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.