The quaternions are the second "number system" other than the real numbers ever discovered, and the first to be found by a deliberate search. Their immediate predecessor was the field of complex numbers, formed by adjoining an imaginary unit, the square root of -1 (usually denoted i), to the real number system. The question of whether the number i "really exists" vexed mathematicians for over 200 years, but by the early nineteenth century it was clear that complex numbers were too useful, for both mathematics and physics, to be ignored.

William Rowan Hamilton, a brilliant Irish astronomer and mathematician, was the first mathematician to wonder if other useful number systems existed "beyond" the complex numbers. His search was reminiscent of astronomers' search for planets beyond Uranus, and was successfully completed at almost the same time. On October 16, 1843, he suddenly realized that such a system could be constructed by adjoining...