The second test in this course will be given in class on Friday, April 4. The test is not cumulative. It covers only material that has been covered in class since the first test. The format will be similar to the first test: several pages of problems and short essay questions. Calculators will be available, in case you need them. This test counts for 17% of the total grade for the course.
The primary readings for the second section of the course were from the book ZERO: The Biography of a Dangerous Idea. This book shows what a long process it was to invent the idea of zero and to have it be accepted and to come into general use. It also makes the argument that zero has always been closely associated with another dangerous idea: infinity.
ZERO introduced several early number systems, giving us the chance to look at the way numbers were represented in the Egyptian, Greek, and Babylonian number systems. We encountered the idea of number bases, since Babylonian numbers used the base 60, and we expanded on this to look at a variety of other number bases. We saw that the same number that we express in one way in our base-ten number system can be represented in many very different ways in other number systems.
As we looked at ancient mathematics, we encountered the association in Greek philosophy between numbers and geometry -- an association that made it difficult to think of zero as a number. But we were also led to the idea of mathematical rigor and proof, including Euclid's beautiful proof that there are infinitely many prime numbers. This is based on the idea of divisibility and factoring and led us to look at the Fundamental Theorem of Arithmetic, the fact that any integer greater than one can be uniquely factored into a product of prime numbers. We used this fact in a few exercises from Problem Solving through Recreational Mathematics.
ZERO covers the long history of the invention in India of the base-10 number system with zero, its spread through the Arab world, and its eventual introduction into Europe. The book claims that there was resistance in Europe to the idea of zero because of zero's association with infinity and the void -- things that could not exist according to the dominant Aristotelian philosophy. In India, on the other hand, zero and infinity fit right in with their religion and philosophy. Ultimately, however, the utility of zero in banking, art, and science forced the West to accept zero. In fact, the literal void was observed in experiments using mercury-filled tubes.
In the last reading from ZERO, we saw how mathematicians claim to have "tamed" infinity, by reducing it to a point just like other points (on the Riemann Sphere) and by coming up with a rigorous theory of infinite sets of various sizes, and we saw an application of this in Cantor's proof that the real numbers form a larger infinity than the integers. But we also read "The Book of Sand," which reflects some of the power that the idea of infinity still has and the ability that it still has to unsettle and frighten.
In addition to all this, there were a few extra topics more loosely related to the major themes. We started the second section of the course with a brief look at the Mandelbrot set and the Chaos Game, which showed how an infinite process can lead to a complex and beautiful result, even if the individual steps in the process are very simple. And near the end, we looked at modular arithmetic (which is related to integers and divisibility) and its application to cryptography in the Caesar, Vigenere, and one-time pad ciphers. Although the cryptography reading also introduced public key encryption and the RSA cryptographic system, those topics will not be covered on the test.