Re: New way of dealing with complex numbers

Le 09/03/2025 à 01:11, Richard Hachel a écrit :

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.
 In the case of Hachel's delirium, the product is obviously associative, thus i^2 = -1 and i^4 = -1 makes no sense.
 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.

True in the sense that it took quite a long time for zero to be considered as a number as any other.

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.

True in the sense that negative numbers were at first treated as "fictitious" quantities, just like square roots of negative quantities a few years (not that much !) later.
What matters is that *now* (for more than one century) zéro, natural and relative integers, fractions, reals, complex numbers, etc. have rock-solid definitions.

If this is an imaginary concept why not imagine it?

You've been explained 1000 times that the word "imaginary" when it comes to complex numbers is a historical remnant of the fact they had no rigorous definitions when they first were considered (to solve degree 3 polynomial equations with real coefficient).
The word "imaginary" stays but not as its usual meaning.

Is it less extravagant, in a mathematical thought, to say that i²=-1 than to say that i^x=-1?

It is. Considering that exists i such as i^2 = -1 leads to no contradiction. This is what puzzled mathematicians and was addressed three centuries later with a lot of debates amongst them.
Considering that exists i such as for all x i^x = -1 leads to immediate contradictions. As you've been shown.

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?

In maths, "no sense" is a synonymous with "inconstant". Your proposal (i^x = -1) is inconsistent.
You proposal contradicts this simple property of equality:
if a is in the domain of a function f, then a = b implies f(a) = f(b)
[pick a = i, b = -1, f:x->x^2 or a = i^2, b = -1 and the same f]