Properties of Natural Numbers

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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. $\N_{>0} = \N \setminus \left\{{0}\right\}$.

The structure $\left({\N_{>0}, \times}\right)$ 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 $\N$:

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


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 $\left({\N, +, \times}\right)$ 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.


Also see