In Number Systems, students see that limits are set by one's creativity, rather than by "the way things are."
To me as a mathematician, one of the most amazing things about my field is its continual openness to new ideas. Math isn’t just about following a set of logical rules to their conclusions; it’s about making up new rules and seeing if anything interesting happens. What if 1+1 were 0? What if negative numbers had square roots? What if we changed the definitions of addition and multiplication? All of these childlike what-ifs turn out to lead to beautiful and useful areas of mathematics—different from everyday math, but consistent within themselves. One might even say that math is more art than science: the limits are set by one’s creativity, rather than by “the way things are."
When I set out to create a post-Algebra II course for Proof School kids, my goal was for students to experience some of this creative freedom—and learn how to follow through from the what-if phase to the investigation phase.
In part, we’ve been doing this by retracing the history of the numbers, trying to see familiar things as they looked when they were new.
There is a diagram which many students have seen in their algebra classes: a circle inside a circle inside a circle inside a circle, showing how the set of whole numbers fits inside the set of integers, which fits inside the set of rational numbers, which sits inside the set of real numbers, and so forth. This diagram doesn’t just convey a simple fact about sets—it tells a story about the history of mathematics! Moving out from the center, each circle represents a newer and richer concept of number, a hard-won expansion of what math is able to speak about. The very names of the newer types of numbers—negative, irrational, imaginary—convey the “outsider” status they once held relative to established number systems.
We’ve worked our way outward through this diagram, figuring out what flaws needed to be corrected in the old number systems and how the new systems accomplish this. We’ve also looked at the axioms, or “ground rules”, for each number system, and learned how to use them to prove what we thought we knew: that 1 > 0, or that the product of two nonzero numbers can’t be zero, or that every integer has a unique prime factorization. It was quite a challenge for many students to cast off assumptions and avoid circular reasoning! Yet this is a vital task if one is going to explore the more exotic realms of mathematics, where all of the above “facts” can fail to be true.
Along the way, we learned that some “familiar” number systems can be much weirder than they appear: for example, students were shocked to figure out that most real numbers cannot be printed out in any form by a computer program, or even described in English.
We’ve also created some of our own unusual number systems—like a system in which numbers are geometric distances, and arithmetic is done not by writing columns of figures, but by drawing with a compass and straightedge. One student is on a mission to create his own number system where 0 = 1!
Each week has had its own disparate theme, but as the course has progressed, we’ve started to see some unexpected links between different areas of math. For example, a tool we originally used to divide complex numbers turned out to also help us find fractions approximating √2, draw right triangles with integer sides, and figure out which integers are sums of two squares! The class is learning that when they explore their own what-if questions, the discoveries they make may help answer other questions as well.