Quaternions As Pairs of Complex Numbers
Quaternions can be represented as pairs of complex numbers. From this perspective, quaternions are the result of applying the Cayley–Dickson construction to the complex numbers. This is a generalization of the construction of the complex numbers as pairs of real numbers.
Let C2 be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements 1 and j. A vector in C2 can be written in terms of the basis elements 1 and j as
If we define j2 = −1 and ij = −ji, then we can multiply two vectors using the distributive law. Writing k in place of the product ij leads to the same rules for multiplication as the usual quaternions. Therefore the above vector of complex numbers corresponds to the quaternion a + bi + cj + dk. If we write the elements of C2 as ordered pairs and quaternions as quadruples, then the correspondence is
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