Infinity is a concept that has both intrigued and confused humanity over the millennia. The ancient Greek philosopher, Zeno of Elea, devised a series of paradoxes about infinity–at least nine of which are known to this day–which lead to seemingly absurd results. More than 2,400 years later, the notion of infinity still holds many mysteries for us.
The set of positive integers or natural numbers \(\mathop{\mathrm{\mathbb{N}}}\) , as is well known, is infinite. Between any two natural numbers on the real axis, there are infinitely many rational numbers (which are expressed as the quotient of two integers). So our intuition might suggest, perhaps reasonably enough, that one of them–namely rational numbers–is more infinite than the other–integers. In that case, our intuition would be wrong, because there is a one-to-one correspondence, or “bijection,” between the integers and the rational numbers. The total number or “cardinality” of the rational numbers \(\mathbb{Q}\) is the same as that of \(\mathop{\mathrm{\mathbb{N}}}\) .
One way to picture this is to represent rational numbers \(\frac{p}{q}\) on a lattice and “wrap” positive integers around it like a spiral (Fig. 50 ). We see that if we start counting points on the lattice as we spiral around, and skip points that are undefined (when \(q=0\) ) or unsimplified (when \(p\) and \(q\) are multiples of an integer), we get a bijection between \(\mathbb{Q}\) and \(\mathop{\mathrm{\mathbb{N}}}\) .
