Quaternions are the elements of a number system, with all the regular properties of an algebra like that of the reals and complex numbersComplex numbers except for the commutativity property with multiplication. William Rowan Hamilton, who discovered the quaternion number system, originally aimed to extend the complex number system as used in the 2-D plane to a new number system of dimension 3 for use in the 3-D plane. This turned out to be impossible, but it did lead to the quaternion number system that exists in four dimensions. While the complex number system has one imaginary component \(\imath \) , Hamilton’s quaternions use three imaginary components \(\imath \) , \(\jmath \) and k. The relationship between Hamilton’s imaginary components are introduced using a visual interpretation in this chapter. The resulting quaternion representations and operations are described including addition, multiplication and division operations as well as the quaternion conjugateQuaternionconjugate operation. The scalar–vector representation is detailed as the practical way to work with quaternions and the chapter ends with the Cayley–Dickson approach to constructing quaternions along with an equivalent matrix representation for quaternions.