\begin{definition}[Definition:Complex Number/Construction from Cayley-Dickson Construction]
The complex numbers can be defined by the Cayley-Dickson construction from the set of real numbers $\R$.
From Real Numbers form Algebra, $\R$ forms a nicely normed $*$-algebra.
Let $a, b \in \R$.
Then $\left({a, b}\right) \in \C$, where:
:$\left({a, b}\right) \left({c, d}\right) = \left({a c - d \overline b, \overline a d + c b}\right)$
:$\overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$
where:
:$\overline a$ is the conjugate of $a$
and 
:$\overline {\left({a, b}\right)}$ is the conjugation operation on $\C$.
From Real Numbers form Algebra, $\overline a = a$ and so the above translate into:
:$\left({a, b}\right) \left({c, d}\right) = \left({a c - d b, a d + c b}\right)$
:$\overline {\left({a, b}\right)} = \left({a, -b}\right)$
It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers.
\end{definition}