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There is far more to mathematical number systems than just different bases, like binary and decimal. Fundamental notions of counting, and various number systems go back thousands of years. But by far the most interesting developments, from the point of view of mathematicians, have been made in the past century due to the advent of mechanized computation.

We'll take a brief look at the history of counting and number Systems: Roman Numerals, Egyptian fractions, etc. Then we'll see how high-speed scientific computations have spurred decades of research into efficient methods of arithmetic, and how many important discoveries rest not only upon clever techniques for duplicating the addition, subtraction, multiplication and division we all learned in grade school, but also upon completely new systems of representing numbers, many quite different from our familiar positional number systems.

Some of these systems have the property that it is possible to add numbers without carrying, and subtract without borrowing. Others allow negative digits, and still others are based on irrational radices (like the golden ratio). Some even permit very fast and convenient column-by-column multiplication and division, with no need to consult adjacent information. But these systems have their drawbacks as we shall see.

We'll also examine the mathematical techniques used by floating-point accelerators, and consider the theory behind how a scientific calculator works, including those special function keys. We'll study how the Fast-Fourier transform yields a great method for multiplication, how the fastest known method for division relates to the distribution of prime numbers, and efficient techniques for computing powers in a finite field.

This class should be of interest to both pure Mathematics majors and Computer Science students, because it will deal with both the mathematical representation of numbers along with the algorithms for their arithmetic. If you've ever wondered how interesting ordinary numbers and their arithmetic can get, take this class.