# Definition:Real Number/Dedekind Cuts

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## Definition

The set of rational numbers are identified with the set of rational cuts.

All other cuts are called, and are identified with, **irrational numbers**.

## 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 $R$.

## 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

- 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.31$. Definition