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$\sqrt{-1}$ 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?

hmakholm left over Monica
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Heisenberg
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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 $

Zach Effman
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    +1. I might add that another problem is that you need a different $h_x$ for every nonzero numerator $x$, and then the situation gets even worse in the case $x=0$. – Ian Mar 17 '16 at 19:46
  • I think you can do it if you also introduce an indeterminate $k$, and define equality appropriately. – user7530 Mar 17 '16 at 19:48
  • @Ian That's true, but we could define $h_x = x\cdot h$ without much (further) issue. – Zach Effman Mar 17 '16 at 19:49
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    @Zach: I don't think so - you've fallen into the associativity trap! –  Mar 17 '16 at 20:31
  • @MikeMiller I don't think I have. $0\cdot (x\cdot h) := x$. We couldn't move the parentheses, but we already knew that. – Zach Effman Mar 18 '16 at 17:22
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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?

Konstantinos Gaitanas
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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.

Vik78
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    That doesn't seem to have anything in particular to do with division by zero. – hmakholm left over Monica Mar 17 '16 at 20:14
  • No, not directly, but seeing as he thought up this idea as an analogue to the complex numbers I thought he might be interested in an analogue to the complex numbers that's actually interesting algebraically. – Vik78 Mar 17 '16 at 21:09
  • You still cannot divide by it. By the way, division is usually prohibited not only by zero but by any zero divisors. – Anixx Sep 29 '21 at 14:11
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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.

Aurey
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  • There is also projective real line with just one added element, which is closer to division by zero. – Anixx Sep 29 '21 at 14:14