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.

Real Number Line
It can be shown (and intuitively understood) that the set of real numbers is isomorphic to any infinite straight line in space.

Thus we can identify any (either physically drawn or imagined) line with the set of real numbers and thereby illustrate truths about the real numbers by means of diagrams.