# 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 $\R$.

### Operations on 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.

### 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.

## Also denoted as

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

## Also see

• Results about real numbers can be found here.