Re: y=f(x)=(x²)²+2x²+3

Op 07/02/2025 om 12:14 schreef Richard Hachel:

Le 07/02/2025 à 01:24, sobriquet a écrit :

Op 06/02/2025 om 23:15 schreef Richard Hachel:

Le 06/02/2025 à 21:52, sobriquet a écrit :

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Ok, but that's a bit like people saying that 3 + 5 = 7 and then claiming that usually mathematicians say that 3 + 5 = 8, but they have different concepts that are more correct.
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Unless you're able to demonstrate that your alternative concepts are superior than the conventional way of defining these concepts, it
seems a bit silly.
The concept of a complex number didn't fall from the sky.. it was developed over many centuries by multiple generations of mathematicians. So it seems unlikely that someone can come along and claim their way to conceive of a complex number is superior or more correct.

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You may be right.
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But I also know that mathematicians and physicists can also be wrong.
Descartes' works are full of errors, Berkeley and Newton did not agree on the calculation of an infinitely small increment (I think Berkeley was right), Lorentz wandered for years on relativistic transformations before Poincaré gave it to him.
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If you observe carefully, you will realize that basically, the notion of complex numbers is very quickly presented, then very quickly skipped. You are given an i²=-1, in order to square an awkward discriminant, and, if the principle is correct (we multiply by 1, then we pose that i²=-1 by convention and this allows us to have a positive root).
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But no one has ever explained where this being comes from, which, in its being, is an entity whose square is equal to -1. It is a convention.
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Personally, I wish that we put this being under the microscope once and for all, he fuuses imaginary.
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It would seem that i is a special entity, which, like 1, can be used in such a way that, whatever its exponent, it remains identical to itself in its abstract being. -i is always equal to -i.
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We can do as with 1^x give all the possible and imaginable powers to x, always, 1^x=1.
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It seems that it is the same thing with i=-1.
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i²=-1.
i^(-1/2)=-1
i^4=-1
i°=-1
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etc...etc...etc...
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So we have to define i, because then, in the calculations, big sign errors can appear. The biggest one can be that, sometimes, depending on the concept i²=-1 or i²=1 if we simplify too quickly.
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Let's take z1=16+9i and z2=14+3i and make a product.
We have: aa'+i(ab'+ba')+(ib')(ib)
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If we make the product as is, we see that if i=-1 then (-9)(-3)=+27,
but if we square it, we have i²=-1 and i²bb'=-27
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The error is then colossal:
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Hachel finds Z=251+174i.
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Mathematicians find Z=197+174i
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We need to think about this and check what is correct and why it is correct.
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R.H.
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But what would be a compelling reason to accept one way to define multiplication of complex numbers over another way?
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With the standard way of defining complex numbers, if we pick any point on the unit circle and we multiply it with any other point on the unit circle, the result will end up on the unit circle. With your method, this would fail (in the sense that the product would not end up on the unit circle if the factors of the product are on the unit circle).
This seems like a compelling reason to me to prefer the standard way to define multiplication of complex numbers over your alternative way.
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You might say it's irrelevant whether complex numbers on the unit
circle multiply in a way that ends up on the unit circle, but other nice properties also fail, like the way we can multiply complex numbers in polar form by multiplying their modulus and adding their argument.

 I think a new approach to complex numbers may be possible, and it starts by redefining what the imaginary i is.
 It is defined in a dramatically stupid way.
 And we say, stuttering: "it is the... the number... uh... that... which... that if you square it, it becomes -1."
 This is not very reasonable.
 But this dramatic and narrow definition turns downright horrific when we say: "Let's square the square".
 Then everything becomes dreadful. We say (i²)(i²)=1 because (-1)(-1)=1.
 And we attribute to an imaginary structure the same property as to a real structure.
 But, hold on tight, friends, this is false.
 (i²)²=-1, and not 1.
 And there, the whole structure that we thought we had defined by a simple i²=-1, which was true, collapses for everything else.
 R.H.

Suppose you go to a store and you buy a product that costs 3 euro and
you pay them with a 10 euro bill. They give you back 6 euro in change and you complain that 10 - 3 = 7 and they say.. no, that's incorrect math. The correct math is 10 - 3 = 6.
You take out your calculator on your phone and show them that 10 - 3 = 7
on your calculator and they say.. well, that calculator uses flawed mathematics. We use correct mathematics in this store.
Would there be any way for you to convince them that their math is incorrect?
This is analogous to this discussion here on complex numbers.
There is an entire mathematical framework that would collapse if you
change the way complex numbers multiply in the way you propose.
Beautiful fundamental results like the power series demonstrating
Euler's formula would no longer hold.
https://en.wikipedia.org/wiki/Euler%27s_formula#Using_power_series