quaternion

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quaternion

(kwətûr`nēən), 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 vectorsvector,
quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.
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 and tensorstensor,
in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates).
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 (later outgrowths of the concept of quaternions), have many important applications in mechanics.

quaternion

[kwə′ter·nē·ən]
(mathematics)
The division algebra over the real numbers generated by elements i, j, k subject to the relations i 2= j 2= k 2= -1 and ij = -ji = k, jk = -kj = i, and ki = -ik = j. Also known as hypercomplex number.