# 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

## Definition

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

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

### Real Number Axioms

## Also defined as

Some sources additionaly specify that $\left({R, \le}\right)$ be densely ordered.

This condition, while conceptually important, is superfluous, by Totally Ordered Field is Densely Ordered.

## Sources

- James R. Munkres:
*Topology*(2nd ed., 2000)... (previous)... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers

## 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.

## Sources

- James M. Hyslop:
*Infinite Series*(1942)... (previous)... (next): $\S 2$: Functions - Murray R. Spiegel:
*Theory and Problems of Complex Variables*(1964)... (previous)... (next): $1$: Complex Numbers: The Real Number System: $4$ - J.A. Green:
*Sets and Groups*(1965)... (previous)... (next): $\S 1.1$: Example $2$ - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 1$ - George McCarty:
*Topology: An Introduction with Application to Topological Groups*(1967)... (previous)... (next): Introduction: Special Symbols - Ian D. Macdonald:
*The Theory of Groups*(1968)... (previous)... (next): Appendix: Elementary set and number theory - C.R.J. Clapham:
*Introduction to Abstract Algebra*(1969)... (previous)... (next): $\S 1.1$ - B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*(1970)... (previous)... (next): $\S 1.2$: Some examples of rings: Ring Example $4$ - Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*(1970)... (previous)... (next): $\text{II}: \ 28$ - Robert H. Kasriel:
*Undergraduate Topology*(1971)... (previous)... (next): $\S 1.8$: Collections of Sets: Definition $8.4$ - T.S. Blyth:
*Set Theory and Abstract Algebra*(1975)... (previous)... (next): $\S 1$ - W.A. Sutherland:
*Introduction to Metric and Topological Spaces*(1975)... (previous)... (next): Notation and Terminology - K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*(1977)... (previous)... (next): $\S 1.2$: The set of real numbers - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): $\S 2 \ \text{(b)}$ - H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*(1996)... (previous)... (next): Appendix $\text{A}.1$: Sets - Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*(2000)... (previous)... (next): $\S 1.2.5$: An aside: proof by contradiction