real number system

Real numbers are a set on which the operations of  addition and multiplication are defined.

[axiom for addition]
[1] If $a,b\in \mathbb{R}$, $a+b\in \mathbb{R}$
[2] $a+b=b+a$
[3] $(a+b)+c=a+(b+c)$
[4] There is a distinct real number $0$ such that $a+0=a$
[5] For each $a$, there is a real number $-a$ such that $a+(-a)=0$

[axiom for multiplication]
[11] If $a,b\in \mathbb{R}$, $ab\in \mathbb{R}$
[12] $ab=ba$
[13] $(ab)c=a(bc)$
[14] There is a distinct real number $1$ such that $a1=a$
[15] For each $a$, there is a real number $1/a$ such that $a(1/a)=a$, where $a\neq 0$

As you have ever seen, operations of [2] and [12] are called commutative laws, and [3] and [13] are called associative laws. The following operation which shall be satisfied in real numbers is called the distributive law.

[distributive law]
[21] $a(b+c)=ab+ac$

A set which is satisfied the operations of addition [1]-[5], multiplication [11]-[15], and the distributive law [21] is called a field. Therefore, real number system has field properties.

And, real numbers are ordered by the relation "<" ( less than ) between every pair of elements.

[ordered relation]
[31] For each pair of real numbers $a$ and $b$, exactly one of the following is true:
       $a=b$,  $a<b$,  or  $b<a$
[32] If $a<b$ and $b<c$, then $a<c$
[33] If $a<b$, then $a+c<b+c$
[34] If $a<b$, then $ac<bc$, whenever $0<c$

$a=b$ in [31] means that $a$ is not less than $b$, and $b$ is not less than $a$. By the property [32], the relation $<$ is called transitive. Hence, real numbers are said to be a ordered field. As rational numbers satisfy above properties, that is also a ordered field. However, the following property is not satisfied in rational numbers. It is called completeness axiom.

[axiom for completeness]
[41] If a set of real numbers is bounded above, then it has a supremum.

Real number system has above all properties. Hence, real numbers are a ordered field satisfied completeness.