Definition:Complex Number/Construction from Cayley-Dickson Construction

Definition

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.

Sources

• John C. Baez: The Octonions (2002): 1 Introduction