Why don't we define "imaginary" numbers for every "impossibility"? Before, the concept of imaginary numbers, the number $i = \sqrt{-1}$ was shown to have no solution among the numbers that we had. So we declared $i$ to be a new type of number. How come we don't do the same for other "impossible" equations, such as
$x = x + 1$, or $x = 1/0$?
Edit:
OK, a lot of people have said that a number $x$ such that $x = x + 1$ would break the rule that $0 \neq 1$. However, let's look at the extension from whole numbers to include negative numbers (yes, I said that I wasn't going to include this) by defining $-1$ to be the number such that $-1 + 1 = 0$. Note that this breaks the "rule" that "if $x \leq y$, then $ax \leq ay$", which was true for all $a, x, y$ before the introduction of negative numbers. So I'm not convinced that "That would break some obvious truth about all numbers" is necessarily an argument against this sort of thing.

i should not be defined as a solution to x^2=-1 (i.e. as an artificial way to find a "solution" to a previously considered "impossible" equation. Complex numbers are defined as orderer pairs (a,b) of real numbers. You define arithmetic rules (+,x,-,/) for such ordered pairs which can be done in a way that makes (a,0) equivalent with the ordinary real number a. Complex numbers (0,b) will be found to satisfy (0,b)^2=(-b^2,0). Voila, we have found a certain kind of numbers (0,b) also called "imaginary" numbers which when squared will give a complex number (-b^2,0) equivalent with the negative real number -b^2. A similar construction of a new type of number x satisfying x=x+1 cannot be made with preservation of established arithmetic rules.   
