# Properties of Rational Numbers

## Rational Numbers are Countably Infinite

The set $\Q$ of rational numbers is countably infinite.

## Rational Numbers under Addition form Abelian Group

Let $\Q$ be the set of rational numbers.

The structure $\struct {\Q, +}$ is a countably infinite abelian group.

The operation of addition on the set of rational numbers $\Q$ is well-defined and closed:

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

The operation of addition on the set of rational numbers $\Q$ is associative:

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

### Rational Addition Identity is Zero

The identity of rational number addition is $0$:

$\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$

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

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

The operation of addition on the set of rational numbers $\Q$ is commutative:

$\forall x, y \in \Q: x + y = y + x$

## Non-Zero Rational Numbers under Multiplication form Abelian Group

Let $\Q_{\ne 0}$ be the set of non-zero rational numbers:

$\Q_{\ne 0} = \Q \setminus \set 0$

The structure $\struct {\Q_{\ne 0}, \times}$ is a countably infinite abelian group.

### Rational Multiplication is Closed

The operation of multiplication on the set of rational numbers $\Q$ is well-defined and closed:

$\forall x, y \in \Q: x \times y \in \Q$

### Rational Multiplication is Associative

The operation of multiplication on the set of rational numbers $\Q$ is associative:

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

### Rational Multiplication Identity is One

The identity of rational number multiplication is $1$:

$\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$

### Inverses for Rational Multiplication

Each element $x$ of the set of non-zero rational numbers $\Q^*$ has an inverse element $\dfrac 1 x$ under the operation of rational number multiplication:

$\displaystyle \forall x \in \Q^*: \exists \frac 1 x \in \Q^*: x \times \frac 1 x = 1 = \frac 1 x \times x$

### Rational Multiplication is Commutative

The operation of multiplication on the set of rational numbers $\Q$ is commutative:

$\forall x, y \in \Q: x \times y = y \times x$

## Strictly Positive Rational Numbers under Multiplication form Countably Infinite Abelian Group

Let $\Q_{> 0}$ be the set of strictly positive rational numbers, i.e. $\Q_{> 0} = \set {x \in \Q: x > 0}$.

The structure $\struct {\Q_{> 0}, \times}$ is a countably infinite abelian group.

## Rational Numbers form Ring

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

## Rational Numbers form Integral Domain

The set of rational numbers $\Q$ forms an integral domain under addition and multiplication: $\struct {\Q, +, \times}$.

### Rational Multiplication Distributes over Addition

The operation of multiplication on the set of rational numbers $\Q$ is distributive over addition:

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

## Rational Numbers form Ordered Integral Domain

The rational numbers $\Q$ form an ordered integral domain under addition and multiplication.

## Rational Numbers form Totally Ordered Field

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

## Substructures and Superstructures

### Additive Group of Integers is Normal Subgroup of Rationals

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

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

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

### 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 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 $\left({\C, +}\right)$.

### 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 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 $\left({\C, \times}\right)$.

### Integers form Subdomain of Rationals

The integral domain of integers $\left({\Z, +, \times}\right)$ forms a subdomain of the field of rational numbers.

### Rational Numbers form Subfield of Real Numbers

The 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 $\struct {\Q, +, \times, \le}$.