Definition:Real Number/Operations on Real Numbers
From ProofWiki
Definition
Let $\R$ be the set of real numbers.
We interpret the following symbols:
- Negative: $\forall a \in \R: \exists ! \left({-a}\right) \in \R: a + \left({-a}\right) = 0$
- Minus: $\forall a, b \in \R: a - b = a + \left({-b}\right)$
- Reciprocal: $\forall a \in \R \setminus \left\{{0}\right\}: \exists ! a^{-1} \in \R: a \times \left({a^{-1}})\right) = 1 = \left({a^{-1}}\right) \times a$ (we often write $1/a$ or $\displaystyle \frac 1 a$ for $a^{-1}$)
- Divided by: $\displaystyle \forall a, b \in \R \setminus \left\{{0}\right\}: a \div b = \frac a b = a / b = a \times \left({b^{-1}}\right)$
The validity of all these operations is guaranteed by the fact that the real numbers form a field‎.
