Abstract: When I was in my freshman year at Stony Brook, a friend of mine was perplexed by an interesting question. Suppose we enumerate the rationals on the interval [0,1] in the “standard” way in which the denominators increase: 0, 1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/7, … Suppose we consider removing the interval of width 1/2^n around the nth number from the interval [0,1] (so, e.g., we remove the interval [1/3 – 1/16, 1/3+1/16]). Can we find a number in [0,1] which is not removed in this procedure? With a little finagling from measure theory, it is clear that an answer must exist, but how can we find one? By the end of the semester, my friend came up with a beautiful solution. It wasn’t until my senior year that I learned that his solution fit into a general framework for discussing a measure for irrationality of a given real number. We will discuss this problem and some of the mathematics behind it. Assuming we have time, along the way, we will discuss one of my favorite MathOverflow posts of all time, as well as a recent video of YouTuber Matt Parker of Stand-up Maths.
