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:
\(\ds \forall x_1, x_2 \in \R: \, \) | \(\ds \) | \(\) | \(\ds \tuple {x_1, x_2} \times \tuple {1, 0}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1 \times 1 - 0 \times x_2, x_1 \times 0 + x_2 \times 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \tuple {x_1, x_2}\) |
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:
\(\ds \forall x, y \in \R: \, \) | \(\ds \norm {x \times y}\) | \(=\) | \(\ds \size {x \times y}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \size x \times \size y\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \norm x \times \norm y\) |
and so $\struct {\R^2, \times}$ is a normed division algebra.
$\blacksquare$
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