Can we treat 'undefined' "like a thing"? $1/0=undefined$, and undefined just means what it says; it's not defined. What happens if we "treat undefined like a thing"? Does that even make sense? I am interested in this since we seem to assign "undefined" to anything whose answer is not defined (correct me if I'm wrong), but can we do anything more with it? I know the tag isn't right, but I don't know what tag would fit, so change it if you can think of a better one.
 A: It is possible to use values such as ERROR, UNDEFINED, N/A or MISSING, and give rules for propagating them through calculations. However, this is more useful as a construct in computer programs than for doing or communicating mathematics between people.
A: Your question is actually a very interesting one; while it is admittedly impossible to assign a real number to the expression 1/0 since it would prove 1=0, it turns out that we can do something close.
Suppose that, instead of $0$, we take a "real number" which is arbitrarily close to it (called an infinitesimal)—a "real number" which is positive, and yet which is smaller than any actual positive real number. Let us call it $\varepsilon$; we can easily deduce that $\varepsilon$ must have the property that $1/\varepsilon$ is greater than any actual positive real number. Hence, if arithmetic is to have any meaning in our new system, we must also introduce a new "real number", $1/\varepsilon$ which is greater than any actual real numbers.
Further reflection shows that, by multiplying real numbers by $\varepsilon$, we'd also need such things as $2\varepsilon,3\varepsilon,\dots$, and also $\varepsilon/2,\varepsilon/3,\dots$, and thus a "copy" of all the positive numbers, all of which would have the same property that they are somehow sandwiched between 0 and all real numbers—and that's not all! If we consider addition and negation as well, we must conclude that we need to have a "copy" of all real numbers in the "neighborhood" of every real number.
Now, one may of course object to all this construction on the basis that we are still not actually dividing by $0$, but merely a number which we willed to be arbitrarily small. It turns out, however, that considering a thing such as $\varepsilon$ has the convenient property that we can simply take it when we need it and turn it into $0$ again when it is not needed. Such a construction is in fact the underpinning of calculus. In traditional treatments of calculus, the role of having something like $\varepsilon$ is played by limits: consider the definition of the derivative:
$$\frac{df}{dx}=\lim_{\varepsilon\to0}\frac{f(x+\varepsilon)-f(x)}{\varepsilon}$$
We can see that "taking the limit" is the operation which makes $\varepsilon$ behave like this infinitesimal number, which we can pretend to not be $0$ when we deal with the division and afterwards make it go away as $0$ again. It turns out, however, that with more advanced techniques one can do away with the limit operation altogether and introduce $\varepsilon$ and its kin directly into the real numbers. An extension of the real numbers in which $\varepsilon$ is an element is called a model of nonstandard analysis; we want to make such an extension elementary (meaning that it satisfies exactly those first-order logical statements which the canonical model $\mathbb{R}$ of actual real numbers satisfies) in order that it also conform to our ideas of how the real numbers ought to behave. It turns out that we can do this by using the Compactness Theorem; an example of this treatment can be found here. More sophisticatedly, one can obtain a model of nonstandard analysis algebraically by taking an ultrapower $\mathbb{R}^\omega/\mathscr{U}$. (Such a construction would admittedly still be non-constructive, given that a non-principal ultrafilter on $\mathbb{R}^\omega$ can only be found by applying an existence lemma.)
Now, given this new structure (call it $\mathbb{R}^*$), even though we still cannot divide by $0$, we now have a proper idea of doing arithmetic with infinitesimals which are essentially $0$. We may show that each nonstandard number (i.e. elements of $\mathbb{R}^*\backslash\mathbb{R}$) is either infinite (i.e. greater than or smaller than all standard real numbers) or is infinitely close to a unique standard real number; this gives us the standard part function $s$ with which we can easily deal with limits. What is traditionally denoted $\lim_{x\to a^+}f(x)=y$ is then replaced by the statement $s(f(a+\varepsilon))=y$, where $\epsilon$ is some infinitesimal.
Coming back to the original question of dealing with such expressions as $1/0$; we may now see that, once we have a model of nonstandard analysis, we can easily just take an infinitesimal $\varepsilon$, and $1/\varepsilon$ behaves exactly as it should be in our model: it is greater than all standard real numbers, though it is still not the greatest: we can yield an even greater infinite number by taking $1/\varepsilon'$ for some "even more infinitesimal" $\varepsilon'$ (i.e. such that $0<\varepsilon'<\varepsilon$). The problem of trying to assign a value to $0/0$ is also disposed of: the value of $\varepsilon_1/\varepsilon_2$ simply depends on the relative "infinitesimalness" of the two infinitesimals; we may choose them in a way that yield any number in return, which corresponds to our intuition of $0/0$. Such a construction is probably the closest you can get to formalizing the idea of "treating division by 0 as a thing" and making the result conform to intuition.
