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.

Instead of being content with having only 1 number having square equal to -1, we have 3 different such numbers. Then we can build the quaternions where $k^2 = j^2 = i^2 = -1$. And why not have 7 different such numbers? Then we could build the octinions and even more complicated can of course be built in a similar way.
Also we can define many different operations on our numbers. We are used to having addition and multiplication. This is usually the case for an algebraic field. But some types of numbers have only one operation and some can have many many operations defined on them.
Also instead of defining operations with numbers to the left or right on a line, we could define operations on a grid. Each operation having not only two elements in a particular order but a whole set of neighbours as an input.
