Re: Hello!

On 01/18/2025 05:53 PM, sobriquet wrote:

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.
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Let z=a+ib
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It is a number that has a real component and an imaginary component.
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I wonder if the terms "certain component" and "possible component"
would not be as appropriate.
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What is i?
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It is an imaginary unit, such that i*i=-1.
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In our universe, this seems impossible, a square can never be negative.
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Except that we are in the imaginary.
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Let's assume that i is a number, or rather a unit, which is both its
number and its opposite.
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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
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I remind you that we are in the imaginary. So why not.
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Let's set z=16+9i
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It then comes that at the same time, z=25 and z=7.
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It is a strange universe, but which can be useful for writing things
in different ways.
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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.
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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?
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We can nevertheless give z a real part, which is the average of the
two classes. a=16.
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And ib then becomes the fluctuation of the average.
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If we set i=1 then ib=+9; if we set i=-1 then ib=-9.
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"i" would therefore be this entity, this unit, equal to both 1 and -1,
depending on how we look at it (Schrodinger's cat).
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But what happens if we square i?
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It is both 1 and -1?
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Can we write i²=(1)*(1)=1?
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No, because i would only be 1.
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Can we write i²=(-1)(-1)=1?
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No, because i is not only -1, it is both -1 and 1.
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We then have i²=(i)*(i)=(1)(-1)=(-1)(1)=-1.
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But here, we will notice something extraordinary, the additions and
products of complex numbers can be determined.
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Z=z1+z2
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Z=(a+ib)+(a'+ib')
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and, Z=(a+a')+i(b+b')
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All this is very simple for the moment.
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But we are going to enter into a huge astonishment concerning the
product of two complexes.
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How do mathematicians practice?
Z=z1*z2
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so, so far it's correct:
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Z=(a+ib)(a'+ib')
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So, and it's still correct for Dr. Hachel (that's me):
Z=aa'+i(ab'+a'b)+(ib)(ib')
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And there, for Dr. Hachel, mathematicians make a huge blunder by
setting (ib)(ib')=i²bb'=-bb'
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Why?
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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.
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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)
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The real part of the product being aa'+bb' and not aa'-bb'
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With a remaining imaginary part where i is equal to both -1 and 1,
which gives two results each time for Z.
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It seems that this is an astonishing blunder, due to the
misunderstanding of the handling of complex and imaginary numbers.
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On the other hand, by going through statistics, statistics confirms
HAchel's ideas, and the results usually proposed by mathematicians
become totally false.
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I wish you a good reflection on this.
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Have a good day.
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R.H.

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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.
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So conj(z1*z2) would not be equal to conj(z1)*conj(z2).
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https://www.desmos.com/calculator/kqzgbliix1
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The definition of division of complex numbers is contrived.
There's a way to make objects like complex-complex numbers
or left-complex and right-complex numbers,
like a + bi and ai + b, with regards to that
the definition of division of complex numbers
is contrived.
There's a consideration of this in my podcasts
under the "Descriptive Differential Dynamics"
about an inner "original analysis" and not so much
about the De Moivre-Euler-Gauss complex analysis.