In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that the product of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions can also be represented as the sum of a scalar and a vector.

Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics and computer vision. They can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them depending on the application.

In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and thus also form a domain. In fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by (Unicode U+210D, ℍ). It can also be given by the Clifford algebra classifications C0,2(R) ≅ Cℓ03,0(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers.

The unit quaternions can therefore be thought of as a choice of a group structure on the 3-sphere that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).

Read more about Quaternion:  History, Definition, Conjugation, The Norm, and Reciprocal, Algebraic Properties, Quaternions and The Geometry of R3, Matrix Representations, Quaternions As Pairs of Complex Numbers, Square Roots of −1, Functions of A Quaternion Variable, Three-dimensional and Four-dimensional Rotation Groups, Generalizations, Quaternions As The Even Part of Cℓ3,0(R), Brauer Group, Quotes

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Classical Hamiltonian Quaternions - Classical Elements of A Quaternion - Quaternion - Right Quaternion
... A right quaternion is a quaternion whose scalar component is zero, The angle of a right quaternion is 90 degrees ... A right quaternion can also be thought of as a vector plus a zero scalar ... Right quaternions may be put in what was called the standard trinomial form ...
Quaternion - Quotes
... I regard it as an inelegance, or imperfection, in quaternions, or rather in the state to which it has been hitherto unfolded, whenever it becomes or seems to become necessary to have ... The mathematical quaternion partakes of both these elements in technical language it may be said to be "time plus space", or "space plus time" and in this sense it has, or at least involves a ... Quaternions came from Hamilton after his really good work had been done and, though beautifully ingenious, have been an unmixed evil to those who have touched ...
... In modern algebra, a versor or unit quaternion is a quaternion of norm one ... When used to represent a rotation, a versor will rotate any quaternion vector v through the angle θ around the unit vector r through the sandwiching product qvq−1 ... of vertere = "to turn", and was introduced by William Rowan Hamilton, in the context of his quaternion theory ...
Classical Hamiltonian Quaternions - Other Operators in Detail - Tensor
... Symbolically A quaternion is by definition the quotient of two vectors and the tensor of a quaternion is by definition the quotient of the tensors of these ... From this definition it can be shown that a useful formula for the tensor of a quaternion is It can also be proven from this definition that another ... The square root of the common norm of a quaternion is equal to its tensor A useful identity is that the square of the tensor of a quaternion is equal to the tensor of the square of a quaternion, so that ...
Classical Hamiltonian Quaternions - Other Operators in Detail - Axis and Angle
... Taking the angle of a non-scalar quaternion, resulted in a value greater than zero and less than π ... When a non-scalar quaternion is viewed as the quotient of two vectors, then the axis of the quaternion is a unit vector perpendicular to the plane of the two vectors in this original ...