Difference between undefined and forbidden (division by zero)

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.

• The division by zero is forbidden to high school students because it is undefined. Feb 11 '18 at 9:04
• Whether something is forbidden is an ethical question, not a mathematical one Feb 11 '18 at 9:06
• So, there's no definition of "forbidden" in mathematics? Feb 11 '18 at 9:09
• "forbidden" is a term one should reserve to young minds and therefore is a more pedagogical term (and maybe a bad one). For someone more mature, he or she should not divide by zero because it is not yet defined, definitely not because someone forbade him/her to do so. So no "forbidden" is not mathematical whereas undefined is. Feb 11 '18 at 9:29
• 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?" The answer to that question is "No": However you define the result of operation $x/0$, you'll lose some of the nice properties. Feb 6 '20 at 16:10

1 Answer

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 $$0 where Inf represents $$\infty$$.
• $$:=-\mathrm{Inf}$$ if $$\mathrm{Inf}.
• $$:=\mathrm{NaN}$$ 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.