The real numbers

Ordered fields

What are real numbers? We use them every day and intuitively think of them as all possible “lengths”. But this notion is somewhat imprecise.

One could alternately think of real numbers as ones with a decimal expansion. This notion is precise, but some work has to be done to ensure that you can perform the basic operations (addition, multiplication, division).

As a result we’re going to take an axiomatic approach and define the real numbers to be a complete ordered field. Recall a field is a set with two binary operations $+$ and $\cdot$ that satisfy the Field axioms (commutativity, distributivity, etc.)

Definition 1. An ordered field is a field $F$, equipped with an ordering $<$, such that:

  1. For every $x, y \in F$, either $x = y$, (exclusive) or $x < y$, (exclusive) or $y < x$.
  2. If $x < y$ and $0 < c$, then $c x < cy$.
  3. If $x < y$ and $c \in F$ then $x + c < y + c$.

Using this we define $x > y$, if $y < x$, define $x \leq y$ if $x < y$ or $x = y$, and $x \geq y$ if $y < x$ or $x = y$.

Remark 2. From the above the standard properties you’re used to can all be deduced. For instance, try and prove $0 < 1$, or $x^2 \geq 0$ for any $x \in F$.

Infimum and Supremum

Definition 3. Let $F$ be an ordered field, and let $A \subseteq F$ be non-empty. We say $u$ is an upper bound of $A$ if for every $a \in A$, we have $a \leq u$. We say $A$ is bounded above if $A$ has an upper bound.

(The notions of lower bound and bounded below are defined similarly.)

Definition 4. We say $\beta \in F$ is the supremum of $A$ if:

  1. $\beta$ is an upper bound of $A$.
  2. If $u$ is an upper bound of $A$, then $\beta \leq u$.

Definition 5. We say $\alpha \in F$ is the infimum of $A$ if:

  1. $\alpha$ is an lower bound of $A$.
  2. If $\ell$ is an lower bound of $A$, then $\alpha \geq \ell$.

Proposition 6 (Uniqueness). If $\beta_1$, $\beta_2$ are two suprema of $A$, then $\beta_1 = \beta_2$.

Order complete fields

Definition 7. We say $F$ is a complete ordered field if every non-empty set which is bounded above has a supremum.

Example 8. $\Q$ and $Q(\sqrt{2}) = \set{ a + b \sqrt{2} \st a, b \in \Q}$ are ordered fields, but not complete ordered fields.

Example 9. $\mathbb{F}_2$ and $\C$ are not ordered fields. (There does not exist an order on these fields that satisfies the conditions in Definition 1.)

Proposition 10. If $F$ is a complete ordered field, every non-empty set which is bounded below has an infimum.

Theorem 11. There exists an order complete field which is unique up to isomorphism.

Proof sketch. For existence, use either Dedekind cuts, or equivalence classes of Cauchy sequences of rationals

For uniqueness, suppose $F, F’$ are two complete ordered fields. To build an isomorphism $\varphi \colon F \to F’$, set $\varphi(0) = 0’$, $\varphi(1) = 1’$, and extend this to all rationals using the axioms for addition / multiplication. Given $\beta \in \R$, define \begin{equation} \varphi(\beta) = \sup \set{\varphi(q) \st q \in \Q, q < \alpha} \,. \end{equation} Now we need to go through and check $\varphi \colon F \to F’$ is a bijection and satisfies the isomorphism conditions ($\varphi(a + b) = \varphi(a) + \varphi(b)$, $\varphi(ab) = \varphi(a) \varphi(b)$, and $a < b \implies \varphi(a) < \varphi(b)$. This is carried out in detail in most standard references for anyone that is curious.

Definition 12. Real numbers are a complete ordered field.

Problems

Problem 13. If $A \subseteq B$ are non-empty, then show that \begin{equation} \inf(B) \leq \inf(A) \leq \sup(A) \leq \sup(B) \,. \end{equation} If $B - A \neq \emptyset$ then must $\inf(B) < \inf(A)$ or $\sup(A) < \sup(B)$? Prove it, or find a counter example.

Problem 14. Suppose $A, B$ are non-empty sets such that for every $a \in A$, $b \in B$ we have $a \leq b$. Show that $\sup(A) \leq \inf(B)$.

Problem 15. Let $(X, d)$ be a metric space, and $A \subseteq X$ be non-empty. Define \begin{equation} d(x, A) = \inf_{a \in A} d(x, a) \,. \end{equation} Show that $d$ is continuous.

Problem 16. If $u$ is an upper bound of $A$, and $u \in A$ then show that $u = \sup(A)$. (If $\sup(A) \in A$ we call $\sup(A)$ the maximum of $A$.)

Problem 17.

  1. If $C \subseteq \R$ is a nonempty closed set which is bounded above, then $\sup(C) \in C$.
  2. If $U \subseteq \R$ is a nonempty open set which is bounded above, then $\sup(U) \not\in U$.

Problem 18. Show that $\sup\set{q \in \Q \st q^2 < 2} = \sqrt{2}$.

Properties of real numbers

At this level of abstractness we need to be careful on what we do / do not assume about arbitrary real numbers. We know $1$ (the multiplicative identity) is an element of $\R$. As a result we get $\N$, $\Z$, $\Q$ to all be subsets of $\R$. We don’t however, apriori know standard properties that we typically take for granted.

Proposition 19 (Archemedian priniciple). The set $\N$ is not bounded above.

Proposition 20. Any real number lies between two consecutive integers. That is, for any $x \in \R$, there exists a unique $n \in \Z$ such that $n \leq x < n+1$.

Remark 21. The unique $n \in \Z$ in Proposition 20 is called the floor of $x$, and denoted by $\floor{x}$.

Proposition 22 (Density of rationals). Between any two reals there is a rational number. That is, for any $x < y \in \R$ there exists $q \in \Q$ such that $x < q < y$.

Proposition 23 (Density of irrationals). Between any two reals there is an irrational number. That is, for any $x < y \in \R$ there exists $z \in \R-\Q$ such that $x < z < y$.

Proof sketch. Show first that $q \sqrt{2} \not\in \Q$ for every $q \in \Q - \set{0}$, and use Proposition 22.

Proposition 24 (Decimal expansions). Any real number has a decimal expansion. That is, for every $\alpha \in \R$ show that there exists unique numbers $a_0 \in \Z$, and $a_1$, $a_2$, … $\in \set{0, \dots, 9}$ such that for every $N \in \N$ we have \begin{equation} \sum_{n=0}^N \frac{a_n}{10^n} \leq \alpha < \frac{1}{10^N} + \sum_{n=0}^N \frac{a_n}{10^n} \,. \end{equation}

Proposition 25 (Uncountability of reals). The set $\R$ is uncountable (i.e. there does not exist a bijection between $\N$ and $\R$).