# 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 $X = \sequence {x_n}, Y = \sequence {y_n}$, define an equivalence relation between the two as:

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

A **real number** is an equivalence class $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. (See Equivalence Relation on Cauchy Sequences.)

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 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.

## Also see

- Results about
**real numbers**can be found here.

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