# Could we “invent” a number $h$ such that $h = {{1}\over{0}}$, similarly to the way we “invented” $i=\sqrt{-1}$? [duplicate]

$\sqrt{-1}$ 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?

## marked as duplicate by Henning Makholm, Matthew Towers, Surb, user147263, user296602 Mar 17 '16 at 22:01

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 $(h\cdot 0)\cdot 0 = 1\cdot 0 =0$

• +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
• @Zach: I don't think so - you've fallen into the associativity trap! – user98602 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

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?

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.

• That doesn't seem to have anything in particular to do with division by zero. – Henning Makholm 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

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.