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There are several equivalent ways to define the quaternions, each of which has its uses.
To understand this, you will need knowledge of the complex numbers, such as is often taught in a high school second-year algebra course.
The quaternions may be described as the algebra consisting of the set of ordered pairs of complex numbers, with addition defined in the component-wise way, and with multiplication defined by
and with an operation called conjugation, defined by
This definition yields a norm, via
and a multiplicative inverse
These pages favor the complex number pair, for its symmetry.
See Component-wise representation for a definition of quaternions in terms of four real numbers.
See Complex numbers and Quaternions as Matrices for a definition of quaternions as 2 × 2 matrices of complex numbers.
Tait used a geometrical definition. In An Elementary Treatise on Quaternions, Chapter II, § 47, he writes:
Thus it appears that the ratio of two vectors, or the multiplier required to change one vector into another, in general depends upon four distinct numbers, whence the name Quaternion.
A quaternion q is thus defined as expressing a relation
β = q αbetween two vectors α, β. By what precedes, the vectors α, β, which serve for the definition of a given quaternion, must be in a given plane, at a given inclination to each other, and with their lengths in a given ratio ; but it is to be noticed that they may be any two such vectors. [Inclination is understood to include sense, or currency, of rotation from a to β.]
This geometrical interpretation is currently used in many applications. See Geometry of Quaternions.
A quaternion may be written in terms of a pair consisting of a scalar and a 3-vector. See representations of quaternions
P. G. Tait An Elementary Treatise on Quaternions, Cambridge U. Press, 1890