Difference between undefined and forbidden (division by zero)

Question

Division by zero is undefined

Division by zero is not allowed / forbidden

I wonder if there is a mathematical difference between undefined and forbidden in the context of division my zero and if so, what's the correct term.

Answer 1

Well, technically it is not undefined because you could define it to anything you like.

The question is however: "Can you extend the familiar arithmetic rules known from integers, rationals or reals to also cover division by zero?"

The answer to that question is "No": However you define the result of operation $x/0$, you'll lose some of the nice properties. This is also true when you try to extend these numbers to also include $\infty$ and / or $-\infty$ or the like.

In some contexts, like computation, it's common to define $x/0$ to:

• $:=\mathrm{Inf}$ if $0<x<\mathrm{Inf}$ where Inf represents $\infty$.
• $:=-\mathrm{Inf}$ if $\mathrm{Inf}<x<0$.
• $:=\mathrm{NaN}$ if $x=0$ where NaN stands for not-a-number. So you can do it technically, but you lose properties like existence of inverse elements of multiplication, inverse elements of addition, associativity, etc.