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:

\((R1)\)   $:$   Negative      \(\displaystyle \forall a \in \R:\) \(\displaystyle \exists ! \paren {-a} \in \R: a + \paren {-a} = 0 \)             
\((R2)\)   $:$   Minus      \(\displaystyle \forall a, b \in \R:\) \(\displaystyle a - b = a + \paren {-b} \)             
\((R3)\)   $:$   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 $a^{-1}$
\((R4)\)   $:$   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.


Sources