Definition:Quaternion/Construction from Complex Pairs

Definition
A quaternion can be defined as an ordered pair $\left({x, y}\right)$ where $x, y \in \C$ are complex numbers, on which the operations of addition and multiplication are defined as follows:

Quaternion Addition of Complex Pairs
Let $x_1, x_2, y_1, y_2$ be complex numbers.

Then $\left({x_1, y_1}\right) + \left({x_2, y_2}\right)$ is defined as:


 * $\left({x_1, y_1}\right) + \left({x_2, y_2}\right):= \left({x_1 + x_2, y_1 + y_2}\right)$

Quaternion Multiplication of Complex Pairs
Let $x_1, x_2, y_1, y_2$ be complex numbers.

Then $\left({x_1, y_1}\right) \left({x_2, y_2}\right)$ is defined as:


 * $\left({x_1, y_1}\right) \left({x_2, y_2}\right) := \left({x_1 x_2 - y_2 \overline {y_1}, \overline {x_1} y_2 + y_1 x_2}\right)$

where $\overline {x_1}$ and $\overline {y_1}$ are the complex conjugates of $x_1$ and $y_1$ respectively.

Also see

 * Quaternions Defined by Ordered Pairs where this definition can be seen to be equivalent to the main definition.