# 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:

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

## Natural Numbers under Addition form Commutative Monoid

The algebraic structure $\left({\N, +}\right)$ 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 ($0$).

## 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:

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

### 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 $\left({\N, +, \times}\right)$ forms a commutative semiring.

### Natural Number Multiplication Distributes over Addition

The operation of multiplication is distributive over addition on the set of natural numbers $\N$:

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

## Natural Numbers form Subsemiring of Integers

The semiring of natural numbers $\left({\N, +, \times}\right)$ forms a subsemiring of the ring of integers $\left({\Z, +, \times}\right)$.

## Natural Numbers Set Equivalent to Ideals of Integers

Let the mapping $\psi: \N \to$ the set of all ideals of $\Z$ be defined as:

$\forall b \in \N: \psi \left({b}\right) = \left({b}\right)$

where $\left({b}\right)$ is the principal ideal of $\Z$ generated by $b$.

Then $\psi$ is a bijection.