There are numerous conventions in mathematics that student continually question.
- Why isn’t 1 a prime number?
- Why is 0! defined to be 1?
- Why is an empty sum 0 and an empty product 1?
- Why can’t you just say 1/0 = ∞?
There are good reasons for the existing conventions, and they usually boil down to this: On the whole, theorems are more simply stated with these conventions than with the alternatives. For example, if you defined 0! to be some other value, say 0, then there would be countless theorems that would have to be amended with language of the form “… except when n is zero, in which case …”
In short, the existing conventions simplify things more than they complicate them. But that doesn’t mean that everything is simpler under the standard conventions. The next post gives an example along these lines.