One of the defining properties of 0 is that anything multiplied by it results in 0.

So in your operation, without being given the actual result, I’d say no, the question is ill-defined.

You get this property in algrabraic structures called “wheels”. The simplest to understand wheel is probably the wheel of fractions, which is a slightly different way of defining fractions that allows division by 0.

The effect of this is to create 2 additional numbers: ∞ = z/0 for z != 0, ⊥, and ⊥ = 0/0.

Just add infinity gives you the real projective line (or Riemen Sphere if you are working with comples numbers). In this structure, 0 * ∞ is undefined, so is not quite what you want

⊥ (bottom) in a wheel can be thought as filling in for all remaining undefined results. In particular, any operation involving ⊥ results in ⊥. This includes the identity: 0 * ⊥ = ⊥.

As far as useful applications go, there are not many. The only time I’ve ever seen wheels come up when getting my math degree was just a mistake in defining fractions.

In computer science however, you do see something along these lines. The most common example is floating point numbers. These numbers often include ∞, -∞ and NaN, where NaN is essentially just ⊥. In particular, 0 * NaN = NaN, also 0 * ∞ = ⊥. The main benefit here is that arithmetic operations are always defined.

I’ve also seen an arbitrary precision fraction library that actually implemented something similar to the wheel of fractions described above (albeit with a distinction between positive and negative infinity). This would also give you 0 * ∞ = ⊥ and 0 * ⊥ = ⊥. Again, by adding ⊥ as a proper value, you could simplify the handling of some computations that might fail.

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?

While introducing a new number that would yield a nonzero result when multiplied by zero would break the logic of arithmetic and algebra, leading to irresolvable contradictions, we do have something kind of similar.

You’re probably familiar with certain things, like 1/0, being undefined: They don’t have any sensible answer, and trying to give them an answer leads to the same sort of irresolvable logical contradictions as making something times zero be nonzero.

There’s a related concept you might also be familiar with, called indeterminate forms. While something like 1/0 is undefined, 0/0 is an example of an indeterminate form - and they’re special because you can sensibly say they equal anything you want.

Let’s say 0/0 = x. If we multiply both sides of that equation by 0, we get 0 = 0 * x. The right side will equal 0 no matter what x is - and so the equation simplifies to 0 = 0. So our choice of x didn’t matter: No matter what value we say 0/0 equals, the logic works out.

This isn’t just a curiosity - pretty much all of calculus works on the principle of resolving situations that give indeterminate forms into sensible results. The expression in the definition of a derivative will always yield 0/0, for example - but we use algebraic and other tricks to work actual sensible answers out of them.

0/0 isn’t the only indeterminate form, though - there are a few. 0^0 is one. So are 1^∞ and ∞ - ∞ and ∞⁰ and ∞/∞ and, most important to your question, 0*∞. 0 times infinity isn’t 0 - it’s indeterminate, and can generally be made to equal whatever value you want depending on the context. The expression that defines integrals works out to 0*infinity, in a sense, in the same way the definition for derivatives gives 0/0.

This doesn’t break the rules or logic of arithmetic or algebra because infinity isn’t an actual number - it’s just a concept. Any time you see infinity being used, what you really have is a limit where some value is increasing without bound - but I thought it was close enough to what you asked to be worth mentioning.

There can be no such actual number that gives a nonzero number that works with the standard axioms and definitions of arithmetic and algebra that we all know and love - they would necessarily break very basic things like the distributive property. You can define other logically consistent systems where you get results like that, though. Wheel algebra is one such example - note that the ‘Algebra of wheels’ section specifically mentions 0*x ≠ 0 in the general case.

Such a number leads to mathematical paradox, it’s beyond useless.