Definition:Real Number

Informal Definition
Any number on the number line is referred to as a real number.

This includes more numbers than the set of rational numbers as $\sqrt{2}$ for example is not rational.

The set of real numbers is denoted $\R$.

Formal Definition
Consider the set of rational numbers, $\Q$.

For any two Cauchy sequences of rational numbers $X = \left \langle {x_n} \right \rangle, Y = \left \langle {y_n} \right \rangle$, define an equivalence relation between the two as:


 * $X \equiv Y \iff \forall \epsilon > 0: \exists n \in \N: \forall i, j > n: \left|{x_i - y_j}\right| < \epsilon$

The real numbers are the set of all equivalence classes $\left[\!\left[{\left \langle {x_n} \right \rangle}\right]\!\right]$ of Cauchy sequences of rational numbers.

The set of real numbers is denoted $\R$.

Also denoted as
Variants on $\R$ are often seen, for example $\mathbf R$ and $\mathcal R$, or even just $R$.