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 *C*ℓ_{0,2}(**R**) ≅ *C*ℓ0_{3,0}(**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 Quaternions: 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 Cℓ_{3,0}(R), Brauer Group, Quotes