Definition:Real Number

The set of real numbers is the set of all numbers which is closed under the taking of Cauchy sequences.

The set of real numbers is denoted $$\mathbb{R}$$.

Operations on Real Numbers
We interpret the following symbols:


 * Negative: $$\forall a \in \mathbb{R}: \exists ! \left({-a}\right) \in \reals: a + \left({-a}\right) = 0$$
 * Minus: $$\forall a, b \in \mathbb{R}: a - b = a + \left({-b}\right)$$
 * Reciprocal: $$\forall a \in \mathbb{R} - \left\{{0}\right\}: \exists ! a^{-1} \in \mathbb{R}: a \times \left({a^{-1}})\right) = 1 = \left({a^{-1}}\right) \times a$$ (we often write $$1/a$$ or $$\frac 1 a$$ for $$a^{-1}$$
 * Divided by: $$\forall a, b \in \mathbb{R} - \left\{{0}\right\}: a \div b = \frac a b = a / b = a \times \left({b^{-1}}\right)$$

The validity of all these operations is guaranteed by the fact that the Real Numbers form a Field‎.