Division Algebras : Octonions Quaternions Complex Numbers and the Algebraic Design of Physics The four real division algebras (reals complexes quaternions and octonions) are the most obvious signposts to a rich and intricate realm of select and beautiful mathematical structures. Using the new tool of adjoint division algebras with respect to which the division algebras themselves appear in the role of spinor spaces some of these structures are developed including parallelizable spheres exceptional Lie groups and triality. In the case of triality the use of adjoint octonions greatly simplifies its investigation. Motivating this work however is a strong conviction that the design of our physical reality arises from this select mathematical realm. A compelling case for that conviction is presented a derivation of the standard model of leptons and quarks. The book will be of particular interest to particle and high energy theorists and to applied mathematicians