# What is a real number

1. Jul 23, 2014

### Greg Bernhardt

Definition/Summary

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

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

Equations

Extended explanation

A Brief History
The real numbers originate from a need to quantify the geometric notion of 'length'. It was known in ancient times that the 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 the 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 proveded. 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".

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