# Properties of Real Numbers

## Real Numbers are Uncountable

The set of real numbers $\R$ is uncountably infinite.

## Real Numbers under Addition form Abelian Group

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

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

The operation of addition on the set of real numbers $\R$ is well-defined.

The set of real numbers $\R$ is closed under addition:

$\forall x, y \in \R: x + y \in \R$

The operation of addition on the set of real numbers $\R$ is associative:

$\forall x, y, z \in \R: x + \left({y + z}\right) = \left({x + y}\right) + z$

### Real Addition Identity is Zero

The identity of real number addition is $0$:

$\exists 0 \in \R: \forall x \in \R: x + 0 = x = 0 + x$

Each element $x$ of the set of real numbers $\R$ has an inverse element $-x$ under the operation of real number addition:

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

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:

$\forall x, y, z \in \R: x \times \left({y \times z}\right) = \left({x \times y}\right) \times z$

### Real Multiplication Identity is One

The identity element of real number multiplication is the real number $1$:

$\exists 1 \in \R: \forall a \in \R_{\ne 0}: a \times 1 = a = 1 \times a$

### Inverses 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, i.e. $\R_{>0} = \left\{{ x \in \R: x > 0}\right\}$.

The structure $\left({\R_{>0}, \times}\right)$ is an uncountable abelian group.

## Real Numbers form Ring

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

### Real Multiplication Distributes over Addition

The operation of multiplication on the set of real numbers $\R$ is distributive over the operation of addition:

$\forall x, y, z \in \R:$
$x \times \left({y + z}\right) = x \times y + x \times z$
$\left({y + z}\right) \times x = y \times x + z \times x$

## Real Numbers form Integral Domain

The set of real numbers $\R$ forms an integral domain under addition and multiplication: $\left({\R, +, \times}\right)$.

## Real Numbers form Ordered Integral Domain

The set of real numbers $\R$ forms an ordered integral domain under addition and multiplication: $\left({\R, +, \times}\right)$.

## Real Numbers form Totally Ordered Field

The set of real numbers $\R$ forms a totally ordered field under addition and multiplication: $\left({\R, +, \times, \le}\right)$.

## 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 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 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 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 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 field $\left({\Q, +, \times, \le}\right)$ of rational numbers forms a subfield of the field of real numbers $\left({\R, +, \times, \le}\right)$.

That is, the field of real numbers $\left({\R, +, \times, \le}\right)$ is an extension of the rational numbers $\left({\Q, +, \times, \le}\right)$.

### 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

### 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.