A Tale of Two Numbers

A few months ago, we had just finished talking about polynomials and were moving into matrices.  Because a lot of matrix concepts have analogs in the real numbers, we kicked things off with a review of some real number topics.  Specifically, I wanted to talk about solving linear equations using multiplicative inverses as a preview of determinants and using inverse matrices for solving linear systems.  For instance:

\begin{array}{ll}    2x=8 & AX=B \\    2^{-1}2x = 2^{-1}8 & A^{-1}AX = A^{-1}B \\    1x = \frac{1}{2}8 & IX = A^{-1}B \\    x=4 & X = A^{-1}B    \end{array}

As an aside, I threw out this series of equations in the hopes of (a) foreshadowing singular matrices, and (b) offering a justification for the lifelong prohibition against dividing by zero:

\begin{array}{l}    0x=1 \\    0^{-1}0x = 0^{-1}1 \\    1x = \frac{1}{0}1 \\    x = \frac{1}{0}    \end{array}

I thought this was just so beautiful.  Why can’t we divide by zero?  Because zero doesn’t have a multiplicative inverse.  There is no solution to 0x = 1, so 0-1 must not exist!  Q.E.D.

As it turns out, Q.E.NOT.  One of my students said, “Why can’t we just invent the inverse of zero?  Like we did with i?”

Again, we had just finished our discussion of polynomials, during which we had conjured the square root of -1 seemingly out of the clear blue sky.  They wanted to do the same thing with 1/0.  What an insightful and beautiful idea!  Consider the following stories, from my students’ perspectives:

  1. When we’re trying to solve quadratic equations, we might happen to run into something like x2 = -1.  Now of course there is no real number whose square is -1, so for convenience let’s just name this creature i (the square root of -1), and put it to good use immediately.
  2. When we’re trying to solve linear equations, we might happen to run into something like 0x = 1.  Now of course there is no real number that, when multiplied by 0, yields 1, so for convenience let’s just name this creature j (the multiplicative inverse of 0), and put it to good use immediately.

Why are we allowed to do the first thing, but not the second?  Why do we spend a whole chapter talking about the first thing, and an entire lifetime in contortions to avoid the second?  Both creatures were created, more or less on the spot, to patch up shortcomings in the real numbers.  What’s the difference?

And this is the tricky part: how do I explain it within the confines of a high school algebra class?  Well, I can tell you what I tried to do…

Let’s suppose that j is a legitimate mathematical entity in good standing with its peers, just like i.  Since we’ve defined j as the number that makes 0j = 1 true, it follows that 0 = 1/j.  Consider the following facts:

\begin{array}{l}    2 \cdot 0 = 0 \\    2\frac{1}{j} = \frac{1}{j} \\    \frac{2}{j} = \frac{1}{j} \\    2 = 1    \end{array}

In other words, I can pretty quickly show why j allows us to prove nonsensical results that lead to the dissolution of mathematics and perhaps the universe in general.  After all, if I’m allowed to prove that 2 = 1, then we can pretty much call the whole thing off.  What I can’t show, at least with my current pedagogical knowledge, is why i doesn’t lead to similar contradictions.

Therein lies the broad problem with proof.  It’s difficult.  If there are low-hanging fruit on the counterexample tree, then I can falsify bad ideas right before my students’ very eyes.  But if there are no counterexamples, then it becomes incredibly tough.  It’s easy to show a contradiction, much harder to show an absence of contradiction.  I can certainly take my kids through confirming examples of why i is helpful and useful.  But in my 50 min/day with them, there’s just no way I can organize a tour through the whole scope and beauty of complex numbers.  Let’s be serious, there’s no way that I can even individually appreciate their scope and beauty.

The complex numbers aren’t just a set, or a group.  They’re not even just a field.  They form an algebra (so do matrices, which brings a nice symmetry to this discussion), and algebras are strange and mysterious beings indeed.  I could spend the rest of my life learning why i leads to a rich and self-consistent system, so how am I supposed to give a satisfactory explanation?

Take it on faith, kids.  Good enough?

Update 3/20/12: My friend, Frank Romascavage, who is currently a graduate student in math at Bryn Mawr College (right down the road from my alma mater Villanova), pointed out the following on Facebook:

“We need to escape integral domains first so that we can have zero divisors!  Zero divisors give a quasi-invertibility condition (with respect to multiplication) on 0.  They aren’t really true inverses, but they are somewhat close!  In Z_{6} we have two zero divisors, 3 and 2, because 3 times 2 (as well as 2 times 3) in Z_{6} is 0.”

In many important ways, an integral domain is a generalization of the integers, which is why they behave very much the same.  An integral domain is just a commutative ring (usually assumed to have a unity), with no zero divisors.  If there are two members of a ring, say a and b, then they are said to be zero divisors if ab = 0.  In other words, to “escape integral domains,” is to move into a ring where the Zero Product Property no longer holds.  This means that, in non-integral domains, we can almost, sort of, a little bit, divide by zero.  Zero doesn’t really have a true inverse, but it’s close.  Frank’s example is the numbers 2 and 3 in the ring of integers modulo 6, since 3 x 2 = 0 (mod 6).  In fact, the ring of integers modulo n fails to be an integral domain in general, unless n is prime.  CTL