# 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

*i*

^{2}=−1. The rules for operating with complex numbers are simply those of operating with the polynomial

*a*+

*bx*except that

*i*

^{2}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

*i*

^{2}=

*j*

^{2}=

*k*

^{2}=

*ijk*=−1. Quaternions, as well as vectors

**vector,**

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.

**.....**Click the link for more information. and tensors

**tensor,**

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).

**.....**Click the link for more information. (later outgrowths of the concept of quaternions), have many important applications in mechanics.

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## 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.McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

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