**Rather than consider the time it takes to breathe, we could consider the time it takes light to travel between one person’s smile and another person’s eyes; this contains the same number of instants as all of eternity.**

This bizarre property is so fundamental to infinite sets, that it in fact characterizes them. In other words, every infinite set exhibits this property, and conversely if you have a set that is the same size as a proper subset of itself, that set must be infinite.

**But where are the rational numbers, on the real line? They're everywhere.** Given any point on the real line, and some arbitrarily small interval near that point, there will be infinitely many rational numbers in that interval. Mathematicians say that the rational numbers are *dense* on the real line.

For example, \(\pi\) is an irrational number (i.e. a real number that isn't rational), approximately equal to \(3.1415926\). So we have fractions

\[\frac{3}{1}, \frac{31}{10}, \frac{314}{100}, \frac{3141}{1000}, \frac{31415}{10000}, \frac{314159}{100000}, ...\]

that get us as close to \(\pi\) as we care to go.