I teach a precalculus course almost every semester, and over these semesters I’ve found various things that work quite well. For example, when talking about polynomials and rational functions, in particular “zeroes” and “vertical asymptotes”, I introduce them as the same thing, only the asymptotes are “points at infinity”. This (projective plane) model helps the students in understanding what multiplicity of a zero/asymptote really means. I later use the same projective plane model to show how all conic sections (circle, ellipse, parabola, hyperbola, lines) are related. I get very nice feedback on this, and the students seem to really enjoy it.
I have not, however, been able to find similar motivating examples for introducing complex numbers. I know there must be similar (pictorial!) arguments to engage the students and pique their curiosity, but I haven’t found it yet. Simply saying “all polynomials have a zero over the complex numbers” doesn’t really do it for them (again, the more pictures involved, the better).
Are there “neat” and “cool” ways of talking about complex numbers for the first time, that are understandable by beginning precalculus students, but also interesting enough to capture their attention and provoke thought?
Edan Maor provides an answer on this post, which gives a nice explanation:
A way to solve polynomials
We came up with equations like x−5=0, what is x?, and the naturals solved them (easily). Then we asked, “wait, what about x+5=0?” So we invented negative numbers. Then we asked “wait, what about 2x=1?” So we invented rational numbers. Then we asked “wait, what about x2=2?” so we invented irrational numbers.
Finally, we asked, “wait, what about x2=−1?” This is the only question that was left, so we decided to invent the complex numbers, in particular “imaginary” numbers, to solve it. All the other numbers, at some point, didn’t exist and didn’t seem “real”, but now they’re fine. Now that we have complex numbers, we can solve every polynomial, so it makes sense that that’s the last place to stop.
Pairs of numbers
This explanation goes the route of redefinition. Tell the listener to forget everything he knows about imaginary numbers. You’re defining a new number system, only now there are always pairs of numbers. Why? For fun. Then go through explaining how addition/multiplication work. Try and find a good “realistic” use of pairs of numbers (many exist).
Then, show that in this system, (0,1)∗(0,1)=(−1,0), in other words, we’ve defined a new system, under which it makes sense to say that √−1=i, when i=(0,1). And that’s really all there is to imaginary numbers: a definition of a new number system, which makes sense to use in most places. And under that system, there is an answer to √−1.
- Along these lines, see André Nicolas’s answer to this post.
The historical explanation
Explain the history of the imaginary numbers. Showing that mathematicians also fought against them for a long time helps people understand the mathematical process, i.e., that it’s all definitions in the end.
I’m a little rusty, but I think there were certain equations that kept having parts of them which used √−1, and the mathematicians kept throwing out the equations since there is no such thing.
Then, one mathematician decided to just “roll with it”, and kept working, and found out that all those square roots cancelled each other out.
Amazingly, the answer that was left was the correct answer (he was working on finding roots of polynomials, I think). Which lead him to think that there was a valid reason to use √−1, even if it took a long time to understand it.
Another idea comes from Byron Schmuland answer on this Math Overflow thread:
Use complex numbers to explain Ptolemy’s Theorem. For a cyclic quadrilateral with vertices A,B,C,D we have
- Thanks to André Nicolas, whom I’ll quote: that mathematician “was Bombelli, showing that for cubics with three distinct real roots, the Cardano Formula involved square roots of negative numbers.” (See André Nicolas’s comment below.)