Properties of Natural Numbers
Natural Numbers are Infinite
The set $\N$ of natural numbers is infinite.
Note that by definition of countability, $\N$ is a countable set.
Natural Number Addition is Closed
The operation of addition on the set of natural numbers $\N$ is closed:
- $\forall x, y \in \N: x + y \in \N$
Natural Number Addition is Associative
The operation of addition on the set of natural numbers $\N$ is associative:
- $\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$
Natural Numbers under Addition form Commutative Monoid
The algebraic structure $\struct {\N, +}$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative monoid whose identity is zero.
Natural Number Addition is Commutative
The operation of addition on the set of natural numbers $\N$ is commutative:
- $\forall m, n \in \N: m + n = n + m$
Identity Element of Natural Number Addition is Zero
The identity element for the natural numbers under addition is zero ($0$):
- $\forall n \in \N: 0 + n = n$
Non-Zero Natural Numbers under Multiplication form Commutative Monoid
Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \set 0$.
The structure $\struct{\N_{>0}, \times}$ forms a commutative monoid.
Natural Number Multiplication is Closed
Let $m$ and $n$ be natural numbers.
Then:
- $m \times n \in \N$
where $\times$ denotes natural number multiplication.
Natural Number Multiplication is Associative
The operation of multiplication on the set of natural numbers $\N$ is associative:
- $\forall x, y, z \in \N: \paren {x \times y} \times z = x \times \paren {y \times z}$
Natural Number Multiplication is Commutative
The operation of multiplication on the set of natural numbers $\N$ is commutative:
- $\forall x, y \in \N: x \times y = y \times x$
Identity Element of Natural Number Multiplication is One
Let $1$ be the element one of $\N$.
Then $1$ is the identity element of multiplication:
- $\forall n \in \N: n \times 1 = n = 1 \times n$
Natural Numbers form Commutative Semiring
The semiring of natural numbers $\struct {\N, +, \times}$ forms a commutative semiring.
Natural Number Multiplication Distributes over Addition
The operation of multiplication is distributive over addition on the set of natural numbers $\N$:
- $\forall x, y, z \in \N:$
- $\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
- $z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$
Non-Zero Natural Numbers form Commutative Semiring
Non-Zero Natural Numbers form Commutative Semiring
Natural Numbers form Subsemiring of Integers
The semiring of natural numbers $\struct {\N, +, \times}$ forms a subsemiring of the ring of integers $\struct {\Z, +, \times}$.
Natural Numbers Set Equivalent to Ideals of Integers
Let $S$ be the set of all ideals of $\Z$.
Let the mapping $\psi: \N \to S$ be defined as:
- $\forall b \in \N: \map \psi b = \ideal b$
where $\ideal b$ is the principal ideal of $\Z$ generated by $b$.
Then $\psi$ is a bijection.