It depends on what area of study you are in. Division by zero tends to lead to a contradiction regardless of what you define it to be. Sometimes it is treated as being infinity (in the compacting of the complex numbers for example), sometimes it's not allowed.
By far the most common scenario I have seen is that one must define infinity times zero as either zero or infinity. For example in measure theory, one will define infinity times zero as 0. But in some other cases I vaguely recall, I believe I have seen it defined as infinity as well.
By far the most common scenario I have seen is that one must define infinity times zero as either zero or infinity. For example in measure theory, one will define infinity times zero as 0. But in some other cases I vaguely recall, I believe I have seen it defined as infinity as well.