Mathematical Uses
Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions and the sedenions; they are also useful tool in number theory, particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group.
The study of integral quaternions began with Rudolf Lipschitz in 1886, whose system was later simplified by Leonard Eugene Dickson; but the modern system was published by Adolf Hurwitz in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions all four of whose coordinates are half-integers. Both systems are closed under subtraction and multiplication, and are therefore rings, but Lipschitz's system does not permit unique factorization, while Hurwitz's does.
Read more about this topic: History Of Quaternions
Famous quotes containing the word mathematical:
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“An accurate charting of the American woman’s progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of time—but like a mathematical curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.”
—Susan Faludi (20th century)