In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right.

Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space[1] or equivalently as the quotient of two vectors.[2] Quaternions can also be represented as the sum of a scalar and a vector. Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and epipolar geometry. .

They have been superseded in many applications by vectors and matrices.

& cut it on a stone of this bridge. Quaternion plaque on Brougham (Broom) Bridge, Dublin, which says: Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton in a flash of genius discovered the fundamental formula for quaternion multiplication i² = j² = k² = i j k = −1 & cut it on a stone of this bridge.