Ghostly Irrational Numbers

Rational numbers can be expressed as ratios and irrational numbers cannot. Hippasus is sometimes credited with the discovery that the length of the diagonal of a square is an irrational number. This is an important discovery in the history of mathematics. Cantor’s diagonal argument also deepened our understanding of rational numbers and irrational numbers. The former are countably infinite and the latter are uncountably infinite. They also have distinct densities in the reals. I have written a poem which describes the differences between the rationals and the irrationals, a bit of history of the discovery, as well as the significance and the density of the irrationals.