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I'm wondering if there's any kind of "imaginary anti-grassmann" (for lack of a better idea) or some strange object or other in math that you can multiply by zero and somehow not get something other than just zero?

For clarity, the answer I'm looking for would be something like: "0m = m Where 'm' is a Foo-number, which is a special object that keeps it's state when multiplied by zero, and here's how it interacts with normal numbers."

I'm looking to see if someone has invented such an object and built an algebra for it, etc.

Mason Cloud
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  • That depends. What does it mean to "multiply by zero"? If this is a ring, then the answer is no. If it's not a ring, then what does "multiply" mean? The basic example I can think of is that in analysis $\infty \cdot 0$ is generally not defined. (Moreover, when it is contextually defined, it is usually zero.) – Ian Mar 22 '15 at 04:21
  • Basically, the meaning of "multiply by zero" doesn't matter with this context, as long as there is some meaning through which there is a valid operation that can be multiplied by zero and not get zero. I guess undefined isn't zero, so that works. I guess I was wondering if there's anything else, like some new object that holds it's state when multiplied through a zero. I guess I'm looking for something like: "0m = m Where 'm' is a Foo-number. And Foo numbers are these things that this guy in russia made up centuries ago, and here's how the algebra of these new types of numbers work..." – Mason Cloud Mar 22 '15 at 04:29
  • Dirac delta function? – Brian Tung Apr 15 '15 at 09:06
  • You asked for an object foo that when multiplied by zero remains foo. Then in your comment you write "as long as there is some meaning through which there is a valid operation that can be multiplied by 0 and not get zero". foo + 0 = foo. Or do you want operation 0 = operation 0? – amWhy Jul 18 '21 at 15:30

3 Answers3

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The name "zero" is usually reserved for in every setting to be the additive identity.

I.e. that $x+0=0+x=x$

In any multiplicative ring we have the following then:

$$\begin{align} x\cdot 0 &= x\cdot 0 & \text{by reflexivity}\\ x\cdot (0+0) & =x\cdot 0 & \text{by additive identity}\\ x\cdot 0 + x\cdot 0 & = x\cdot 0 &\text{by distributivity}\\ x\cdot 0 & =0 & \text{by cancellation} \end{align}$$

JMoravitz
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This is not exactly what you want but I think it is worth to note (and is too long to be a comment).

By convention we have $$\large\color{red}{0^0=1}$$

Somehow exponentiation (by integers) is a kind of multiplication and here $0$ multiplied $0$ times by himself gives $1$. I have to say that the first time I encountered this, it was a very weird feeling and is probably the same as the one I would have by seeing an example which would answers positively your question. Anyway this stays a convention and other answers showed that the axioms we are currently using in mathematics don't leave space for such aliens. Moreover, I'm not sure how interesting would be a theory which allows the existence of this kind of objects.

Surb
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Distributivity of multiplication over addition gives you the fact that the additive identity (zero) multiplied by any other element is $0$: $$0x=\left(0+0\right)x=0x+0x=0$$ See Example of "ring" without the distributive property? or near-ring.

Community
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parsiad
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