Could dividing by zero actually be useful?

If you study complex analysis, you see that the additional mathematics stemming from the use of i = √-1 is incredible and beautiful. The real numbers form a beautiful structured called a field, but in that field you can't solve equations like x^2 + 1 = 0. If you extend that field by including i, you can solve it. The field is now "algebraically complete." And now you can say that every polynomial of degree n has exactly n zeroes (counting multiplicity). That's just one example of the incredible things you get from i.

e^iπ = 1 = 0 is another.

However, you cannot extend division to 0. Division is multiplication by the inverse. And you cannot have an inverse for zero where 0(x) = 1. That's because 0x = (1-1)x = x-x = 0.

Trying to extend division to 0 would require altering the fundamental axioms of the foundation of our number systems. If you tried to do that, you could come up with some sort of new algebraic system, but it wouldn't resemble anything we now use. It would be a fundamentally new structure.

The very definition of division prevents dividing by zero from making sense.

EXCEPT if you have 0/0. Depending on the function(s) that lead to a division of 0/0, there MAY be a way (usually, using limits) to determine if that division has a value.

The definition of a square root does not forbid the square root of a negative number (or even a complex number or a quaternions or...).

It's just that in REAL NUMBERS, there is no real number that can be used to express the square root of a negative number.

If you move away from real numbers, then there can be solutions.

Division by ZERO is undefined in our mathematics !

Allowing the operation results in CONTRADICTIONS !

That is NOT the result when finding the square root of negative numbers...

No. No. No.