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 $R$.

Also known as
When the term number is used in general discourse, it is often tacitly understood as meaning real number, but depending on the context, it may also mean integer or natural number.