What is a Real Number? A 5 Minute Introduction

Definition/Summary

The real numbers are the most commonly encountered number system, familiar to the layman via the number line, and as the number system lying behind decimal notation.

Because real numbers have much nice arithmetic and geometric properties, they feature prominently in many fields of mathematics.

Extended explanation

This is a preliminary article. More information will be added in the future.

A Brief History

The real numbers originate from a need to quantify the geometric notion of ‘length’. It was known in ancient times that rational numbers are not adequate, culminating with Eudoxus’s ‘method of exhaustion.

Formal Definition

With the development of modern calculus, it became increasingly clear that a rigorous definition of real numbers was required. Cantor provided the first definition, by identifying a real number with the set of Cauchy sequences of rational that ought to converge to that real number. Since then, many other equivalent definitions have been provided. The following definition is the one traditionally used today, except that the Dedekind completeness axiom has been replaced with an equivalent axiom:

Language

The real numbers consist of:

A set $\mathbb{R}$ whose elements are called real numbers (also written R)
A distinguished real number $0$ (zero)
A distinguished real number $1$ (one)
A binary operation $+$ (addition)
A binary operation $\cdot$ (multiplication)
A unary operation $-$ (negation)
A unary partial operation ${}^{-1}$ (reciprocal)
A relation $\leq$ (less than or equal to)

In what follows, the symbols $a, b, c$ denote real numbers, $d$ denotes a nonzero real number (meaning $d \neq 0$).

Field Axioms

$a+b$ is a real number
$a \cdot b$ is a real number
$-a$ is a real number
$d^{-1}$ is a real number
$a+b = b+a$ (commutativity of addition)
$a \cdot b = b \cdot a$ (commutativity of multiplication)
$a+(b+c) = (a+b)+c$ (associativity of addition)
$a\cdot (b\cdot c) = (a\cdot b)\cdot c$ (associativity of multiplication)
$a\cdot (b+c) = (a\cdot b) + (a \cdot c)$ (distributivity of multiplication over addition)
$a + 0 = a$ (0 is the additive identity)
$a \cdot 1 = a$ (1 is the multiplicative identity)
$a + (-a) = 0$ (additive inverses)
$d \cdot (d^{-1}) = 1$ (multiplicative inverses)

Ordering axioms

If $a \leq b$ and $b \leq a$ then $a = b$
If $a \leq b$ and $b \leq c$ then $a \leq c$
$a \leq b$ or $b \leq a$
If $a \leq b$ then $a + c \leq b + c$
If $0 \leq c$ and $a \leq b$ then $a \cdot c \leq b \cdot c$

Completeness axiom

The completeness axiom is significantly more complicated. One way to state it is via the least upper bound property of the calculus of sequences. Let $\{ f_n \}$ be a sequence of real numbers

If $f_n \leq f_{n+1} \leq a$ for every natural number $n$, then $\lim_{n \rightarrow +\infty} f_n$ exists

Other Operations and Identities

These axioms are sufficient to derive all of the familiar properties of the real numbers. Some examples:

Subtraction is defined by $a – b = a + (-b)$
Division is defined by $a / d = a \cdot (d^{-1})$

Common Errors

The word “real” in “the real numbers” is often mistaken for the ordinary English word. In actuality, it is simply a name, and is used to distinguish the number system from other familiar systems like “the integers” or “the rational numbers” or “the Galois field of 49 elements”.

Extra

There are two main ways to introduce real numbers. Cauchy sequences are one possibility, Dedekind cuts are another. It is surprising how such us common thing as real numbers needs some elaborated mathematics in order to properly define them. Even the approach by Cauchy sequences needs the concept of equivalence classes.

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