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This question already has an answer here:

$\sqrt{-1}$ 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?

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marked as duplicate by Henning Makholm, Matthew Towers, Surb, user147263, T. Bongers Mar 17 '16 at 22:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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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 $

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    $\begingroup$ +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 $x=0$. $\endgroup$ – Ian Mar 17 '16 at 19:46
  • $\begingroup$ I think you can do it if you also introduce an indeterminate $k$, and define equality appropriately. $\endgroup$ – user7530 Mar 17 '16 at 19:48
  • $\begingroup$ @Ian That's true, but we could define $h_x = x\cdot h$ without much (further) issue. $\endgroup$ – Zach Effman Mar 17 '16 at 19:49
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    $\begingroup$ @Zach: I don't think so - you've fallen into the associativity trap! $\endgroup$ – user98602 Mar 17 '16 at 20:31
  • $\begingroup$ @MikeMiller I don't think I have. $0\cdot (x\cdot h) := x$. We couldn't move the parentheses, but we already knew that. $\endgroup$ – 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?

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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.

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  • $\begingroup$ That doesn't seem to have anything in particular to do with division by zero. $\endgroup$ – Henning Makholm Mar 17 '16 at 20:14
  • $\begingroup$ 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. $\endgroup$ – Vik78 Mar 17 '16 at 21:09
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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.

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