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Date: Sun, 19 Jan 25 07:41:39 +0000
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From: Richard Hachel <r.hachel@liscati.fr.invalid>
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Le 19/01/2025 à 02:53, sobriquet a écrit :
> Op 18/01/2025 om 11:34 schreef Richard Hachel:
>> Hello friends of mathematics.
>> I was recently thinking, because of a poster named Python, about what 
>> complex numbers were, wondering if teaching them was so important and 
>> useful, especially in kindergarten where children are only learning to 
>> read.
>> What is a complex number? Many have difficulty answering, especially 
>> girls, whose minds are often more practical than abstract.
>> 
>> Let z=a+ib
>> 
>> It is a number that has a real component and an imaginary component.
>> 
>> I wonder if the terms "certain component" and "possible component" would 
>> not be as appropriate.
>> 
>> What is i?
>> 
>> It is an imaginary unit, such that i*i=-1.
>> 
>> In our universe, this seems impossible, a square can never be negative.
>> 
>> Except that we are in the imaginary.
>> 
>> Let's assume that i is a number, or rather a unit, which is both its 
>> number and its opposite.
>> 
>> Thus, if we set z=9i we see that z is both, as in this story of 
>> Schrödinger's cat, z=9 and z=-9
>> 
>> I remind you that we are in the imaginary. So why not.
>> 
>> Let's set z=16+9i
>> 
>> It then comes that at the same time, z=25 and z=7.
>> 
>> It is a strange universe, but which can be useful for writing things in 
>> different ways.
>> 
>> Explanations: We ask Mrs. Martin how many students she has in her class, 
>> and she is very bored to answer because she does not know if 
>> Schrödinger's cat is dead.
>> 
>> It has two classes, and depending on whether we imagine the morning 
>> class or the evening class (catch-up classes for adults), the answer 
>> will not be the same. There is no absolute answer. What is z?
>> 
>> We can nevertheless give z a real part, which is the average of the two 
>> classes. a=16.
>> 
>> And ib then becomes the fluctuation of the average.
>> 
>> If we set i=1 then ib=+9; if we set i=-1 then ib=-9.
>> 
>> "i" would therefore be this entity, this unit, equal to both 1 and -1, 
>> depending on how we look at it (Schrodinger's cat).
>> 
>> But what happens if we square i?
>> 
>> It is both 1 and -1?
>> 
>> Can we write i²=(1)*(1)=1?
>> 
>> No, because i would only be 1.
>> 
>> Can we write i²=(-1)(-1)=1?
>> 
>> No, because i is not only -1, it is both -1 and 1.
>> 
>> We then have i²=(i)*(i)=(1)(-1)=(-1)(1)=-1.
>> 
>> But here, we will notice something extraordinary, the additions and 
>> products of complex numbers can be determined.
>> 
>> Z=z1+z2
>> 
>> Z=(a+ib)+(a'+ib')
>> 
>> and, Z=(a+a')+i(b+b')
>> 
>> All this is very simple for the moment.
>> 
>> But we are going to enter into a huge astonishment concerning the 
>> product of two complexes.
>> 
>> How do mathematicians practice?
>> Z=z1*z2
>> 
>> so, so far it's correct:
>> 
>> Z=(a+ib)(a'+ib')
>> 
>> So, and it's still correct for Dr. Hachel (that's me):
>> Z=aa'+i(ab'+a'b)+(ib)(ib')
>> 
>> And there, for Dr. Hachel, mathematicians make a huge blunder by setting 
>> (ib)(ib')=i²bb'=-bb'
>> 
>> Why?
>> 
>> Because at this point in the calculation, we impose that i will 
>> indefinitely remain
>> both positive and negative, and the correct formula
>> Z=aa'+i(ab'+a'b)+(ib)(ib') will become incorrect written in the form
>> Z=aa'+i(ab'+a'b)+(i²bb') and a sign error will appear.
>> 
>> We must therefore write, for the product of two complexes:
>> Z=aa'+bb'+i(ab'+a'b) and not aa'-bb'+i(ab'+a'b)
>> 
>> The real part of the product being aa'+bb' and not aa'-bb'
>> 
>> With a remaining imaginary part where i is equal to both -1 and 1, which 
>> gives two results each time for Z.
>> 
>> It seems that this is an astonishing blunder, due to the 
>> misunderstanding of the handling of complex and imaginary numbers.
>> 
>> On the other hand, by going through statistics, statistics confirms 
>> HAchel's ideas, and the results usually proposed by mathematicians 
>> become totally false.
>> 
>> I wish you a good reflection on this.
>> 
>> Have a good day.
>> 
>> R.H.
> 
> If we define complex multiplication in the way you suggest instead of 
> the conventional way, that would mean that the operation of conjugation 
> would no longer be a homomorphism with respect to the field of complex 
> numbers under multiplication.
> 
> So conj(z1*z2) would not be equal to conj(z1)*conj(z2).
> 
> https://www.desmos.com/calculator/kqzgbliix1

Thank you for your answer.

But nevertheless, I continue to certify that there is an extremely fine 
mathematical error, at the moment when physicists pose
i²=-1 to quickly simplify what seems a convenient operation.

Because as long as we do not know what i is worth, which can be BOTH equal 
to 1 or -1 in this imaginary mathematics, we must pose i²=-1.

But once we pose i=1, it is no longer possible to say i²=-1; and in the 
same way, when we pose i=-1, it is no longer possible to say 1²=-1.

It is necessary, at this instant where we have defined i (whether it is 1 
or -1 but defined at this instant, it is necessary to set Z=z1*z2 such 
that:
Z=(a+ib)(a'+ib')=aa'+bb'+i(ab'+a'b) to have the correct result, otherwise 
the real part becomes very incorrect.

You tell me: yes, but it does not work with the conjugate.

Of course it does.

If it does not work, it is because you make a sign error, and the computer 
does the same because it is not formatted on the right concept giving the 
right real part.

Mathematical proof that Z(conj)=z1(conj)*z2(conj)

We set:
z1=16+9i
z2= 14+3i

Z (equation correct)=aa'+bb'-i(ab'+a'b)

Z=251+174i.

Let z1(conj)=16-9i and z2(conj)=14-3i
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