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 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 Never Real, $\C$ is not a real algebra.

Proof of Normed Division Algebra

Consider the element $\left({1, 0}\right)$ of $\R^2$.

We have:

\(\ds \)

\(\)

\(\ds \left({x_1, x_2}\right) \times \left({1, 0}\right)\)

\(\ds \)

\(=\)

\(\ds \left({x_1 \times 1 - 0 \times x_2, x_1 \times 0 + x_2 \times 1}\right)\)

\(\ds \)

\(=\)

\(\ds \left({x_1, x_2}\right)\)

As $\times$ has already been shown to be commutative, it follows that $\left({1, 0}\right) \times \left({x_1, x_2}\right) = \left({x_1, x_2}\right)$.

So $\left({1, 0}\right) \in \R^2$ functions as a unit.

That is, $\left({\R^2, \times}\right)$ is a unitary algebra.

We define a norm on $\left({\R^2, \times}\right)$ by:

$\forall \mathbf a = \left({a_1, a_2}\right) \in \R^2: \left \Vert {\mathbf a} \right \Vert = \sqrt {a_1^2 + a_2^2}$

This is a norm because:

$(1): \quad \left \Vert \mathbf x \right \Vert = 0 \iff \mathbf x = \mathbf 0$

$(2): \quad \left \Vert \lambda \mathbf x \right \Vert = \left \vert \lambda \right \vert \left \Vert x \right \Vert$

$(3): \quad \left \Vert x - y \right \Vert \le \left \Vert x - z \right \Vert + \left \Vert z - y \right \Vert$

It also follows that:

$\left \Vert x \times y \right \Vert = \left \vert x \times y \right \vert = \left \vert x \right \vert \times \left \vert y \right \vert = \left \Vert x \right \Vert \times \left \Vert y \right \Vert$

and so $\left({\R^2, \times}\right)$ is a normed division algebra.

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$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem