Abstract
IN 1833 HAMILTON PRESENTED a paper before the Irish Academy in which he introduced a formal algebra of real number pairs whose rules of combination are precisely those for complex numbers. The important rule for multiplication of these pairs, corresponding to the rule
is
which he interpreted as an operation involving rotation. Hamiltion’s paper provided the definitive formulation of complex numbers as pairs of real numbers.
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Notes
It is worth noting in this connection that Hamilton’s mathematical work was strongly influenced by his philosophical views, which were derived in the main from Kant. Kant had maintained that space and time were the two essential forms of sensuous intuition, and Hamilton went so far as to proclaim that, just as geometry is the science of pure space, so algebra must be the science of pure time. were the two essential forms of sensuous intuition, and Hamilton went so far as to proclaim that, just as geometry is the science of pure space, so algebra must be the science of pure time.
Extensive quantities are to be contrasted with intensive quantities. Quantities such as mass or volume are extensive in the sense that they are defined over extended regions of space and are therefore additive: thus two pounds + two pounds = 4 pounds. Vector quantities such as velocity or acceleration, being additive in this sense, also count as extensive quantities. On the other hand, quantities such as temperature or density are intensive in that they are defined at a point and are not additive: thus on mixing two buckets of water each having a uniform temperature of 50 degrees one obtains a quantity of water at a temperature of 50, rather than 100, degrees.
It was the elder Peirce who, in 1870, formulated the well-known definition: Mathematics is the science that draws necessary conclusions.
A system of first-degree equations in the unknowns x, y, z, â‹Ż is said to be homogeneous if each is of the form ax + by + cz +â‹Ż = 0.
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© 1999 Springer Science+Business Media Dordrecht
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Bell, J.L. (1999). The Evolution of Algebra, II. In: The Art of the Intelligible. The Western Ontario Series in Philosophy of Science, vol 63. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4209-0_5
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DOI: https://doi.org/10.1007/978-94-011-4209-0_5
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