One Divided by Zero

Think logically about division. What does division mean? How do you divide a number by another number? Think, "repeated subtraction and revision until the quantity to subtract can no longer be subtracted". Now, what happens when you try to divide a number by zero?

 

In what sense do you wish to divide by zero? In the real numbers, it makes no sense to divide by zero. In other situations symbols such as 1/0 are perfectly well defined (but they still don't mean you can cancel a zero off in a multiplication).

 

What in the world is "ERR09"? A calculator notation?

If you are talking about dividing 1 (or any other non-zero number) in the Complex number system or any of its subfields, then "1/ 0" is just an error- you don't do it. It is true that the limit of 1/x, as "x goes to infinity" (which, in the real number system, is 'code' for "gets larger without bound"), is 0. I can't think of any situation in which it would make sense to say that 1 "divided by 0" is 0.

 

Dividing anything by zero is undefined (see the axioms of a field). However, as HallsofIvy pointed out, the limit of something like 1/x as x approaches 0 tends to either positive or negative infinity.

 

I'm going to get pedantic again. Sorry.

First, a priori, the axioms of a field (at least those normally given) do not state that 0 does not have a multiplicative inverse. It is, however, easy to deduce from the axioms that one must define 0*x=0 for all x, and one cannot cancel zeroes.

But there are more things than just fields. In the extended complex plane the symbols x/0 are defined for all non-zero x (and are equal to the symbol [itex]\infty[/itex]. Division by zero is still not the inverse of multiplication, though.