quaternion kwətûr´nēən [key], in mathematics, a type of higher complex number first suggested by Sir William R. Hamilton in 1843. A complex number is a number of the form a+bi when a and b are real numbers and i is the so-called imaginary unit defined by the equation i2=−1. The rules for operating with complex numbers are simply those of operating with the polynomial a+bx except that i2 is replaced by −1 whenever it occurs. A quaternion, an extension of this concept, is a number of the form a+bi+cj+dk when a, b, c, and d are real numbers and i, j, and k are imaginary units defined by the equations i2=j2=k2=ijk=−1. Quaternions, as well as vectors and tensors (later outgrowths of the concept of quaternions), have many important applications in mechanics.

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