A few months ago, we had just finished talking about polynomials and were moving into matrices.\u00a0 Because a lot of matrix concepts have analogs in the real numbers, we kicked things off with a review of some real number topics.\u00a0 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.\u00a0 For instance:<\/p>\n
$latex begin{array}{ll}<\/p>\n
2x=8 & AX=B \\<\/p>\n
2^{-1}2x = 2^{-1}8 & A^{-1}AX = A^{-1}B \\<\/p>\n
1x = frac{1}{2}8 & IX = A^{-1}B \\<\/p>\n
x=4 & X = A^{-1}B<\/p>\n
end{array}&s=2$<\/p>\n
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:<\/p>\n
$latex begin{array}{l}<\/p>\n
0x=1 \\<\/p>\n
0^{-1}0x = 0^{-1}1 \\<\/p>\n
1x = frac{1}{0}1 \\<\/p>\n
x = frac{1}{0}<\/p>\n
end{array}&s=2$<\/p>\n
I thought this was just so beautiful.\u00a0 Why can't we divide by zero?\u00a0 Because zero doesn't have a multiplicative inverse.\u00a0 There is<\/strong> no solution to 0x = 1, so 0-1<\/sup> must not exist!\u00a0 Q.E.D.<\/p>\n As it turns out, Q.E.NOT.\u00a0 One of my students said, \"Why can't we just invent<\/strong> the inverse of zero?\u00a0 Like we did with<\/em><\/strong> i<\/strong><\/em>?\"<\/p>\n 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.\u00a0 They wanted to do the same thing with 1\/0.\u00a0 What an insightful and beautiful idea!\u00a0 Consider the following stories, from my students' perspectives:<\/p>\n Why are we allowed to do the first thing, but not the second?\u00a0 Why do we spend a whole chapter<\/strong> talking about the first thing, and an entire lifetime<\/strong> in contortions to avoid the second?\u00a0 Both creatures were created, more or less on the spot, to patch up shortcomings in the real numbers.\u00a0 What's the difference?<\/p>\n And this is the tricky part: how do I explain it within the confines of a high school algebra class?\u00a0 Well, I can tell you what I tried<\/strong> to do...<\/p>\n Let's suppose that j<\/em> is a legitimate mathematical entity in good standing with its peers, just like i<\/em>.\u00a0 Since we've defined j<\/em> as the number that makes 0j<\/em> = 1 true, it follows that 0 = 1\/j<\/em>.\u00a0 Consider the following facts:<\/p>\n $latex begin{array}{l}<\/p>\n 2 cdot 0 = 0 \\<\/p>\n 2frac{1}{j} = frac{1}{j} \\<\/p>\n frac{2}{j} = frac{1}{j} \\<\/p>\n 2 = 1<\/p>\n end{array}&s=2$<\/p>\n In other words, I can pretty quickly show why j<\/em> allows us to prove nonsensical results that lead to the dissolution of mathematics and perhaps the universe in general.\u00a0 After all, if I'm allowed to prove that 2 = 1, then we can pretty much call the whole thing off.\u00a0 What I can't<\/strong><\/em> show, at least with my current pedagogical knowledge, is why i<\/em> doesn't lead to similar contradictions.<\/p>\n Therein lies the broad problem with proof.\u00a0 It's difficult.\u00a0 If there are low-hanging fruit on the counterexample tree, then I can falsify bad ideas right before my students' very eyes.\u00a0 But if there are<\/strong> no counterexamples, then it becomes incredibly tough.\u00a0 It's easy to show a contradiction, much harder to show an absence of contradiction<\/strong>.\u00a0 I can certainly take my kids through confirming examples of why i<\/em> is helpful and useful.\u00a0 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.\u00a0 Let's be serious, there's no way that I can even individually appreciate<\/strong> their scope and beauty.<\/p>\n The complex numbers aren't just a set, or a group.\u00a0 They're not even just a field.\u00a0 They form an algebra<\/strong> (so do matrices, which brings a nice symmetry to this discussion), and algebras are strange and mysterious beings indeed.\u00a0 I could spend the rest of my life learning<\/strong> why i<\/em> leads to a rich and self-consistent system, so how am I supposed to give a satisfactory explanation?<\/p>\n Take it on faith, kids.\u00a0 Good enough?<\/p>\n Update<\/span> 3\/20\/12<\/span><\/strong>: My friend, Frank Romascavage, who is currently a graduate student in math at Bryn Mawr College (right down the road from my alma mater<\/em> Villanova), pointed out the following on Facebook:<\/p>\n \"We need to escape integral domains first so that we can have zero divisors!\u00a0 Zero divisors give a quasi-invertibility condition (with respect to multiplication) on 0.\u00a0 They aren't really true inverses, but they are somewhat close!\u00a0 In $latex Z_{6}$ we have two zero divisors, 3 and 2, because 3 times 2 (as well as 2 times 3) in $latex Z_{6}$ is 0.\"<\/p>\n<\/blockquote>\n In many important ways, an integral domain is a generalization of the integers, which is why they behave very much the same.\u00a0 An integral domain is just a commutative ring (usually assumed to have a unity), with no zero divisors.\u00a0 If there are two members of a ring, say a<\/em> and b<\/em>, then they are said to be zero divisors if ab<\/em> = 0.\u00a0 In other words, to \"escape integral domains,\" is to move into a ring where the Zero Product Property no longer holds.\u00a0 This means that, in non-integral domains, we can almost, sort of, a little bit, divide by zero.\u00a0 Zero doesn't really have a true inverse, but it's close.\u00a0 Frank's example is the numbers 2 and 3 in the ring of integers modulo 6, since 3 x 2 = 0 (mod 6).\u00a0 In fact, the ring of integers modulo n<\/em> fails to be an integral domain in general, unless n<\/em> is prime. \u00a0CTL<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":" A few months ago, we had just finished talking about polynomials and were moving into matrices.\u00a0 Because a lot of matrix concepts have analogs in the real numbers, we kicked things off with a review of some real number topics.\u00a0 Specifically, I wanted to talk about solving linear equations using multiplicative inverses as a preview 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