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Field and order structure
The real numbers are introduced as a structure characterised by a combination of algebraic and order properties. These properties determine their behaviour and distinguish it from all other numerical fields. The real numbers form a field under addition and multiplication. This means that both operations are associative and commutative, multiplication distributes over addition, and every nonzero real number admits a multiplicative inverse. The additive identity is $0$ and the multiplicative identity is $1$. The algebraic axioms underlying this structure are discussed in detail in Properties of Real Numbers. Beyond its algebraic structure, $\mathbb{R}$ carries a total order relation, denoted $<$: for any two elements $x, y \in \mathbb{R}$, exactly one of the following three relations holds:
The order is compatible with the field operations.
- If $x < y$, then $x + z < y + z$ for every $z \in \mathbb{R}$
- If $x < y$ and $z > 0$ then $xz < yz$
A field equipped with a total order satisfying these compatibility conditions is an ordered field. Both $\mathbb{Q}$ and $\mathbb{R}$ are ordered fields (what separates the two is the completeness property introduced below).
The real line
The real numbers admit a geometric interpretation that makes their order and completeness clear. Fix an arbitrary point on a straight line and label it $0$. Fix a second point to its right and label it $1$. Every real number $x$ then corresponds to a unique point on the line: positive numbers lie to the right of $0$, negative numbers to the left, at a distance from the origin equal to the absolute value $|x|$. This correspondence is a bijection between $\mathbb{R}$ and the points of the line and it preserves the order. $x < y$ holds if and only if the point corresponding to $x$ lies to the left of the point corresponding to $y$.
The completeness axiom
The property that distinguishes $\mathbb{R}$ from $\mathbb{Q}$ is completeness. It expresses the absence of gaps. Every position on the number line that could be approached by a sequence of rational numbers is actually occupied by a real number. The rational numbers, by contrast, leave the line with infinitely many holes, one for each irrational value.
The formulation relies on the notion of an upper bound. A subset $S \subseteq \mathbb{R}$ is said to be bounded above if there exists a real number $M$ such that $x \leq M$ for every $x \in S$. Such a number $M$ is called an upper bound of $S$. When a smallest upper bound exists, it is called the supremum of $S$, or least upper bound, and is denoted $\sup S$.
The completeness axiom of the real numbers can be stated as follows: every non-empty subset of $\mathbb{R}$ that is bounded above has a supremum in $\mathbb{R}$. This statement is known as the least upper bound property. The rational numbers fail to satisfy it. To see why, consider the following set:
This set is non-empty and bounded above within $\mathbb{Q}$, yet it has no least upper bound in $\mathbb{Q}$. The value $\sqrt{2}$, which plays the role of $\sup S$, is irrational and therefore absent from $\mathbb{Q}$. In $\mathbb{R}$, the number $\sqrt{2}$ exists and one has $\sup S = \sqrt{2}$.
A symmetric notion applies to sets bounded below. A subset $S \subseteq \mathbb{R}$ is bounded below if there exists $m \in \mathbb{R}$ such that $x \geq m$ for all $x \in S$. The greatest lower bound, or infimum, is denoted $\inf S$. The completeness axiom implies that every non-empty subset of $\mathbb{R}$ bounded below has an infimum in $\mathbb{R}$.
The Archimedean property
A consequence of completeness is the Archimedean property of $\mathbb{R}$. It states that for every real number $x$, there exists a natural number $n$ such that $n > x$. Equivalently, the set of natural numbers $\mathbb{N}$ is not bounded above in $\mathbb{R}$. The argument runs as follows.
- Suppose, for contradiction, that $\mathbb{N}$ were bounded above in $\mathbb{R}$.
- By the completeness axiom, $\mathbb{N}$ would then have a supremum that we call $s = \sup \mathbb{N}$.
- Since $s - 1 < s$, the number $s - 1$ is not an upper bound of $\mathbb{N}$, so there exists $n \in \mathbb{N}$ with $n > s - 1$.
- It follows that $n + 1 > s$. Since $n + 1 \in \mathbb{N}$, this contradicts $s$ being an upper bound of $\mathbb{N}$.
To illustrate the property concretely, consider the real number $x = 7.4$. The Archimedean property guarantees the existence of a natural number greater than $x$: since $8 > 7.4$, the smallest such natural number is $n = 8$. The result, elementary as it appears, depends on the completeness of $\mathbb{R}$ and fails in ordered fields that do not satisfy it.
Dedekind cuts
A Dedekind cut is a subset $A \subseteq \mathbb{Q}$ satisfying three conditions: $A$ is non-empty and $A \neq \mathbb{Q}$. If $q \in A$ and $p < q$ then $p \in A$, and $A$ has no greatest element. The set $\mathbb{R}$ is then defined as the collection of all Dedekind cuts of $\mathbb{Q}$.
To illustrate, the rational number $r \in \mathbb{Q}$ corresponds to the cut $A_r = \{ q \in \mathbb{Q} : q < r \}$. An irrational number such as $\sqrt{2}$ corresponds instead to the cut:
This set satisfies all three conditions, yet has no rational supremum in $\mathbb{Q}$: the cut carves out a position on the rational line where no rational number sits, and it is precisely this gap that the construction fills by declaring $A$ itself to be a real number.
The algebraic structure of $\mathbb{R}$ is then built directly from set-theoretic operations on cuts. Addition is defined by setting:
The order is given by inclusion: $A \leq B$ if and only if $A \subseteq B$. One verifies that these definitions make $\mathbb{R}$ into a totally ordered field. The completeness of $\mathbb{R}$ in this construction has the following proof: given a non-empty collection $S$ of cuts that is bounded above, the supremum is the union $\bigcup_{A \in S} A$, which is itself a cut and is the least upper bound of $S$ by construction.
The Dedekind construction builds the real numbers from the order structure of $\mathbb{Q}$ alone. Its main contribution is to show that the completeness of $\mathbb{R}$ is a consequence of filling in all the gaps that the rational order leaves open.
Cauchy sequence construction
A second construction of $\mathbb{R}$ starts from a limitation of $\mathbb{Q}$. Not every Cauchy sequence of rational numbers converges to a rational number. A sequence $(x_n)_{n \in \mathbb{N}}$ in $\mathbb{Q}$ is a Cauchy sequence if for every $\varepsilon \in \mathbb{Q}^+$ there exists $N \in \mathbb{N}$ such that:
The terms of the sequence cluster together without the sequence needing to refer to a limit, which may not yet exist in $\mathbb{Q}$. The sequence of rational approximations to $\sqrt{2}$ is a standard example:
It is Cauchy in $\mathbb{Q}$, yet its limit lies outside $\mathbb{Q}$. Two Cauchy sequences $(x_n)$ and $(y_n)$ in $\mathbb{Q}$ are declared equivalent if their difference tends to zero:
One verifies that $\sim$ is an equivalence relation. The real numbers are then defined as the set of equivalence classes:
$C$ denotes the collection of all Cauchy sequences in $\mathbb{Q}$. Addition and multiplication are defined termwise:
These operations are well-defined, meaning independent of the choice of representative. The order is given by $[(x_n)] \leq [(y_n)]$ when either $(x_n) \sim (y_n)$ or there exists $N \in \mathbb{N}$ such that $x_n < y_n$ for all $n \geq N$. The rational numbers embed into $\mathbb{R}$ via $q \mapsto [(q, q, q, \ldots)]$ preserving both operations and order.
The real number $\sqrt{2}$, for instance, is the equivalence class of any Cauchy sequence of rationals converging to it, such as $(1, 1.4, 1.41, 1.414, \ldots)$. Two such sequences satisfy $(x_n) \sim (y_n)$ and therefore define the same real number. Completeness in this construction means that every Cauchy sequence of real numbers converges to a real number. The Dedekind and Cauchy constructions are isomorphic as complete ordered fields (they describe exactly the same mathematical object).
Consequences of completeness
The rational numbers are dense in $\mathbb{R}$: between any two distinct real numbers there exists a rational number. In more formal terms for every $x, y \in \mathbb{R}$ with $x < y$, there exists $q \in \mathbb{Q}$ such that $x < q < y$. This follows from the Archimedean property. Given $x < y$, one finds a natural number $n$ satisfying $n(y - x) > 1$ and among the integers $m$ with $m > nx$ one can identify one for which $x < m/n < y$ holds.
Despite the density of $\mathbb{Q}$ in $\mathbb{R}$, the two sets differ in cardinality. The rational numbers are countable, meaning their elements can be placed in a one-to-one correspondence with the natural numbers. The real numbers are uncountable since no such correspondence exists. This result implies that the irrational numbers, which form the set $\mathbb{R} \setminus \mathbb{Q}$, constitute the overwhelming majority of the real line.
The Bolzano-Weierstrass theorem is a further consequence of completeness. It states that every bounded sequence of real numbers has a convergent subsequence. This result guarantees that bounded infinite sets cannot spread indefinitely without accumulating somewhere, and it underpins the theory of limits, continuous functions, and compactness in $\mathbb{R}$.
Uniqueness of $\mathbb{R}$
The real number system is the unique complete ordered field. Any two complete ordered fields are isomorphic as ordered fields, and the isomorphism between them is unique. This means that $\mathbb{R}$ is not merely one among many possible completions of $\mathbb{Q}$ but the only one up to a structure-preserving bijection.
Intervals
Among the subsets of $\mathbb{R}$, intervals occupy a central role. An interval is a subset $I \subseteq \mathbb{R}$ with the property that, whenever two points belong to it, every point lying between them also belongs to it. Intervals may be bounded, such as the open interval $(a, b)$ or the closed interval $[a, b]$, or unbounded, such as $[a, +\infty)$ or $(-\infty, b)$. The entire real line is itself an interval, denoted $(-\infty, +\infty)$.