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 three-dimensional 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 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 matrices, or as an alternative to them depending on the application.

In modern 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, â„). It can also be given by the Clifford algebra classifications Câ„“0,2(R) = Câ„“03,0(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 S3, the group Spin(3), the group SU(2), or the universal cover of SO(3). wikipedia - quaternion

See Also

**Hamiltonian**
**Quantum Arithmetic**
**Scalar**
**Schrodinger equation**
**Vector**
**Wavefunction**