![]() | The Abstractor | Elementary Notions | A Fourth Dimension | A Division Algebra | Closure | An Example of Hamilton's Work | Bibliography | Back to the front page Elementary Notions Joseph Johnson Hamilton began his attempt to beautify algebra around 1830. By 1843, his task was completed, though his goal was not fully realized. During this decade or so, he argued that a lower-level definition of algebra was more applicable to the ``real" world. He even aligned himself against the seemingly abstract laws that De Morgan had put forth in 1841. Of course Hamilton's abstract approach characterizes our modern algebra. For example, De Morgan posited that commutativity was essential in any algebra. But general function composition (including matrix multiplication), one of the most important notions in current mathematics, is non-commutative. In this quest to buttress algebra with better foundations, Hamilton set out to discover a three-dimensional complex number system in the vein of number triplets. Because the world is three-dimensional, he reasoned, the undiscovered system may be a model for our universe. Definitions of equality and addition were easy:
Multiplication, on the other hand, was difficult.
Realizing that the employment of ``imaginary" units was more practical computationally, Hamilton was
willing to concentrate on the non-coordinate version of his idea. The use of i for (0,1,0) and j
for (0,0,1) facilitated the attempt at multiplication definition.
In pondering the definition of his new multiplication, Hamilton first considered how to define ii and jj. He was working under the assumption that the undiscovered structure would be reducible to the ordinary complex number system. That is, the new algebra should be a superset of the already established C. Hence both x+yi+0j and x+0i+zj (two-dimensional elements from the new algebra) should behave as they did in the old algebra. That is, he forced the following equations to be true:
Whence, ii=-1 and jj=-1. With this, he had a notion of how to multiply the triplets whose coefficients were not restricted to zero.
How was Hamilton to define ij (ji)? He initially assumed commutativity for his system, which would imply ij=ji; however, the following computation convinced him that commutativity would have to be abandoned.
Thus, if the undiscovered system was to be commutative, ij would have to be 1 or -1. Either case was unacceptable to
Hamilton, because he was working under a special assumption about the lengths of his vectors. His system was supposed to be
natural, so he took the length (modulus) of a general vector x+yi+zj to be
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