Are there infinitely many prime numbers? How can we know? How do we know for certain that the infinitely many digits in the decimal expansion of the square root of 2 never repeat? Can we ever have definite knowledge about abstractions like infinite sets or the fourth dimension? These questions are typical of ‘pure’ mathematics: mathematics studied for its own sake rather than for any particular application. Pure mathematical questions are usually not only about how to compute something (e.g. how to find prime numbers), but also about how we know something for certain (e.g. that there are infinitely many prime numbers). However, pure mathematics often leads to important applications. This class is an introduction to this type of reasoning. We will look at some fundamental ideas of mathematics: rational and irrational numbers, infinite sets, geometric axioms, and some classic questions about them. This course is intended to serve as a foundation, and it will be a prerequisite for many other advanced mathematics courses. Students will be expected to have a good facility with high school algebra. Students without this solid background can still take the course if they are willing to work on this as the course progresses.