"Quantum theory may be formulated using s over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. ...[P]roblems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". ... This three-fold classification sheds light on the physics of time reversal symmetry, and it already plays an important role in particle physics. ...There are precisely four 'normed division algebras': the real numbers \mathbb{R}, the complex numbers \mathbb{C}, the quaternions \mathbb{H} and the octonions \mathbb{O}. Roughly speaking, these are the number systems extending the reals that have an ‘absolute value’ obeying the equation |xy| = |x| |y|. Since the octonions are nonassociative [their use] proves difficult... except in a few special cases. ...[I]nstead of being distinct alternatives, real, complex and quaternionic quantum mechanics are three aspects of a single unified structure."