Definition:Real Number

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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.

RealNumberLine.png


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 $k$
$\ds z = \sum_{j \mathop = 0}^n a_j b^j + \sum_{k \mathop = 1}^\infty d_k b^{-k}$

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      \(\ds \forall x, y \in \R:\) \(\ds x + y \in \R \)      
\((\R \text A 1)\)   $:$   Associativity of addition      \(\ds \forall x, y, z \in \R:\) \(\ds \paren {x + y} + z = x + \paren {y + z} \)      
\((\R \text A 2)\)   $:$   Commutativity of addition      \(\ds \forall x, y \in \R:\) \(\ds x + y = y + x \)      
\((\R \text A 3)\)   $:$   Identity element for addition      \(\ds \exists 0 \in \R: \forall x \in \R:\) \(\ds x + 0 = x = 0 + x \)      
\((\R \text A 4)\)   $:$   Inverse elements for addition      \(\ds \forall x: \exists \paren {-x} \in \R:\) \(\ds x + \paren {-x} = 0 = \paren {-x} + x \)      
\((\R \text M 0)\)   $:$   Closure under multiplication      \(\ds \forall x, y \in \R:\) \(\ds x \times y \in \R \)      
\((\R \text M 1)\)   $:$   Associativity of multiplication      \(\ds \forall x, y, z \in \R:\) \(\ds \paren {x \times y} \times z = x \times \paren {y \times z} \)      
\((\R \text M 2)\)   $:$   Commutativity of multiplication      \(\ds \forall x, y \in \R:\) \(\ds x \times y = y \times x \)      
\((\R \text M 3)\)   $:$   Identity element for multiplication      \(\ds \exists 1 \in \R, 1 \ne 0: \forall x \in \R:\) \(\ds x \times 1 = x = 1 \times x \)      
\((\R \text M 4)\)   $:$   Inverse elements for multiplication      \(\ds \forall x \in \R_{\ne 0}: \exists \frac 1 x \in \R_{\ne 0}:\) \(\ds x \times \frac 1 x = 1 = \frac 1 x \times x \)      
\((\R \text D)\)   $:$   Multiplication is distributive over addition      \(\ds \forall x, y, z \in \R:\) \(\ds x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z} \)      
\((\R \text O 1)\)   $:$   Usual ordering is compatible with addition      \(\ds \forall x, y, z \in \R:\) \(\ds x > y \implies x + z > y + z \)      
\((\R \text O 2)\)   $:$   Usual ordering is compatible with multiplication      \(\ds \forall x, y, z \in \R:\) \(\ds x > y, z > 0 \implies x \times z > y \times z \)      
\((\R \text O 3)\)   $:$   $\struct {\R, +, \times, \le}$ is Dedekind complete      


These are called the real number axioms.


Examples

The numbers:

$2$
$\dfrac 1 3 = 0 \cdotp 3333 \ldots$
$-1.5$
$\pi = 3 \cdotp 14159 \ldots$

are all real 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 $R$.


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      \(\ds \forall a \in \R:\) \(\ds \exists ! \paren {-a} \in \R: a + \paren {-a} = 0 \)      
\((\text R 2)\)   $:$   Minus      \(\ds \forall a, b \in \R:\) \(\ds a - b = a + \paren {-b} \)      
\((\text R 3)\)   $:$   Reciprocal      \(\ds \forall a \in \R \setminus \set 0:\) \(\ds \exists ! a^{-1} \in \R: a \times \paren {a^{-1} } = 1 = \paren {a^{-1} } \times a \)      it is usual to write $1/a$ or $\dfrac 1 a$ for $a^{-1}$
\((\text R 4)\)   $:$   Divided by      \(\ds \forall a \in \R \setminus \set 0:\) \(\ds a \div b = \dfrac a b = a / b = a \times \paren {b^{-1} } \)      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.


Sources

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