Path: ...!eternal-september.org!feeder3.eternal-september.org!news.eternal-september.org!eternal-september.org!.POSTED!not-for-mail From: "Chris M. Thomasson" Newsgroups: sci.math Subject: Re: Division of two complex numbers Date: Mon, 20 Jan 2025 13:45:35 -0800 Organization: A noiseless patient Spider Lines: 150 Message-ID: References: <1f331uj8cjsge$.rox7zzvx5o63$.dlg@40tude.net> MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit Injection-Date: Mon, 20 Jan 2025 22:45:36 +0100 (CET) Injection-Info: dont-email.me; posting-host="f83ba5790de8b51aa60b613f63b035eb"; logging-data="3613930"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18zzY1tnWXQWq9caqgSSWxPJguHwvZR/s8=" User-Agent: Mozilla Thunderbird Cancel-Lock: sha1:rMLCKxx39xytPukNHsuvK+sCWLo= Content-Language: en-US In-Reply-To: Bytes: 6821 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*. > > >