HC 241H: The Necessity for New Numbers

Professor: Chris Sinclair

4.00 credits

  • CRN 26170: Monday & Wednesday, 1415-1545 @ REMOTE

The counting, or natural, numbers seem pretty fundamental to the nature of the universe. The rational numbers, or fractions, arise fairly naturally too, since some things humans might count—like apples—can be divided and one might sensibly need to capture the idea of ’two and a half apples’. But there are far more numbers/quantities/measurements that we might want to capture than can be accommodated by the rationals. For instance, the length of the diagonal of a square of side length one is the square root of 2. The square root of 2 is not rational (a fact that, at least apocryphally, drove the ancient Greeks nuts), and hence the need for numbers beyond the rationals. We will explore the need and construction of new kinds of numbers. We will do this through both historical and ahistorical lenses, using examples and basic proofs. No prerequisites required, though projects may be assigned based on mathematical (etc) background.