Editing Quaternion (section)

== Quaternions as pairs of complex numbers ==
{{Main|Cayley–Dickson construction}}
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 <math>\mathbb C^2</math> be a two-dimensional vector space over the complex numbers. Choose a basis consisting of two elements {{math|1}} and {{math|'''j'''}}. A vector in <math>\mathbb C^2</math> can be written in terms of the basis elements {{math|1}} and {{math|'''j'''}} as

<math display=block>(a + b i)1 + (c + d i)\mathbf j. </math>

If we define {{math|1='''j'''<sup>2</sup> = −1}} and {{math|1=''i'' '''j''' = −'''j''' ''i''}}, then we can multiply two vectors using the distributive law. Using {{math|'''k'''}} as an abbreviated notation for the product {{math|''i'' '''j'''}} leads to the same rules for multiplication as the usual quaternions. Therefore, the above vector of complex numbers corresponds to the quaternion {{math|''a'' + ''b i'' + ''c'' '''j''' + ''d'' '''k'''}}. If we write the elements of <math>\mathbb C^2</math> as ordered pairs and quaternions as quadruples, then the correspondence is

<math display=block>(a + bi,\,c + di) \leftrightarrow (a,\,b,\,c,\,d).</math>