Re: Sign and complex.

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)