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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.

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    $\begingroup$ The division by zero is forbidden to high school students because it is undefined. $\endgroup$ Feb 11 '18 at 9:04
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    $\begingroup$ Whether something is forbidden is an ethical question, not a mathematical one $\endgroup$ Feb 11 '18 at 9:06
  • $\begingroup$ So, there's no definition of "forbidden" in mathematics? $\endgroup$
    – Scriptim
    Feb 11 '18 at 9:09
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    $\begingroup$ "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. $\endgroup$ Feb 11 '18 at 9:29
  • $\begingroup$ 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. $\endgroup$ Feb 6 '20 at 16:10
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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.
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