Properties of Rational Numbers
Rational Numbers are Countably Infinite
The set $\Q$ of rational numbers is countably infinite.
Rational Numbers under Addition form Infinite Abelian Group
Let $\Q$ be the set of rational numbers.
The structure $\struct {\Q, +}$ is a countably infinite abelian group.
Rational Addition is Closed
The operation of addition on the set of rational numbers $\Q$ is well-defined and closed:
- $\forall x, y \in \Q: x + y \in \Q$
Rational Addition is Associative
The operation of addition on the set of rational numbers $\Q$ is associative:
- $\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$
Rational Addition Identity is Zero
The identity of rational number addition is $0$:
- $\exists 0 \in \Q: \forall a \in \Q: a + 0 = a = 0 + a$
Inverse for Rational Addition
Each element $x$ of the set of rational numbers $\Q$ has an inverse element $-x$ under the operation of rational number addition:
- $\forall x \in \Q: \exists -x \in \Q: x + \paren {-x} = 0 = \paren {-x} + x$
Rational Addition is Commutative
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 Infinite 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:
- $\forall x, y, z \in \Q: x \times \paren {y \times z} = \paren {x \times y} \times z$
Rational Multiplication Identity is One
The identity of rational number multiplication is $1$:
- $\exists 1 \in \Q: \forall a \in \Q: a \times 1 = a = 1 \times a$
Inverse for Rational Multiplication
Each element $x$ of the set of non-zero rational numbers $\Q_{\ne 0}$ has an inverse element $\dfrac 1 x$ under the operation of rational number multiplication:
- $\forall x \in \Q_{\ne 0}: \exists \dfrac 1 x \in \Q_{\ne 0}: x \times \dfrac 1 x = 1 = \dfrac 1 x \times x$
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: $\struct {\Q, +, \times}$.
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:
- $\forall x, y, z \in \Q: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$
- $\forall x, y, z \in \Q: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$
Rational Numbers form Ordered Integral Domain
The rational numbers $\Q$ form an ordered integral domain under addition and multiplication.
Rational Numbers form Ordered Field
The set of rational numbers $\Q$ forms an ordered field under addition and multiplication: $\struct {\Q, +, \times, \le}$.
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 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 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 $\struct {\C, +}$.
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 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 $\struct {\C, \times}$.
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 (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 $\struct {\Q, +, \times, \le}$.