User:Silly rabbit/Sandbox/Hypernumber History

Historical motivation and development

During the early 19th century, it was generally known that complex numbers could be represented geometrically as a real and imaginary axis in the Euclidean plane, along with a parallelogram law for addition and a multiplication law based on similar triangles. Algebraically, complex numbers were represented as pairs of real numbers written additively as a + ib which were added and multiplied exactly as one would do with polynomials, but for the single relation i² = -1. The algebraic and geometrical viewpoints of the complex numbers would then lead William Rowan Hamilton and Hermann Grassmann in the 1840s to simultaneously and independently develop the two first hypercomplex number systems: the quaternions and the exterior algebra. Each of these new developments was concerned with a way to do geometry algebraically (or algebra geometrically) in more than the two dimensions afforded by the system of complex numbers. Of the two, it was Hamilton's quaternions which would set the trend for the development of hypernumbers over the next 50 years or so, owing to the relative philosophical obscurity and "moral solitude" of Grassmann's exterior algebra.

Hamilton's significant insight was that it was possible to construct a number system with four separate components (drawn from one real axis and three non-real axes), obeying most of the usual laws of arithmetic: Associativity, distributivity, and multiplicative inversion. However, he saw that in order to construct such a system, he had to eliminate the commutativity requirement. The quaternions were represented by additive combinations

$a+{\mathbf {i} }b+{\mathbf {j} }c+{\mathbf {k} }d$

which were added as ordinary polynomials in several variables, but multiplied modulo the relations

${\mathbf {i} }^{2}={\mathbf {j} }^{2}={\mathbf {k} }^{2}=-1,{\mathbf {i} }{\mathbf {j} }={\mathbf {k} },{\mathbf {j} }{\mathbf {k} }={\mathbf {i} },{\mathbf {k} }{\mathbf {i} }={\mathbf {j} }.$

The new numbers i, j, and k, together with the real unit 1, were termed the units of the quaternions.

The quaternions were an enormous success in both mathematics and physics at the time. In 1845, Arthur Cayley generalized the quaternions to a similar system containing eight units (now known as the octonions). These satisfied one less property: they were non-associative under multiplication. Other than the lack of commutativity and associativity, however, the other familiar laws of arithmetic held. In particular, division — except by zero — also continued to be possible, a fact not proven until 1912 by Leonard Dickson. In 1873, William Kingdon Clifford introduced another type of hypernumber: the Clifford biquaternions. These were

History

In the 18th century, due in large part to the efforts of Leonard Euler and others, the theory of complex numbers was formally introduced to the mathematical world from its obscure beginnings in the works of John Wallis and Roger Cotes. These new numbers had many striking new geometrical properties, as later explored in detail in Jean Robert Argand's 1806 treatise on the subject. Later, in 1831, Carl Friedrich Gauss consolidated the arithmetic and geometrical notions surrounding the complex numbers, and laid the foundations for the development of hypercomplex number systems.

By the mid 19th century, it was generally known that complex numbers could be represented geometrically as a real and imaginary axis in the Euclidean plane, along with a parallelogram law for addition and a law for multiplication based on similar triangles. Algebraically, complex numbers were represented as pairs of real numbers written additively as a + ib which were added and multiplied exactly as one would do with polynomials, but for the single relation i² = -1. The algebraic and geometrical viewpoints of the complex numbers would then lead William Rowan Hamilton and Hermann Grassmann to simultaneously and independently develop the two first hypercomplex number systems: the quaternions and the exterior algebra.

Hamilton spent some years of effort trying to develop a generalization of the complex numbers to three dimensions, by expressions formed as combinations such as a + b i' + c j. Ultimately, Hamilton was forced to go up to four dimensions. Furthermore, and no less significantly, he had to abandon the commutativity which had previously been naturally associated with the idea of "number." The result was the quaternions. These new numbers were represented as additive combinations

$a+{\mathbf {i} }b+{\mathbf {j} }c+{\mathbf {k} }d.$

These were added much as polynomials in several variables, but multiplied subject to the relations:

${\mathbf {i} }^{2}={\mathbf {j} }^{2}={\mathbf {k} }^{2}=-1,{\mathbf {i} }{\mathbf {j} }={\mathbf {k} },{\mathbf {j} }{\mathbf {k} }={\mathbf {i} },{\mathbf {k} }{\mathbf {i} }={\mathbf {j} }.$