# Definition:Real Number

## Contents

## 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:

\((R1):\) | Negative | \(\displaystyle \forall a \in \R:\) | \(\displaystyle \exists ! \left({-a}\right) \in \R: a + \left({-a}\right) = 0 \) | ||||

\((R2):\) | Minus | \(\displaystyle \forall a, b \in \R:\) | \(\displaystyle a - b = a + \left({-b}\right) \) | ||||

\((R3):\) | Reciprocal | \(\displaystyle \forall a \in \R \setminus \left\{ {0}\right\}:\) | \(\displaystyle \exists ! a^{-1} \in \R: a \times \left({a^{-1} })\right) = 1 = \left({a^{-1} }\right) \times a \) | it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ | |||

\((R4):\) | Divided by | \(\displaystyle \forall a \in \R \setminus \left\{ {0}\right\}:\) | \(\displaystyle a \div b = \dfrac a b = a / b = a \times \left({b^{-1} }\right) \) | it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ |

The validity of all these operations is justified by Real Numbers form Field.

### Real Number Line

From Set of Real Numbers is Equivalent to Infinite Straight Line, the **set of real numbers** is isomorphic to any infinite straight line.

The **real number line** is an arbitrary infinite straight line each of whose points is identified with a **real number** such that the distance between any two **real number** is consistent with the length of the line between those two points.

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.

## Axiomatic Definition

Let $\left({R, +, \times, \le}\right)$ be a Dedekind complete totally ordered field.

Then $R$ is called the **(field of) real numbers**.

### Real Number Axioms

The properties of the field of real numbers $\left({R, +, \times, \le}\right)$ are as follows:

\((\R A0):\) | Closure under addition | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x + y \in \R \) | ||||

\((\R A1):\) | Associativity of addition | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle \left({x + y}\right) + z = x + \left({y + z}\right) \) | ||||

\((\R A2):\) | Commutativity of addition | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x + y = y + x \) | ||||

\((\R A3):\) | Identity element for addition | \(\displaystyle \exists 0 \in \R: \forall x \in \R:\) | \(\displaystyle x + 0 = x = 0 + x \) | ||||

\((\R A4):\) | Inverse elements for addition | \(\displaystyle \forall x: \exists \left({-x}\right) \in \R:\) | \(\displaystyle x + \left({-x}\right) = 0 = \left({-x}\right) + x \) | ||||

\((\R M0):\) | Closure under multiplication | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x \times y \in \R \) | ||||

\((\R M1):\) | Associativity of multiplication | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle \left({x \times y}\right) \times z = x \times \left({y \times z}\right) \) | ||||

\((\R M2):\) | Commutativity of multiplication | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x \times y = y \times x \) | ||||

\((\R M3):\) | Identity element for multiplication | \(\displaystyle \exists 1 \in \R, 1 \ne 0: \forall x \in \R:\) | \(\displaystyle x \times 1 = x = 1 \times x \) | ||||

\((\R M4):\) | Inverse elements for multiplication | \(\displaystyle \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:\) | \(\displaystyle x \times \frac 1 x = 1 = \frac 1 x \times x \) | ||||

\((\R D):\) | Multiplication is distributive over addition | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle x \times \left({y + z}\right) = \left({x \times y}\right) + \left({x \times z}\right) \) | ||||

\((\R O1):\) | Usual ordering is compatible with addition | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle x > y \implies x + z > y + z \) | ||||

\((\R O2):\) | Usual ordering is compatible with multiplication | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle x > y, z > 0 \implies x \times z > y \times z \) | ||||

\((\R O3):\) | $\left({R, +, \times, \le}\right)$ is Dedekind complete |

These are called the **real number axioms**.

## 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.

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