Path: ...!news.mixmin.net!proxad.net!feeder1-2.proxad.net!usenet-fr.net!pasdenom.info!from-devjntp Message-ID: JNTP-Route: nemoweb.net JNTP-DataType: Article Subject: Re: Division of two complex numbers References: <3OCPm1fExu73aCqhbosWeWpssJM@jntp> <1f331uj8cjsge$.rox7zzvx5o63$.dlg@40tude.net> Newsgroups: sci.math JNTP-HashClient: UPg18dchw1P6nLJr6tC2OwsAEi4 JNTP-ThreadID: EDRXv_p0dplN4woC0vmk_fQazIU JNTP-Uri: http://www.nemoweb.net/?DataID=XmWkVwNVTy8QHqeS0TsrPZNuk_c@jntp User-Agent: Nemo/1.0 JNTP-OriginServer: nemoweb.net Date: Mon, 20 Jan 25 20:51:40 +0000 Organization: Nemoweb JNTP-Browser: Mozilla/5.0 (X11; Linux x86_64; rv:109.0) Gecko/20100101 Firefox/115.0 Injection-Info: nemoweb.net; posting-host="7a226b0f6c2d664790534afd7e847737d5df5486"; logging-data="2025-01-20T20:51:40Z/9181360"; posting-account="190@nemoweb.net"; mail-complaints-to="julien.arlandis@gmail.com" JNTP-ProtocolVersion: 0.21.1 JNTP-Server: PhpNemoServer/0.94.5 MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: 8bit X-JNTP-JsonNewsGateway: 0.96 From: Python Bytes: 4111 Lines: 77 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?