Information on the Second Test

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.