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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in threedimensional space. A striking feature of quaternions is that the product of two quaternions is noncommutative, meaning that the product of two quaternions depends on which factor is to the left of the multiplication sign and which factor is to the right. Hamilton defined a quaternion as the quotient of two directed lines in a threedimensional space^{[1]} or equivalently as the quotient of two vectors.^{[2]} 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 threedimensional rotations such as in threedimensional computer graphics and computer vision. They can be used alongside other methods, such as Euler angles and matrices, or as an alternative to them depending on the application.
In modern language, quaternions form a fourdimensional normed division algebra over the real numbers. In fact, the quaternions were the first noncommutative division algebra to be discovered.^{[3]} 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 Cℓ_{0,2}(R) = Cℓ^{0}_{3,0}(R). The algebra H holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finitedimensional 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 3sphere S^{3}, the group Spin(3), the group SU(2), or the universal cover of SO(3).
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