Complex Numbers form Algebra

Theorem
The set of complex numbers $\C$ forms an algebra over the field of real numbers.

This algebra is:
 * $(1): \quad$ An associative algebra.
 * $(2): \quad$ A commutative algebra.
 * $(3): \quad$ A normed division algebra.
 * $(4): \quad$ A nicely normed $*$-algebra.

However, $\C$ is not a real $*$-algebra.

Proof
The complex numbers $\C$ are formed by the Cayley-Dickson Construction from the real numbers $\R$.

From Real Numbers form Algebra, we have that $\R$ forms:
 * $(1): \quad$ An associative algebra.
 * $(2): \quad$ A commutative algebra.
 * $(3): \quad$ A normed division algebra.
 * $(4): \quad$ A nicely normed $*$-algebra whose $*$ operator is the identity mapping.
 * $(5): \quad$ A real $*$-algebra.

From Cayley-Dickson Construction forms Star-Algebra, $\C$ is a $*$-algebra.

From Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed, $\C$ is a nicely normed $*$-algebra.

From Cayley-Dickson Construction from Real Star-Algebra is Commutative, $\C$ is a commutative algebra.

From Cayley-Dickson Construction from Commutative Associative Algebra is Associative, $\C$ is an associative algebra.

However, from Algebra from Cayley-Dickson Construction is not Real Star-Algebra, $\C$ is not a real $*$-algebra.

Proof of Normed Division Algebra
Consider the element $\tuple {1, 0}$ of $\R^2$.

We have:

As $\times$ has already been shown to be commutative, it follows that:
 * $\tuple {1, 0} \times \tuple {x_1, x_2} = \tuple {x_1, x_2}$.

So $\tuple {1, 0} \in \R^2$ functions as a unit.

That is, $\struct {\R^2, \times}$ is a unitary algebra.

We define a norm on $\struct {\R^2, \times}$ by:
 * $\forall \mathbf a = \tuple {a_1, a_2} \in \R^2: \norm {\mathbf a} = \sqrt { {a_1}^2 + {a_2}^2}$

This is a norm because:


 * $(1): \quad \forall \mathbf x \in \R^2: \norm {\mathbf x} = 0 \iff \mathbf x = \mathbf 0$
 * $(2): \quad \forall \mathbf x \in \R^2: \norm {\lambda \mathbf x} = \size \lambda \norm x$
 * $(3): \quad \forall x, y, z \in \R: \norm {x - y} \le \norm {x - z} + \norm {z - y}$

It also follows that:

and so $\struct {\R^2, \times}$ is a normed division algebra.