Re: New way of dealing with complex numbers

On 3/8/2025 4:11 PM, Richard Hachel wrote:

Le 09/03/2025 à 00:43, efji a écrit :

I just pointed out the fact that the notation x^n is never used in the case of non associative operators because it is ambiguous without further definition. Think about the vector product in R^3 for example, which is not associative, and not commutative too. Nobody would write x^3 for (x \wedge x)\wedge x.
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In the case of Hachel's delirium, the product is obviously associative, thus i^2 = -1 and i^4 = -1 makes no sense.
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And of course, even with your recursive definition, it makes no sense.

 Why would it not make sense?
When zero was introduced into mathematics, perhaps some people said, it's absurd, since zero is nothing.
When negative numbers were introduced, perhaps some people thought the idea was stupid, and that in a field you couldn't have a herd of minus three sheep, or in a basket, minus three apples to go and sell them on the market in Baghdad.
The first relativistic physicists thought that relative time was absurd, and that a second was worth a second for everyone, in short that it was impossible for Stella to be 18 years old, and Terrence 30 years old.
Newton wondered what all the abstract manipulations based on the imaginary could be for, and how we could use them in a concrete way in our universe.
 Here, we have a much more precise definition than what we have been taught for several centuries. We simply pose:
i is this imaginary unit such that it can never be made positive by any power. Similarly for 1, which is constant and similar to itself whatever its exponent, we have, for any exponent x: i^x=-1.
 If this is an imaginary concept why not imagine it?

The imaginary component is a real thing. Heck, we can plot it on graph paper, gosh darn it. ;^)
i = (0 + 1i) = (0, 1)
Plot the sucker!

 Is it less extravagant, in a mathematical thought, to say that i²=-1 than to say that i^x=-1?
 If the natural law wants an imaginary to have its own law when we join positive or negative signs to it, how would this make no sense?
 R.H.