Simply from the title of the course called "Fundamentals" I wouldn't know what material would be covered. The course description says it is about number theory and combinatorics. While these two topics are not necessarily interrelated, they do have some things in common. The most interesting is that both topics do not require very much background in order to encounter very difficult problems.

What we find early on in this course that there are still things that are not known about the integers. For instance, while we know for sure that there are an infinite number of primes, we do not know if there are an infinite number of pairs of prime numbers of the form p and p+2 (e.g. (3,5), (5,7), (11, 13), (17,19), (29, 31), ...). Primes themselves are very hard to pin down and describe. It is still very surprising to me that there doesn't seem to exist a formula for an infinite sequence of primes.

One of the main applications of number theory is that of cryptography (sending coded messages so that an interceptor is unable to read them). One of the commonly used codes in modern cryptography is based theorems about integers that were discovered hundreds of years before the computer age. It still amazes me that the pieces of number theory that were discovered by mathematicians that could not dream of a modern computer became so relevant to making sure that your credit card number is not stolen when you send it over the internet. Although we only briefly touch on this application of number theory (the textbook was written before its invention!), most of what we learn is relevant for further study in cryptography.

Combinatorics is similar to number theory in that most people are familiar with the basic ideas of counting. However, it is easy to describe a set of things which is hard to count. For instance, how many ways are there of seating 10 couples (or more generally, n couples) around a table if no person is allowed to sit next to his or her spouse? To learn some of the basic rules of counting we build up arguments out of simple rules which readers accept and agree upon.

What makes these two topics "fundamental?" The main thing is that they provide a perfect setting for practicing a skill which clearly is fundamental to all of mathematics, writing and expressing yourself clearly and basing your arguments on principles that your reader accepts. This course takes some fundamental ideas (that explanation, justification, logic, proof and reason are behind mathematics) and ties them into some classical mathematics. Number theory is at its base addition, multiplication and division of integers. Combinatorics is about enumerating sets of objects. Both subjects involve familiar concepts to students and are taught at all levels. We will go deeper into these subjects, but what will make them 'fundamental' is we will also practice explaining why one applies basic rules to build up clear arguments.

You can look back at past web pages for this course to see what happened in past years. I've taught this course 3 times already but I tend to adjust it slightly each time.

## Monday, August 24, 2009

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