the same result holds if you restrict to some subset of the real line, say the closed unit interval `[0, 1]`. although there are infinitely many rationals and

the same result holds if you restrict to some subset of the real line, say the closed unit interval `[0, 1]`. although there are infinitely many rationals and infinitely many irrationals in the unit interval [0, 1], from the perspective of cardinality, there are a lot more irrationals. there are countably many rationals in [0, 1], meaning you can put those rationals into correspondence with the natural numbers 1, 2, 3, ... , but there are uncountably many irrationals in that same interval. the quote about "100 percent chance" is probably alluding to measure theory. measure theory is used to define integration. in this 1 dimensional case of the real line, or a subset of the real line, you can think of the measure of a set being something like defining its length. the measure of the whole set [0, 1] is 1. the subset of all rationals in [0, 1] has measure zero, i.e. it has the same measure as a single point. there's no measurable length to it. not all subsets can be assigned a sensible notion of a length. if you think of random sampling a point from the set [0, 1] according to a uniform distribution, then you have a probability of 0 < delta <= 1 of randomly sampling a point that belongs to any pre-defined subset [-0.5 * delta + x, x + 0.5 * delta] of [0, 1] . Informally, if we make delta smaller and smaller, the chance of randomly sampling a point that is arbitrarily close to x is probability delta that becomes arbitrarily close to zero. when you randomly sample from uniform distribution on [0, 1] the chance of sampling any particular pre-specified real number is probability zero. probability zero isn't the same as impossible! see https://en.wikipedia.org/wiki/Almost_surely https://en.wikipedia.org/wiki/Almost_surely on another hand, despite this, the rationals are dense in [0, 1]. this means that for any real number x in [0, 1], for any (small) radius r > 0 we can find some rational y in [0, 1] that is within a ball of radius r from x, i.e. there is some rational y such that |x-y|<r. this means we can pick some sequence y_1, y_2, y_3, ... of rational numbers that become better and better approximations to the target x (which in the interesting case, x is irrational). think a sequence of decimal approximations to the target number x. despite this ability to approximate any irrational with a rational arbitrarily well, there are still a lot more irrationals, in both the cardinality and a crude measure-theoretic sense (if we delete them all it doesnt change the amount of length we started with). > The asterisk at the end doesn't seem to lead to an actual footnote. What is meant by this? perhaps it is meant to provoke all people who like to see everything clearly defined