This depends on what properties you want your number system to satisfy. Usually you want for any three numbers a,b,c to satisfy

1. Associativity of addition: a+(b+c)=(a+b)+c This is quite useful, so we don’t want to give this up

2. Commutativity of addition: a+b=b+a Also useful but you could get around that if you really want to, but for our purposes let’s keep it

3. An additive identity (or zero): 0+a=a=a+0 You want a zero, so this is needed

4. Additive inverses: There exists x such that a+x=0 (here x=-a); you also want this

5. Associativity of multiplication: a*(bc)=(ab)*c Same as above, you want this property

6. Commutativity of multiplication: Useful but not necessary

7. A multiplicative identity (or one): 1a=a=a1 Usually with 1=/=0, also useful

8. Multplicative inverses for nonzero elements: Not that necessary, there are useful number systems without this (like the integers …,-1,0,1,…)

9. Distributivity: a(b+c)=ab+ac, (a+b)c=ac+bc You ant this, as it links addition and multiplication and this is quite desirable.

If you assume 4. and 9., you get 0a = (0+0)a=0a+0a, hence 0=0a. This means that you would have to give up distributivity wihin your number system, however distributivity is what links addition and multiplication together, hence your question would just be “what if we have two binary operations that don’t really interact with each other?” and the answer is: Maybe there are useful cases?

Edit: I forgot about losing property 4, in which case some examples are found in the following math stackexchange post

Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn’t make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.

the following image is the Mathematical Abstract for the book Quantum Theoretic Machines by August Stern

i honestly have no idea what any of it means, but i read the book (it’s fascinating) and remembered that the square root of zero is defined in his theory.

interesting, no?

This is a question I see from time to time, and it’s a good question to ask.

Your question as I understand it can be phrased another way as:

The square root of -1 has no defined answer. So we put a mask on it and pretend that’s the answer. We do math with the masked number and suddenly everything is fine now. Why can’t we do the same thing to division by zero?

The difference is that, if you try to put a funny mask on the square root of -1 and treat it like a number, nothing breaks, but if you try the same thing with a division by zero, all sorts of things break.

If you define i = √-1, that is the only thing i can ever be. That specific quantity. You can factor it out of stuff, raise it to that exponent, whatever. And if it is ever convenient to do so, you can always unmask it back into that thing, e.g. i^2 = (√-1)^2 = -1. All the while, all the already existing rules of math stay true.

A division by zero isn’t like this, because if you tried it, every number divided by zero would equal to the same thing. If we give it a name, say, 1 / 0 = z, then it would also be true that 2 / 0 = z. We could then solve both sides for zero:

1 / z = 0

2 / z = 0

then set them equal:

1 / z = 2 / z

then multiply both sides by z:

1 = 2

which is a contradiction.

i doesn’t have this problem.