  We compute the mod two homology Hopf rings of the spectra KO and KU. The
spaces in these spectra are the infinite classical groups and their coset
spaces, and their homology was first calculated in the Cartan seminars, but the
Hopf ring structure was first determined in the second author's unpublished PhD
thesis. The presentation given here serves as an introduction to the first
author's much more intricate work on the connective spectrum bo. The Hopf ring
viewpoint turns out to be very convenient for understanding the homological
effect of various maps between classical groups and fibrations of their
connective covers.
