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.
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.