|A digital representaion of a Julia set quaternion|
i2 = j2 = k2 = ijk = -1
Hamilton named a quadruple of this type a quaternion and he devoted the rest of his career to their study. The simplest definition of such a number is a four-dimensional number. Quaternions are still studied today, and present a number of uses in modern mathematics. According to The Math Book by Clifford Pickover, they've been used to describe the dynamics of motion in three dimensions, and have been applied to fields including virtual reality computer graphics, game programming, robots, bioinformatics, and flight software of spacecrafts.
The invention, or discovery, of quaternions represents a moment of great ingenuity in math history. You can actually still visit Brougham (Broom) Bridge today, and see the plaque commemorating Hamilton's discovery. There is even an annual commemorative walk following Hamilton and his wife's path on that fateful morning. Perhaps we should have added this location to our list of math destinations!
One more fun fact about quaternions is that they generated such polarizing responses from the math community. Scottish physicist William Thomson(1824 - 1907) considered them an "unmixed evil to those who have touched them in any way," while mathematician Oliver Heaviside thought of their invention as a feat of human ingenuity. Curiously, one mathematician (who gained his fame from a harshly different circumatance) who studied quaternions was Theodore Kaczynski, the "Unabomber."