email@example.com (Mikstu3141) writes: >Any suggestions as to how to approach 0!=1 with an Algebra II class? Thanks-
I think the key point is that whenever a mathematical idea is extended beyond its original range, (1) the extension is to some extent arbitrary, but (2) one wants to preserve as much as possible the old rules about that idea.
So, for example, when we invent negative integers we are confronted with the question of what (-1)*(-1) should be. In a sense we could pick any answer we want, but it turns out that only one answer preserves the associative and distributive laws.
Similarly, in extending the idea of factorial down to 0!, we'd like to preserve the defining recurrence relation for factorial: n! = n * (n-1)! Plug in n=1 and you see that 0! must be 1 to preserve this rule.
You can then try n=0 to see that there can't be a (-1)! preserving the rule.