noncommutative extension of the real numbers
Quaternion multiplication table
1 i j k
1 1 i j k
i i −1 k j
j j k −1 i
k k j i −1

In mathematics, the quaternion number system extends the complex numbers into four dimensions. They were first described by Irish mathematician William Rowan Hamilton in 1843.[1][2] They are often used in computer graphics to compute 3-dimensional rotations.

Cayley Q8 graph showing the 6 cycles of multiplication by i, j and k. (In the SVG file, hover over or click a cycle to highlight it.)


  1. "On Quaternions; or on a new System of Imaginaries in Algebra". Letter to John T. Graves. 17 October 1843.
  2. Rozenfelʹd, Boris Abramovich (1988). The history of non-euclidean geometry: Evolution of the concept of a geometric space. Springer. p. 385. ISBN 9780387964584.

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