Baking a Better

Real numbers are scary! There are a lot of them, most of which can’t even be written down! They are extremely hard to understand, and you would have to be crazy to attempt to do anything with these so-called “real” numbers! So, what is a mathematician to do?

It turns out that any real number can be arbitrarily well approximated by fractions (or rational numbers). That is, if you hand me a real number and a margin of error, I can find some fraction that differs from your real number by less than the margin of error. This is great, because we actually know how to add, subtract, multiply, and divide fractions!

For example, you may know that

π ≈ 22/7, and π ≈ 3.142 = 3142/1000 = 1571/500.

Is one of these approximations “better” than the other? In one sense, the second approximation is better, since 3.1412 is closer to π than 22/7 (there is a smaller margin of error). In another sense, 22/7 is a better approximation, as it has a smaller denominator and so should be easier to compute with. Finding “good” approximations of π is then a balancing act: on the one hand we want to get as close to π as possible, but on the other hand we want to work with small denominators.

This quarter, we will explore techniques for finding “good” approximations of irrational numbers in a way that gives us both small margins of error and small denominators. We will then discuss techniques of simultaneous rational approximation (that is, approximating collections of irrational numbers in a “good” way). Our main goal is to study the LLL algorithm (named for mathematicians A. K. Lenstra, H. W. Lenstra Jr., and L. Lovaśz), and its application to approximation. If time allows, we hope to address one of the open questions about the algorithm.