Definition:Complex Number/Construction from Cayley-Dickson Construction

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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 $\tuple {a, b} \in \C$, where:

$\tuple {a, b} \tuple {c, d} = \tuple {a c - d \overline b, \overline a d + c b}$
$\overline {\tuple {a, b} } = \tuple {\overline a, -b}$

where:

$\overline a$ is the conjugate of $a$

and

$\overline {\tuple {a, b} }$ is the conjugation operation on $\C$.

From Real Numbers form Algebra, $\overline a = a$ and so the above translate into:

$\tuple {a, b} \tuple {c, d} = \tuple {a c - d b, a d + c b}$
$\overline {\tuple {a, b} } = \tuple {a, -b}$


It is clear by direct comparison with the formal definition that this construction genuinely does generate the complex numbers.


Sources