Text: € For Hamilton a complex number was just an ordered pair of real numbers. € What Hamilton sought to do next was to generalize his idea of ordered pairs of real numbers, and to consider the possibility of hypercomplex numbers which would contain the real and complex systems as proper subsets just as the complex aggregate contains the reals and the imaginaries. € Hamilton employed the quadruple of real numbers (a, b, c, d) instead of the complex pair (a+bi, c+di). Therefore Hamilton called the hypercomplex numbers he created quaternions. After fifteen years of thought on the subject [of quaternions], he found that he could not formulate a quaternion algebra in which the hypercomplex numbers woud simultaneously satisfy both the traditional arithmetic laws and the requirements of a physical science of space. Finally, in 1843, he released a revolutionary but logically consistent algebra in which the commutative law of multiplication was abandoned.

See Also: QUANTUM ARITHMETIC stack.; SCALAR; POLARITY

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