quaternion

Irish stamp commemorating the discovery of quaternions by William Hamilton

An ordered set of four numbers. Quaternions, first introduced by William Hamilton, can also be written in the form a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are imaginary numbers defined by the equations i² = j² = k² = -1 and -ji = k. Quaternions are similar to complex numbers, but whereas complex numbers can be represented by points of a two-dimensional plane, quaternions can be viewed as points in the fourth dimension.

For a while, quaternions were very influential: they were taught in many mathematics departments in the United States in the late 1800s, and were a mandatory topic of study at Dublin, where Hamilton ran the observatory. William Clifford developed the theory of them further. But then they were driven out by the vector notation of William Gibbs and Oliver Heaviside. Had quaternions come along later, when theoretical physicists were trying to understand patterns among subatomic particles, they may have found a place in modern science; after all, the unit quaternions form the group SU(2), which is perfect for studying spin-½ particles. But the way things turned out, quaternions had fallen from favor by the 20th century and Wolfgang Pauli used 2 × 2 complex matrices instead to describe the generators of SU(2).

Related category

• TYPES OF NUMBERS

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