# Properties of Complex Numbers

## Complex Numbers are Uncountable

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

## Complex Numbers under Addition form Infinite Abelian Group

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

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

### Complex Addition is Closed

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

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

### Complex Addition is Associative

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

- $\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$

### Complex Addition Identity is Zero

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

The identity element of $\struct {\C, +}$ is the complex number $0 + 0 i$.

### Inverse for Complex Addition

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:

- $\forall z \in \C: \exists -z \in \C: z + \paren {-z} = 0 = \paren {-z} + z$

### Complex Addition is Commutative

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 Infinite 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:

- $\forall z_1, z_2, z_3 \in \C: z_1 \paren {z_2 z_3} = \paren {z_1 z_2} z_3$

### Complex Multiplication Identity is One

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

The identity element of $\struct {\C_{\ne 0}, \times}$ 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 $z^{-1}$ under the operation of complex multiplication:

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

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: $\struct {\C, +, \times}$.

### Complex Multiplication Distributes over Addition

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

- $\forall z_1, z_2, z_3 \in \C:$
- $z_1 \paren {z_2 + z_3} = z_1 z_2 + z_1 z_3$
- $\paren {z_2 + z_3} z_1 = z_2 z_1 + z_3 z_1$

## 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 Normal Subgroup of Complex

Let $\struct {\Q, +}$ be the additive group of rational numbers.

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

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

### Additive Group of Reals is Normal 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 normal subgroup of $\struct {\C, +}$.

### Multiplicative Group of Rationals is Normal Subgroup of Complex

Let $\struct {\Q, \times}$ be the multiplicative group of rational numbers.

Let $\struct {\C, \times}$ be the multiplicative group of complex numbers.

Then $\struct {\Q, \times}$ is a normal subgroup of $\struct {\C, \times}$.

### Multiplicative Group of Reals is Normal Subgroup of Complex

Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers.

Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.

Then $\struct {\R_{\ne 0}, \times}$ is a normal subgroup of $\struct {\C_{\ne 0}, \times}$.

### Rational Numbers form Subfield of Complex Numbers

Let $\struct {\Q, +, \times}$ denote the field of rational numbers.

Let $\struct {\C, +, \times}$ denote the field of complex numbers.

$\struct {\Q, +, \times}$ is a subfield of $\struct {\C, +, \times}$.

### Real Numbers form Subfield of Complex Numbers

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

## Further Structural Properties

### Complex Numbers form Vector Space over Reals

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.