# One Divided by Zero

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

symbolipoint
Homework Helper
Gold Member
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
Homework Helper
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
Homework Helper
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.

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
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.