What is zero times zero

Question

What is zero zeros? What are no nothings? From a mathematical point of view it would be my thing, but none of us are educated that much in math, so I am curious to hear an expert opinion.

Answer 1

It entirely depends on what you mean by 'zero'. Usually in algebra the zero element of some set, with some 'addition' operation has the property that $a + 0 = a$ where $a$ can be any element of your set. That is, we use addition to define what zero is. If your set of things (affectionately denoted $G$) also has a 'multiplication' operation, and that operation behaves like multiplication on $\mathbb R$ then it can be shown that $a \cdot 0 = 0$ for all $a$ in $G$. Since $0$ is also in $G$ it follows that $0 \cdot 0 = 0$. An example of an algebraic structure where this is all true is a field (http://en.wikipedia.org/wiki/Field_(mathematics)).

So, the way I rationalise this, no lots of nothing is still nothing, because you are counting the amount of nothing you have. But a cleaner way to seeing this is to realise where $0$ came from in the first place, and derive the fact that $0 \cdot 0 = 0$ using axioms.