Properties of Real Numbers
Real Numbers are Uncountable
The set of real numbers $\R$ is uncountably infinite.
Real Numbers under Addition form Infinite Abelian Group
Let $\R$ be the set of real numbers.
The structure $\struct {\R, +}$ is an infinite abelian group.
Real Addition is Well-Defined
The operation of addition on the set of real numbers $\R$ is well-defined.
Real Addition is Closed
The set of real numbers $\R$ is closed under addition:
- $\forall x, y \in \R: x + y \in \R$
Real Addition is Associative
The operation of addition on the set of real numbers $\R$ is associative:
- $\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$
Real Addition Identity is Zero
The identity of real number addition is $0$:
- $\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$
Inverse for Real Addition
Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition:
- $\forall x \in \R: \exists -x \in \R: x + \paren {-x} = 0 = \paren {-x} + x$
Real Addition is Commutative
The operation of addition on the set of real numbers $\R$ is commutative:
- $\forall x, y \in \R: x + y = y + x$
Non-Zero Real Numbers under Multiplication form Abelian Group
Let $\R_{\ne 0}$ be the set of real numbers without zero:
- $\R_{\ne 0} = \R \setminus \set 0$
The structure $\struct {\R_{\ne 0}, \times}$ is an uncountable abelian group.
Real Multiplication is Well-Defined
The operation of multiplication on the set of real numbers $\R$ is well-defined.
Real Multiplication is Closed
The operation of multiplication on the set of real numbers $\R$ is closed:
- $\forall x, y \in \R: x \times y \in \R$
Real Multiplication is Associative
The operation of multiplication on the set of real numbers $\R$ is associative:
- $\forall x, y, z \in \R: x \times \paren {y \times z} = \paren {x \times y} \times z$
Real Multiplication Identity is One
The identity element of real number multiplication is the real number $1$:
- $\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$
Inverse for Real Multiplication
Each element $x$ of the set of non-zero real numbers $\R_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of real number multiplication:
- $\forall x \in \R_{\ne 0}: \exists \dfrac 1 x \in \R_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$
Real Multiplication is Commutative
The operation of multiplication on the set of real numbers $\R$ is commutative:
- $\forall x, y \in \R: x \times y = y \times x$
Strictly Positive Real Numbers under Multiplication form Uncountable Abelian Group
Let $\R_{>0}$ be the set of strictly positive real numbers:
- $\R_{>0} = \set {x \in \R: x > 0}$
The structure $\struct {\R_{>0}, \times}$ is an uncountable abelian group.
Real Numbers form Ring
The set of real numbers $\R$ forms a ring under addition and multiplication: $\struct {\R, +, \times}$.
Real Multiplication Distributes over Addition
The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:
- $\forall x, y, z \in \R:$
- $x \times \paren {y + z} = x \times y + x \times z$
- $\paren {y + z} \times x = y \times x + z \times x$
Real Numbers form Integral Domain
The set of real numbers $\R$ forms an integral domain under addition and multiplication: $\struct {\R, +, \times}$.
Real Numbers form Ordered Integral Domain
The set of real numbers $\R$ forms an ordered integral domain under addition and multiplication: $\struct {\R, +, \times, \le}$.
Real Numbers form Ordered Field
The set of real numbers $\R$ forms an ordered field under addition and multiplication: $\struct {\R, +, \times, \le}$.
Substructures and Superstructures
Additive Group of Integers is Normal Subgroup of Reals
Let $\struct {\Z, +}$ be the additive group of integers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.
Additive Group of Rationals is Normal Subgroup of Reals
Let $\struct {\Q, +}$ be the additive group of rational numbers.
Let $\struct {\R, +}$ be the additive group of real numbers.
Then $\struct {\Q, +}$ is a normal subgroup of $\struct {\R, +}$.
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 Reals
Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers.
Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers.
Then $\struct {\Q_{\ne 0}, \times}$ is a normal subgroup of $\left({\R_{\ne 0}, \times}\right)$.
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 Real Numbers
The (ordered) field $\struct {\Q, +, \times, \le}$ of rational numbers forms a subfield of the field of real numbers $\struct {\R, +, \times, \le}$.
That is, the field of real numbers $\struct {\R, +, \times, \le}$ is an extension of the rational numbers $\struct {\Q, +, \times, \le}$.
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
Real Numbers form Vector Space
The set of real numbers $\R$, with the operations of addition and multiplication, forms a vector space.
Real Numbers form Algebra
The set of real numbers $\R$ 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 whose $*$ operator is the identity mapping.
- $(5): \quad$ A real $*$-algebra.