\begin{definition}[Definition:Real Number/Operations on Real Numbers]
Let $\R$ be the set of real numbers.
We interpret the following symbols:
{{begin-axiom}}
{{axiom | n = \text R 1
        | lc= Negative
        | q = \forall a \in \R
        | m = \exists ! \paren {-a} \in \R: a + \paren {-a} = 0
}}
{{axiom | n = \text R 2
        | lc= Minus
        | q = \forall a, b \in \R
        | m = a - b = a + \paren {-b}
}}
{{axiom | n = \text R 3
        | lc= Reciprocal
        | q = \forall a \in \R \setminus \set 0
        | m = \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a
        | rc= it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$
}}
{{axiom | n = \text R 4
        | lc= Divided by
        | q = \forall a \in \R \setminus \set 0
        | m = a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} }
        | rc= it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$
}}
{{end-axiom}}
The validity of all these operations is justified by Real Numbers form Field.
\end{definition}