0!rganized Emptiness

On the back of the fundamental counting principle, my class has just established the fact that we can use n! to count the number of possible arrangements of n unique objects.  This is fantastic, but we don’t always want to arrange all of the n things available to us, which is okay.  We’ve also been introduced to the permutation function, which has the very nice property of counting ordered arrangements of r-sized subsets of our n objects.  Handy indeed.

Today we made an interesting observation: we now have not one, but two ways to count arrangements of, let’s say, 7 objects.

  1. We can fall back on our old friend, the factorial, and compute 7!
  2. We can use our new friend, the permutation function, and compute \bf{_7P_7}

Since both expressions count the same thing, they ought to be equal, but then we run into this interesting tidbit when we evaluate (2):

_7P_7 = \frac{7!}{(7-7)!} = \frac{7!}{0!},

which seems to imply that 0! = 1.  To say this is counterintuitive for my kids would be a severe understatement.  And in this moment of philosophical crisis, when the book might present itself as a palliative ally, students are instead met with this:

To prevent inconsistency?  How in the world are kids supposed to trust a mathematical resource that paints itself into a corner, only tacitly admits such, and then drops a bomb of a deus ex machina in order to save face?  I haven’t been so angry since the ending of Lord of the Flies.  Especially when this problem appears two pages later:

Okay, 8!.  So how many ways can I arrange my bookshelf with a zero-volume reference set?  One: I can arrange an empty shelf in exactly one way.  And, since we already know that n! counts the ways I can arrange n objects, it follows naturally that this 1 way of arranging 0 things must also be represented by 0!.

There are a lot of good proofs/justifications available for the willing Googler, but this one, to me, seems like the most natural and straightforward for a high school classroom.  At a bare minimum, it’s much, much better than, “Because I need it to be true for my own convenience.”

Only a math textbook could take something so lovely and make it seem dirty.

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4 thoughts on “0!rganized Emptiness

  1. I am totally with you, man. Only one objection to the flavor of your objection…

    Isn’t it possible that the answer to “How many ways can I arrange an empty bookshelf” might be zero (since there are no books to arrange)? Or that it might be “Undefined” (again, since there are no books to arrange; making it nonsensical to talk about arranging them)?

    I would vote for laying these possibilities out on the table and asking which of these answers is the “best” one. I think we can trust students to discern that the “best” one will ultimately have to do with consistency. They’re going to want the two computations to come out the same, right?

    But dropping that sentence in before having the debate? I agree that stinks.

    We could also work backwards, couldn’t we? 2! is \frac{3!}{3}, and 1! is \frac{2!}{2}, so 0! is \frac{1!}{1}.

    • Re working backwards, I did consider using the recursive argument you outlined above, much in the same way I tried to justify n0 = 1 (which, I suppose, is especially fitting since they both turn on the idea of a nullary product), but my students did not react particularly well to that justification the first time around, and so I was nervous about repeating the performance.

      I agree that consistency needs to make an appearance in the discussion. Ultimately, regardless of how (un)intuitive an explanation is, consistent results must be preserved. Consider this actual (though considerably shortened) discussion we had a few months ago:

      In the course of evaluating quadratic equations, we might run into x2 = -1. Now of course there is no real number whose square is -1, so for convenience we will name that creature i (the square root of -1) and put it to good use immediately. In the course of evaluating linear equations, we might run into 0x = 1. Now of course there is no real number that, when multiplied by 0, yields 1, so for convenience we will name that creature j (the multiplicative inverse of 0) and put it to good use immediately.

      We allow the first scenario, but don’t allow the second one? This seemed incredibly suspicious to my students. After all, both of those numbers were made up on the spot to fill a gap in the real numbers. The true distinction, which is largely hidden from student view, is that imaginary numbers lead to a rich, useful, and above all else consistent number system, whereas permitting division by zero allows us to prove nonsensical results and more or less leads to the dissolution of mathematics and possibly the universe.

      So yes, consistency is legitimately important, but I fear it seems too much like handwaving for my students to stomach, so I tried to make a direct appeal to intuition instead. It’s not wonderful mathematics, but I think it’s better pedagogy.

      I think, though, in the future, I’ll take your suggestion and use this as an opportunity to debate the possibilities. I’m not sure how it will go. Since these students have no foundation in set theory (in even the most basic sense), I’m worried that they will have much trouble distinguishing the empty set (a thing containing nothing) from just plain nothing, which might prove problematic.

      • Now we’re getting somewhere, though, right? Debating distinctions between the empty set and just plain nothing? Seriously? That’s some good stuff, right there.

        And that business of the distinction between sqrt(-1) and 1/0? Also brilliant. And should absolutely not be hidden from student view.

        One thing I really dig about your writing is your ability to go to these places mentally-to know the landscape of the mathematics in a rich and abstract way, and also to identify with what it must be like to be a student lost in this landscape (or worse, to be a student being trucked through this landscape on a superhighway with nary a rest area and then being told she has visited it). Keep that up, man.

      • Oh yeah, and consider this a request/demand to write that conversation up in fuller detail. I’m dying to know which parts of the dialogue were yours and which theirs.

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