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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: Sign and complex.
Date: Mon, 3 Mar 2025 15:22:08 -0800
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On 3/3/2025 3:19 PM, Chris M. Thomasson wrote:
> On 3/3/2025 3:16 PM, Chris M. Thomasson wrote:
>> On 3/3/2025 3:10 PM, Richard Hachel wrote:
>>> 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)
>>>
>>> <http://nemoweb.net/jntp?myUYryqxSTftKUa3owdcVRxGpRA@jntp/Data.Media:1>
>>>
>>> R.H.
>>
>> So, you try to embed it all in the real line? 
> 
> If so, it kinds of sounds akin to storing a 2d array inside of a 1d 
> array? Is that somewhat similar?
> 
> 
>> So, why even have your y axis showing in your graphic? Show me a 
>> single point, color it yellow, on your real line that shows the 
>> plotted point 1+.5i.
> 

Humm... I never used Cantor Pairing with real numbers, I don't think it 
would work, but it sure works with unsigned integers. A pairing where a 
complex number is stored as a single unique real number. Then we can get 
back at the complex number from said unique real.