# Definition:Real Number/Number Line Definition

## Definition

A **real number** is defined as a **number** which is identified with a point on the **real number line**.

#### Real Number Line

From the Cantor-Dedekind Hypothesis, 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 numbers is consistent with the length of the line between those two points.

## Notation

While the symbol $\R$ is the current standard symbol used to denote the **set of real numbers**, variants are commonly seen.

For example: $\mathbf R$, $\RR$ and $\mathfrak R$, or even just $R$.

## Equality of Real Numbers

Two real numbers are defined as being **equal** if and only if they correspond to the same point on the real number line.

## Also known as

When the term **number** is used in general discourse, it is often tacitly understood as meaning **real number**.

They are sometimes referred to in the pedagogical context as **ordinary numbers**, so as to distinguish them from **complex numbers**

However, depending on the context, the word **number** may also be taken to mean **integer** or **natural number**.

Hence it is wise to be specific.

## Also see

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

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets: Example $2$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.2$: The set of real numbers