# Definition:Real Number

## Definition

A working definition of the **real numbers** is as the set $\R$ which comprises the set of rational numbers $\Q$ together with the set of irrational numbers $\R \setminus \Q$.

It is admitted that this is a circular definition, as an irrational number is defined as a **real number** which is not a **rational number**.

More formal approaches are presented below.

### Number Line

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.

### Cauchy Sequences

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.

### Digit Sequence

Let $b \in \N_{>1}$ be a given natural number which is greater than $1$.

The set of **real numbers** can be expressed as the set of all sequences of digits:

- $z = \sqbrk {a_n a_{n - 1} \dotsm a_2 a_1 a_0 \cdotp d_1 d_2 \dotsm d_{m - 1} d_m d_{m + 1} \dotsm}$

such that:

- $0 \le a_j < b$ and $0 \le d_k < b$ for all $j$ and $k$
- $\displaystyle z = \sum_{0 \mathop \le j \le n} a_j b^j + \sum_{k \mathop \ge 1} d_k b^{-k}$

It is usual for $b$ to be $10$.

### Dedekind Cuts

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

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

### Dedekind Completion of Rationals

Definition:Real Number/Dedekind Completion of Rationals

### Axiomatic Definition

Let $\struct {R, +, \times, \le}$ be a Dedekind complete ordered field.

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

#### Real Number Axioms

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

\((\R \text A 0)\) | $:$ | Closure under addition | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x + y \in \R \) | ||||

\((\R \text A 1)\) | $:$ | Associativity of addition | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle \paren {x + y} + z = x + \paren {y + z} \) | ||||

\((\R \text A 2)\) | $:$ | Commutativity of addition | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x + y = y + x \) | ||||

\((\R \text A 3)\) | $:$ | Identity element for addition | \(\displaystyle \exists 0 \in \R: \forall x \in \R:\) | \(\displaystyle x + 0 = x = 0 + x \) | ||||

\((\R \text A 4)\) | $:$ | Inverse elements for addition | \(\displaystyle \forall x: \exists \paren {-x} \in \R:\) | \(\displaystyle x + \paren {-x} = 0 = \paren {-x} + x \) | ||||

\((\R \text M 0)\) | $:$ | Closure under multiplication | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x \times y \in \R \) | ||||

\((\R \text M 1)\) | $:$ | Associativity of multiplication | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle \paren {x \times y} \times z = x \times \paren {y \times z} \) | ||||

\((\R \text M 2)\) | $:$ | Commutativity of multiplication | \(\displaystyle \forall x, y \in \R:\) | \(\displaystyle x \times y = y \times x \) | ||||

\((\R \text M 3)\) | $:$ | Identity element for multiplication | \(\displaystyle \exists 1 \in \R, 1 \ne 0: \forall x \in \R:\) | \(\displaystyle x \times 1 = x = 1 \times x \) | ||||

\((\R \text M 4)\) | $:$ | Inverse elements for multiplication | \(\displaystyle \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:\) | \(\displaystyle x \times \frac 1 x = 1 = \frac 1 x \times x \) | ||||

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

\((\R \text O 1)\) | $:$ | Usual ordering is compatible with addition | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle x > y \implies x + z > y + z \) | ||||

\((\R \text O 2)\) | $:$ | Usual ordering is compatible with multiplication | \(\displaystyle \forall x, y, z \in \R:\) | \(\displaystyle x > y, z > 0 \implies x \times z > y \times z \) | ||||

\((\R \text O 3)\) | $:$ | $\struct {R, +, \times, \le}$ is Dedekind complete |

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

## 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.

## Operations on Real Numbers

We interpret the following symbols:

\((\text R 1)\) | $:$ | Negative | \(\displaystyle \forall a \in \R:\) | \(\displaystyle \exists ! \paren {-a} \in \R: a + \paren {-a} = 0 \) | ||||

\((\text R 2)\) | $:$ | Minus | \(\displaystyle \forall a, b \in \R:\) | \(\displaystyle a - b = a + \paren {-b} \) | ||||

\((\text R 3)\) | $:$ | Reciprocal | \(\displaystyle \forall a \in \R \setminus \set 0:\) | \(\displaystyle \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a \) | it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$ | |||

\((\text R 4)\) | $:$ | Divided by | \(\displaystyle \forall a \in \R \setminus \set 0:\) | \(\displaystyle a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} } \) | 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.

## 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.

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