 Dual numbers are often written in the form , where ε is the dual unit that commutes with "i", "j", "k" and has the property .The result is that a dual quaternion can be written as an ordered pair of quaternions .Two dual quaternions add componentwise and multiply by the rule,

It is convenient to write a dual quaternion as the sum of a dual scalar and a dual vector, , where and is the dual vector that defines a screw.