Could we "invent" a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we "invented" $i=\sqrt{-1}$? $\sqrt{-1}$ was completely undefined in the world before complex numbers. So we came up with $i$. 
$1\over0$ is completely undefined in today's world; is there a reason we haven't come up with a new unit to define it? Is it even possible, or would it create logical inconsistencies?
 What would be the effect on modern math if we did so?

There are the dual numbers, which is another two dimensional associative algebra over the reals like the complex numbers. The basis elements are 1 and h, where we define h as a nonzero number whose square is zero. 
