On quaternions and octonions: their geometry, arithmetic, and symmetry, A K

Conway and Smith’s book is a wonderful introduction to the normed division algebras: the real numbers (R), the complex numbers (C), the quaternions (H), and the octonions (O). The first two are well-known to every mathematician. In contrast, the quaternions and especially the octonions are sadly neglected, so the authors rightly concentrate on these. They develop these number systems from scratch, explore their connections to geometry, and even study number theory in quaternionic and octonionic versions of the integers. Conway and Smith warm up by studying two famous subrings of C: the Gaussian integers and Eisenstein integers. The Gaussian integers are the complex numbers x+ iy for which x and y are integers. They form a square lattice: