It's undefined because there isn't an obvious choice:
- At x:=0, every real is a reasonable limit for some path through the euclidean plane to (0,0). Ignore the 0/0 problem, then x/0 still has either positive or negative infinity as reasonable choices. Suppose you choose a different extension like only having a single point at infinity; then you give up other properties like being able to add and subtract all members of the set you're considering.
- However you define x/0, it still doesn't work as a multiplicative inverse for arbitrary x.
A good question to ask is why you want to compute x/0 in the first place. It is, e.g., sometimes true that doing so allows you to compute results arithmetically and ignore intermediate infinities. Doing so is usually dangerous though, since your intuition no longer lines up with the actual properties of the system, and most techniques you might want to apply are no longer valid. Certain definitions are more amenable to being situationally useful (like the one-point compactification of the reals), but without a goal in mind the definition you choose is unlikely to be any better than either treating division as a non-total function (not actually defined on all of ℝ), or, equivalently, considering its output to be ℝ extended with {undefined}.
Not that it directly applies to your question, ℝ typically does not include infinity. ℝ⁺ is one symbol sometimes used for representing the reals with two infinities, though notation varies both for that and the one-point compactification.
- At x:=0, every real is a reasonable limit for some path through the euclidean plane to (0,0). Ignore the 0/0 problem, then x/0 still has either positive or negative infinity as reasonable choices. Suppose you choose a different extension like only having a single point at infinity; then you give up other properties like being able to add and subtract all members of the set you're considering.
- However you define x/0, it still doesn't work as a multiplicative inverse for arbitrary x.
A good question to ask is why you want to compute x/0 in the first place. It is, e.g., sometimes true that doing so allows you to compute results arithmetically and ignore intermediate infinities. Doing so is usually dangerous though, since your intuition no longer lines up with the actual properties of the system, and most techniques you might want to apply are no longer valid. Certain definitions are more amenable to being situationally useful (like the one-point compactification of the reals), but without a goal in mind the definition you choose is unlikely to be any better than either treating division as a non-total function (not actually defined on all of ℝ), or, equivalently, considering its output to be ℝ extended with {undefined}.
Not that it directly applies to your question, ℝ typically does not include infinity. ℝ⁺ is one symbol sometimes used for representing the reals with two infinities, though notation varies both for that and the one-point compactification.