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Subject: Re: Sign and complex.
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Date: Mon, 03 Mar 25 23:10:49 +0000
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From: Richard Hachel <r.hachel@tiscali.fr>
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Le 03/03/2025 à 23:38, "Chris M. Thomasson" a écrit :
> On 3/3/2025 1:37 PM, Richard Hachel wrote:
>> Complex numbers and products of different complex signs.
>> 
>> What is a complex number?
>> 
>> It is initially an imaginary number which is a duality.
>> 
>> The two real roots of a quadratic curve, for example, are a duality.
>> 
>> If we find as a root x'=2 and x"=4 we can include these two roots in a 
>> single expression: Z=3(+/-)i.
>> 
>> Z is this dual number which will split into x'=3+i and x"=3-i.
>> 
>> As in Hachel i^x=-1 whatever x, we have: x'=2 and x"=4.
>> 
>> Be careful with the signs (i=-1). If we add i, we subtract 1.
>> 
>> If we subtract 5i, we add 5.
>> 
>> But let's go further.
>> A small problem arises in the products of complexes.
>> 
>> Certainly, if we take complexes of inverse spacings, that is to say 
>> (+ib) for one and (-ib) for the other, everything will go very well.
>> 
>> Let's set z1=3-i and z2=4+2i.
>> 
>> We have z1*z2=12+6i-4i-2i²=14+2i
>> 
>> Let's set the inverse by permuting the signs of b:
>> 
>> z1=3+i and z2=4-2i.
>> 
>> We have z1*z2=12-6i+4i-2i²=14-2i
>> 
>> We notice that each time, we did:
>> Z=(aa')-(bb)+i(ab'+a'b)
>> and that it works.
>> 
>> Question: Why does this formula become incorrect for complexes of the 
>> same sign in b?
>> 
>> Example Z=(3+i)(4+2i) or Z=(3-i)(4-2i)
>> 
>> The formula given by mathematicians is incorrect.
>> I am not saying that it does not give a result.
>> I am saying that it is incorrect.
> 
> Where would you plot say, 1+.5i on the plane? I would say at 2-ary point 
> (1, .5), right where x = 1 and y = .5. Say draw a little filled circle 
> at said coordinates in the 2-ary plane where:
> 
>        (+y)
>          ^
>          |
>          |
>          |
> (-x)<---0--->(+x)
>          |
>          |
>          |
>          v
>        (-y)

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R.H.