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?
 A: You could do this, but you'd have to sacrifice associativity of multiplication. Presumably $h\cdot 0$ should equal $1$, but then $h\cdot(0\cdot 0) = h\cdot 0 = 1$, while $(h\cdot 0)\cdot 0 = 1\cdot 0 =0 $
A: The imaginary unit was "invented" because we wanted to solve algebraic equations with formulas involving roots.
There seems to be no reason for "inventing" such a number.
This is the main reason (I think) why we don't invent such a number.
Whould it be usefull anywhere at all?
A: 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. 
A: It isn't mathematically consistent with our concepts of multiplication. However, in a similar vein of reasoning, mathematicians sometimes employ the extended real numbers, which are basically the real numbers with two new elements, positive and negative infinity. Although it still isn't 1/0, defining these two numbers has a lot of use in measure theory.
