This is discussed in countless places on the web. Here for example
I have been trying to learn Latex and I used your post for practice inspiration.
I finally got it to look almost how I wanted so this is what I wanted to say.
Unfortunately I don't know how to transfer the program that I built in TEX to this site. Maybe someone can help me with that, so I had to snip the output and attach it as a thumbnail.
It might be what you were after.
You don't have to use a "complicated definition". For this question, it is sufficient to use the definition " , for n positive integer, is a multiplied by itself n times" From that, it follows immediately that . To get , we have to extend that original definition to 0, which is not a "positive integer". We are, of course, free to define something however we want, but from that previous definition, it is easy to show that . That is such a nice property that we would like to define in such a way as to have that still true. That is, we want . On the left, n+ 0= n so . That mean that, to have , we must define . (Part of showing that requires that a not be equal to 0. So we really have " as long as a is not 0" and is not defined.)