The Construction and Analysis of Real Numbers

The real numbers $\mathbb{R}$ constitute the “continuum,” a mathematical structure and a complete ordered field that underlies the vast majority of analysis, calculus, and physics. While the rational numbers $\mathbb{Q}$ provide a dense set of ratios, they are fundamentally “holey”—failing to contain the limits of many convergent sequences (such as the sequence defining $\sqrt{2}$). This lesson explores the rigorous formalizations used to “fill the gaps” in $\mathbb{Q}$ to arrive at $\mathbb{R}$.

The Supremum Property (Completeness Axiom)

The defining characteristic of the real numbers that distinguishes them from the rationals is the Least Upper Bound Property (or Supremum Property).

• Definition: A set $S\subseteq \mathbb{R}$ is bounded above if there exists $M\in \mathbb{R}$ such that $s\le M$ for all $s\in S$.
• Axiom: Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in $\mathbb{R}$.

Consider the set $S=\{q\in \mathbb{Q}\mid q^2<2\}$. In $\mathbb{Q}$, this set is bounded above by $1.5$ or $2$, but it has no least upper bound because $\sqrt{2}\notin \mathbb{Q}$. In $\mathbb{R}$, $\sup (S)=\sqrt{2}$.

Construction 1: Dedekind Cuts

Richard Dedekind (1872) proposed a construction where real numbers are defined as partitions of the rational numbers.

Definition of a Dedekind Cut

A Dedekind Cut is a subset $L$ of $\mathbb{Q}$ satisfying:

1. Non-triviality: $L\ne \varnothing$ and $L\ne \mathbb{Q}$.
2. Downward Closure: If $p\in L$ and $q<p$, where $q\in \mathbb{Q}$, then $q\in L$.
3. No Maximum: For every $p\in L$, there exists $p^{\prime }\in L$ such that $p^{\prime }>p$.

The set of all such cuts defines $\mathbb{R}$. We identify each rational $r\in \mathbb{Q}$ with the cut $\{q\in \mathbb{Q}\mid q<r\}$. Irrational numbers are cuts that do not correspond to any rational. For instance, $\sqrt{2}$ is defined by the cut $\{q\in \mathbb{Q}\mid q<0\text{ or }{q}^2<2\}$. Addition and multiplication are defined set-theoretically on these cuts.

Construction 2: Cauchy Sequences

Georg Cantor and Charles Méray independently formulated $\mathbb{R}$ using the concept of completion.

Cauchy Sequences in $\mathbb{Q}$

Recall that a sequence $(a_n)$ is Cauchy if for every $ϵ>0$, there exists $N\in \mathbb{N}$ such that for all $n,m>N$: $|a_n-a_m|<ϵ$ In $\mathbb{Q}$, some Cauchy sequences do not converge to a rational number.

Equivalence Classes of Sequences

We define $\mathbb{R}$ as the set of all Cauchy sequences of rational numbers, under an equivalence relation $\sim$. Two sequences $(a_n)$ and $(b_n)$ are equivalent if: ${\lim }_{n\to \infty }(a_n-b_n)=0$ A real number is thus an equivalence class of Cauchy sequences. This formalizes the idea that a real number is “anything that a sequence of rationals can converge to.”

Algebraic and Topological Properties

An Ordered Field

$\mathbb{R}$ is a field under addition and multiplication, satisfying the standard field axioms (associativity, commutativity, inverses). It is an ordered field because there is a total ordering $\le$ consistent with the field operations:

1. If $a\le b$, then $a+c\le b+c$.
2. If $a\le b$ and $0\le c$, then $ac\le bc$.

The Archimedean Property

For any $x\in \mathbb{R}$, there exists an integer $n\in \mathbb{Z}$ such that $n>x$. This implies that the set of integers is not bounded above in $\mathbb{R}$, and that for any $ϵ>0$, there exists $n\in \mathbb{N}$ such that $1/n<ϵ$.

Density

Both $\mathbb{Q}$ (rational numbers) and $\mathbb{R}\setminus \mathbb{Q}$ (irrational numbers) are dense in $\mathbb{R}$. This means that between any two distinct real numbers $a<b$, there exists a rational $q$ and an irrational $i$ such that $a<q<b$ and $a<i<b$.

Topological Structure: The Metric Space

We view $\mathbb{R}$ as a metric space with the distance function $d(x,y)=|x-y|$.

1. Open Sets: A subset $U\subseteq \mathbb{R}$ is open if for every $x\in U$, there exists $ϵ>0$ such that $(x-ϵ,x+ϵ)\subseteq U$.
2. Compactness: A subset $K\subseteq \mathbb{R}$ is compact if and only if it is closed and bounded (Heine-Borel Theorem).
3. Connectedness: $\mathbb{R}$ is connected, meaning it cannot be partitioned into two disjoint non-empty open sets. This reflects the “no gaps” nature of the continuum.

Cardinality: The Uncountability of R

While the set of rationals $\mathbb{Q}$ is countable ($|\mathbb{Q}|={\aleph }_0$), the set of real numbers $\mathbb{R}$ is uncountable. By Cantor’s Diagonal Argument, it can be shown that there is no bijection between $\mathbb{N}$ and $\mathbb{R}$. The cardinality of $\mathbb{R}$ is denoted by $c=2^{{\aleph }_0}$.

Computational Considerations: The Gap Between Model and Machine

In pure mathematics, $\mathbb{R}$ consists of infinite precision numbers. In computer systems, we approximate $\mathbb{R}$ using Floating-Point Representation (IEEE 754).

# Demonstrating the limitations of machine precision in representing Real Numbers
a = 0.1
b = 0.2
print(f"Mathematical 0.1 + 0.2 = 0.3")
print(f"Machine Output: {a + b}")
# Output: 0.30000000000000004

This discrepancy arises because $0.1$ and $0.2$, while rational, have infinite repeating representations in binary, and must be truncated. Understanding the topology of $\mathbb{R}$ (specifically error propagation and limits) is critical for numerical analysis and scientific computing.

Exercise