Definition:Real Number/Operations on Real Numbers

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Definition

Let $\R$ be the set of real numbers.

We interpret the following symbols:

 $(\text R 1)$ $:$ Negative $\displaystyle \forall a \in \R:$ $\displaystyle \exists ! \paren {-a} \in \R: a + \paren {-a} = 0$ $(\text R 2)$ $:$ Minus $\displaystyle \forall a, b \in \R:$ $\displaystyle a - b = a + \paren {-b}$ $(\text R 3)$ $:$ Reciprocal $\displaystyle \forall a \in \R \setminus \set 0:$ $\displaystyle \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a$ it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ $(\text R 4)$ $:$ Divided by $\displaystyle \forall a \in \R \setminus \set 0:$ $\displaystyle a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} }$ it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$

The validity of all these operations is justified by Real Numbers form Field.