Properties of Complex Numbers

Complex Numbers are Uncountable

The set of complex numbers $\C$ is uncountably infinite.

Complex Numbers under Addition form Abelian Group

Let $\C$ be the set of complex numbers.

The structure $\struct {\C, +}$ is an infinite abelian group.

The set of complex numbers $\C$ is closed under addition:

$\forall z, w \in \C: z + w \in \C$

The operation of addition on the set of complex numbers $\C$ is associative:

$\forall z_1, z_2, z_3 \in \C: z_1 + \left({z_2 + z_3}\right) = \left({z_1 + z_2}\right) + z_3$

Let $\C$ be the set of complex numbers.

The identity element of $\left({\C, +}\right)$ is the complex number $0 + 0 i$.

Let $z = x + i y \in \C$ be a complex number.

Let $-z = -x - i y \in \C$ be the negative of $z$.

Then $-z$ is the inverse element of $z$ under the operation of complex addition:

$\forall z \in \C: \exists -z \in \C: z + \left({-z}\right) = 0 = \left({-z}\right) + z$

The operation of addition on the set of complex numbers is commutative:

$\forall z, w \in \C: z + w = w + z$

Non-Zero Complex Numbers under Multiplication form Abelian Group

Let $\C_{\ne 0}$ be the set of complex numbers without zero, that is:

$\C_{\ne 0} = \C \setminus \set 0$

The structure $\struct {\C_{\ne 0}, \times}$ is an infinite abelian group.

Complex Multiplication is Closed

The set of complex numbers $\C$ is closed under multiplication:

$\forall z, w \in \C: z \times w \in \C$

Complex Multiplication is Associative

The operation of multiplication on the set of complex numbers $\C$ is associative:

$\forall z_1, z_2, z_3 \in \C: z_1 \left({z_2 z_3}\right) = \left({z_1 z_2}\right) z_3$

Complex Multiplication Identity is One

Let $\C_{\ne 0}$ be the set of complex numbers without zero.

The identity element of $\left({\C_{\ne 0}, \times}\right)$ is the complex number $1 + 0 i$.

Inverse for Complex Multiplication

Each element $z = x + i y$ of the set of non-zero complex numbers $\C_{\ne 0}$ has an inverse element $\dfrac 1 z$ under the operation of complex multiplication:

$\forall z \in \C_{\ne 0}: \exists \dfrac 1 z \in \C_{\ne 0}: z \times \dfrac 1 z = 1 + 0i = \dfrac 1 z \times z$

This inverse can be expressed as:

$\dfrac 1 z = \dfrac {x - i y} {x^2 + y^2} = \dfrac {\overline z} {z \overline z}$

where $\overline z$ is the complex conjugate of $z$.

Complex Multiplication is Commutative

The operation of multiplication on the set of complex numbers $\C$ is commutative:

$\forall z_1, z_2 \in \C: z_1 z_2 = z_2 z_1$

Complex Numbers form Ring

The set of complex numbers $\C$ forms a ring under addition and multiplication: $\left({\C, +, \times}\right)$.

The operation of multiplication on the set of complex numbers $\C$ is distributive over the operation of addition.

$\forall z_1, z_2, z_3 \in \C:$
$z_1 \left({z_2 + z_3}\right) = z_1 z_2 + z_1 z_3$
$\left({z_2 + z_3}\right) z_1 = z_2 z_1 + z_3 z_1$

Complex Numbers form Field

Consider the algebraic structure $\struct {\C, +, \times}$, where:

$\C$ is the set of all complex numbers
$+$ is the operation of complex addition
$\times$ is the operation of complex multiplication

Then $\struct {\C, +, \times}$ forms a field.

Substructures and Superstructures

Additive Group of Integers is Normal Subgroup of Complex

Let $\struct {\Z, +}$ be the additive group of integers.

Let $\struct {\C, +}$ be the additive group of complex numbers.

Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\C, +}$.

Additive Group of Rationals is Subgroup of Complex

Let $\left({\Q, +}\right)$ be the additive group of rational numbers.

Let $\left({\C, +}\right)$ be the additive group of complex numbers.

Then $\left({\Q, +}\right)$ is a normal subgroup of $\left({\C, +}\right)$.

Additive Group of Reals is Subgroup of Complex

Let $\struct {\R, +}$ be the additive group of real numbers.

Let $\struct {\C, +}$ be the additive group of complex numbers.

Then $\struct {\R, +}$ is a subgroup of $\struct {\C, +}$.

Multiplicative Group of Rationals is Subgroup of Complex

Let $\left({\Q, \times}\right)$ be the multiplicative group of rational numbers.

Let $\left({\C, \times}\right)$ be the multiplicative group of complex numbers.

Then $\left({\Q, \times}\right)$ is a normal subgroup of $\left({\C, \times}\right)$.

Multiplicative Group of Reals is Subgroup of Complex

Let $\left({\R_{\ne 0}, \times}\right)$ be the multiplicative group of real numbers.

Let $\left({\C_{\ne 0}, \times}\right)$ be the multiplicative group of complex numbers.

Then $\left({\R_{\ne 0}, \times}\right)$ is a normal subgroup of $\left({\C_{\ne 0}, \times}\right)$.

Rational Numbers form Subfield of Complex Numbers

Let $\left({\Q, +, \times}\right)$ be the field of rational numbers.

Let $\left({\C, +, \times}\right)$ be the field of complex numbers.

Then $\left({\Q, +, \times}\right)$ is a subfield of $\left({\C, +, \times}\right)$.

Real Numbers form Subfield of Complex Numbers

The field of real numbers $\left({\R, +, \times}\right)$ forms a subfield of the field of complex numbers $\left({\C, +, \times}\right)$.

Further Structural Properties

Complex Numbers form Vector Space

Let $\R$ be the set of real numbers.

Let $\C$ be the set of complex numbers.

Then the $\R$-module $\C$ is a vector space.

Complex Numbers form Algebra

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.