Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail
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 13:28:35 -0800
Organization: A noiseless patient Spider
Lines: 169
Message-ID: <vmmf64$3e97a$3@dont-email.me>
References: <zMjaMvWZUkHX6SOb195JTQnVpSA@jntp>
 <pn08rF_5GGMziz3K6TnJrhlBJek@jntp> <vmm8do$3cetq$2@dont-email.me>
 <1f331uj8cjsge$.rox7zzvx5o63$.dlg@40tude.net>
 <hJUop495V-7lOEA98AQwiM7l6-Q@jntp> <vmmcjk$3dtpt$1@dont-email.me>
 <XmWkVwNVTy8QHqeS0TsrPZNuk_c@jntp> <vmmden$3e97a$1@dont-email.me>
 <GKkg17e0FiGdEqKCQvNrtvN0OLY@jntp> <vmmdso$3e97a$2@dont-email.me>
 <q-ufmQf4sCFn4XF8BtUedW5gSA8@jntp>
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8; format=flowed
Content-Transfer-Encoding: 8bit
Injection-Date: Mon, 20 Jan 2025 22:28:36 +0100 (CET)
Injection-Info: dont-email.me; posting-host="f83ba5790de8b51aa60b613f63b035eb";
	logging-data="3613930"; mail-complaints-to="abuse@eternal-september.org";	posting-account="U2FsdGVkX1+FpfuuGVMdVeAn3dSIrSLsQcCdvi7Lg8M="
User-Agent: Mozilla Thunderbird
Cancel-Lock: sha1:Nek5JL4IOGw2DiraVgGNaPQ5t28=
Content-Language: en-US
In-Reply-To: <q-ufmQf4sCFn4XF8BtUedW5gSA8@jntp>
Bytes: 6441

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.


> 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.
_____________________________
#include <stdio.h>


struct vec2
{
     float x;
     float y;
};


struct vec2
ct_cmul(
     struct vec2 p0,
     struct vec2 p1
){
     struct vec2 result = {
         p0.x * p1.x - p0.y * p1.y,
         p0.x * p1.y + p0.y * p1.x
     };

     return result;
}


int main()
{
     struct vec2 z = { 0, 1 };
     struct vec2 zmul = ct_cmul(z, z);

     printf("z = (%f, %f)\n", z.x, z.y);
     printf("zmul = (%f, %f)\n", zmul.x, zmul.y);

     return 0;
}
_____________________________


Let me run it on a C99 compiler... Ok, it works:

z = (0.000000, 1.000000)
zmul = (-1.000000, 0.000000)

I thought it might help out the OP.


> 
>>>
>>> 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*.
> 
> 
>