TITLE: What is $0\div0\cdot0$?
QUESTION [11 upvotes]: We all know that multiplication is the inverse of division, and therefore
$x\div{x}\cdot{x}=x$
But what if $x=0$? $0\div0$ is undefined so $0\div0\cdot0$ should be too, but whatever happens when we divide that first $0$ by $0$ should be reversed when we multiply it by $0$ again, so what is the right answer? 0, undefined, or something else entirely?

REPLY [3 votes]: The problem with $\frac{0}{0}$ is that $0$ is a multiple of itself. In fact, $0$ can be several multiples of itself. Some zeros are 3x bigger than other zeros, others are $\frac{\pi}{e}$ times bigger. The numerator and denominator of $\frac{0}{0}$ are both different flavors of $0$, but there is no indication in the expression what they taste like. Without that information, such as "the numerator is twice as big as the denominator", we can only assume that result is the universal omnivalue, a value that is equal to  every number simultaneously, even infinite ones.
This concept isn't as ridiculous as it sounds. Consider possible  replies to the question "Where are you right now?". Possible replies may include 704 Hausser Street, Queens, NYC, New York State, America, or simply just the Universe. People all over the world may give a different street address, but all would agree that they live in the universe. This begs the question, "Where, precisely, is the Universe?". The short answer: it's everywhere. It's a blur of all locations. It's an Omnivalue. Similarly, $t=\frac{0}{0}=[-\infty,\infty]$ is the solution to the equation
$$0t=0$$
Before continuing. lets consider the omnivalue $\sin\infty$. Because $\mid\sin x\mid\le1$, this would evaluate to the omnivalue $[-1,1]$. If we multiply that value by $0$, we could just multiply each of it's components by $0$ and return the singularity $0$.
$$0\sin\infty=0\cdot[-1,1]=[0,0]=0$$
Now that we can see that multiplying certain omnivalues by zero can merge the result back into a singuality (i.e. $0$), we would like to know if
$$\frac{0}{0}\cdot 0 = 0$$
I don't think that it does because while the real components of the omnivalue $\frac{0}{0}=[-\infty,\infty]$, the infinite components reblur the result when they are multiplied by zero. So I guess my answer is that the result is indeterminate (another omnivalue).