In this student-driven class, we are starting from axiomatic foundations to explore both familiar number systems and exotic number-like objects.
If you ask a mathematician to explain what a "number" is, chances are they'll ask you to be more specific--there are a variety of mathematical objects to which the name "number" is applied. The natural numbers are the collection of non-negative whole numbers we use for counting things. The integers include negative whole numbers as well. The rational numbers allow us to use fractions, and the real numbers allow us to use numbers like π and √ 2 , which cannot be neatly described as fractions. Then there are the mysterious complex numbers, which allow us to factor otherwise unfactorable polynomials using i, an imaginary number whose square is -1.
So let's be more specific. What is a natural number? How would we formally define these counting numbers that we've been using since kindergarten? One way to define the natural numbers is with a formal system called the Peano Axioms. These axioms attempt to fully describe the behavior of the natural numbers in as few statements as possible. Specifically:
- There is a natural number called 0.
- For every natural number n, there is a "next" natural number S(n). We call this number the "successor" of n.
- Two natural numbers n and m are equal if and only if their successors are equal.
- The number 0 is not the successor of any natural number.
- If a set contains 0, and contains the successor of all of its elements, then the set contains all of the natural numbers.
This seems like a lot of work to describe a system of numbers which we all intuitively understand, but we need some sort of a solid base for building rigorous proofs. And the students in this class have been writing lots of proofs. By the second day of class, they had already verified several fundamental facts about addition and multiplication in the natural numbers. For instance, they showed that a(b+c) is always equal to ab+ac.
This is a student-driven class. While we have some full-class discussion, the vast majority of our learning is done through guided worksheets. Students work in groups, check their answers with their friends, and help each other out when they get stuck. And the vast majority of the mathematical content comes straight from the students. They decided how addition and multiplication should be defined. They discovered the tool we needed to handle fractions that are essentially "the same," like 3/2 and 9/6. Right now they are working on a formal write-up describing the construction of the integers in detail.
After this project, we will be able to forge ahead to discuss polynomials, complex numbers, and maybe even some more exotic number-like objects like the quaternions or the ordinals. Next time someone asks these students what a "number" is, they'll have one heck of an answer ready!
-- Susan Durst