One Divided by Zero: What Is the Answer?

[xaviertidus]

Is it zero, undefined, infinity, or ERR09 :)

 

[symbolipoint]

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?

 

[matt grime]

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).

 

[HallsofIvy]

Is it zero, undefined, infinity, or ERR09 :)

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.

 

[maverick_starstrider]

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.

 

[matt grime]

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 $\infty$. Division by zero is still not the inverse of multiplication, though.