X]/X^2 (dual numbers a + b*epsilon where 
>>>>>>>>> epsilon is such as
>>>>>>>>> epsilon =/= 0 and epsilon^2 0) we do have :
>>>>>>>>>
>>>>>>>>> (0, 1) ^ 2 = 0
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>> vec2 ct_cmul(in vec2 p0, in vec2 p1)
>>>>>>>> {
>>>>>>>>      return vec2(p0.x * p1.x - p0.y * p1.y, p0.x * p1.y + p0.y * 
>>>>>>>> p1.x);
>>>>>>>> }
>>>>>>>
>>>>>>> So what? This is not an application of the binomial formula...
>>>>>>>
>>>>>>> What's you point?
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> It's a way I multiply two vectors together as if they are complex 
>>>>>> numbers.
>>>>>>
>>>>>> Another one:
>>>>>>
>>>>>> #define cx_mul(a, b) vec2(a.x*b.x - a.y*b.y, a.x*b.y + a.y*b.x)
>>>>>>
>>>>>> I can pass in normal vectors to this in GLSL. vec2's
>>>>>
>>>>> Good! You know how to write a C program. :-) (pun intended)
>>>>
>>>> Fwiw, that is not is C, it's from one of my GLSL shaders. ;^)
>>>
>>> It is also C.
>>
>> No. GLSL is not C at all, it has a similar style, but is different for 
>> sure.
> 
> It is exactly the same syntax. *facepalm*.
> Ok, let's say so, if you wish, so you can implement complex 
> multiplication in a GLSL shader.

No. C and GLSL are completely different languages. Have you ever even 
used GLSL? You can do fun things in GLSL that C cannot do at all.

> 
> Again: SO WHAT? ? ? This is NOT THE POINT of the discussion.

I thought it might help the OP.


> 
>>> Again what's *your* point? Your posts makes absolutely no sense in 
>>> the context of this thread!
>>
>> Just a way to multiply two 2-ary vectors as if they were complex 
>> numbers. Now, here is a little C99 program I just typed in the 
>> newsreader. It should compile.
>> _____________________________
>> [snip irrelevant triviality]
> 
> So what? ? ?
>> I thought it might help out the OP.
> 
> In which way? ? ? Hachel didn't write that it cannot be done (he's not 
> that silly), he claimed (wrongly) that it is the wrong way to define 
> multiplication between complex numbers.
> 
>>>>>
>>>>> This is quite off-topic to point out that multiplication of complex 
>>>>> numbers in C/C++ can be done.
>>>>>
>>>>> The discussion is not about that it can be done, even crank Hachel 
>>>>> would admit this. It is *why* it makes sense to define 
>>>>> multiplication *that way*.
> 
> 
>