Hamilton's quaternions

Abstract

Publisher Summary This chapter discusses the Hamilton's Quaternions. Hamilton's original motivation was to find an algebraic formalism for the points (x1, x2, x3) in 3-space, in generalization of the formalism of the complex numbers C as pairs of real numbers. A matrix model of the quaternion algebra H is discussed. In modern algebra, a division ring is a ring K with identity 1 ≠ 0 in which every nonzero element is an inverse. These rings are the most “perfect” algebraic systems: as one can add, subtract, multiply, and divide. Most of the applications of quaternions to physics are applications of biquaternions. Several other important mathematical developments––namely, Cayley Algebra, Frobenius' Theorem, and Vector Analysis are also described. The fundamental theorem of algebra says that C is an algebraically closed field; that is, any non-constant polynomial in C[x] has a zero in C. The main point of passing from R to C is exactly to get an “algebraic closure” of the reals. The features of the quaternions are the role they play in the understanding and representation of the rotations of the low-dimensional Euclidean spaces.