 A class of associative algebras of interest to physicists is that of the Clifford algebras. Clifford algebras over the reals (which as vector spaces are Euclidean spaces) generalize complex numbers and quaternions by permitting any number of formal quantities \(e\) analogous to \(i = \sqrt{-1}\) to be adjoined to the field of reals. The common feature of these quantities is that each satisfies either \(e^2 = -1\) or \(e^2 = 1\). Whereas there are a great many associative algebras of low dimension, only a few of them arise as Clifford algebras. The reals form the only one-dimensional Clifford algebra, while the hyperbolic plane, defined by \(e^2 = 1\), and the complex plane, defined by \(e^2 = -1\), are the two two-dimensional Clifford algebras. The hyperbolic plane is just the direct square of the real field, meaning that its product is coordinatewise, \((a, b)(c, d) = (ac, bd)\), unlike that of the complex plane where it defined by \((a, b)(c, d) = (ac - bd, ad+bc)\). The two four-dimensional Clifford algebras are the \(2\times 2\) matrices and the quaternions. Whereas the \(2\times 2\) matrices contain zero divisors (nonzero matrices whose product is zero), and so form only a ring, the quaternions contain no zero divisors and so form a division ring. Unlike the complex numbers however, the quaternions do not form a field because their multiplication is not commutative. Complex multiplication however makes the complex plane a commutative division ring, that is, a field.