The real numbers

The real numbers $\mathbb{R}$ have three defining properties: they are a field with defined operations of addition, scalar multiplication, and non-zero division; they are ordered; and they have no “missing points.” Informally, these properties make the real numbers visualizable as a line.

Formally, $\mathbb{R}$ is the unique complete ordered field. As an upshot of the uniqueness of $\mathbb{R}$, every real number has a unique decimal representation.

Related: Complex numbers, conjugates, and absolute value

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Formal definition of the reals §

Definitions and notation from Modern Analysis I.

Definition: Real numbers

The field of real numbers is a tuple $(\mathbb{R},+,\cdot ,\le )$ consisting of a set $\mathbb{R}$, the maps $+,\cdot :\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ defined by

$$+(x,y):=x+y\cdot (x,y)=xy$$

respectively, and a order relation $\le$ satisfying the following:

• (A1), (M1) Existence of a neutral element. See (F4) Identities.
• (A2), (M2) Existence of inverses. See (F5) Additive inverses and (F6) Multiplicative inverses.
• (A3), (M3) Associativity. See (F1) Associativity.
• (A4), (M4) Commutativity. See (F2) Commutativity.
• (D) Distributivity of addition over multiplication. For all $x,y,z\in \mathbb{R}$, we have $(x+y)\cdot z=x\cdot z+y\cdot z$.

Moreover, the reals satisfy the order relation axioms—(O1) reflexivity, (O2) antisymmetry, (O3) transitivity, and (O4) linearity/trichotomy—as well as the compatibility axioms:

• (OC1) Compatibility of $+$ and $\le$. For all $x,y,z\in \mathbb{R}$, $x\le y⟹x+z\le y+z$.
• (OC2) Compatibility of $\cdot$ and $\le$. For all $x,y\in \mathbb{R}$, $0\le x\wedge 0\le y⟹0\le x\cdot y$.

Finally, the reals uniquely satisfy the completeness axiom.

Axiom: Completeness of \mathbb R

Let $X,Y\subseteq \mathbb{R}$ such that $x\le y$ for all $x\in X$ and $y\in Y$. Then for all $x\in X$ and $y\in Y$, there exists $c\in \mathbb{R}$ such that $x\le c\le y$.

This can be restated in terms of the least upper bound, or supremum.

Theorem: Completeness of \mathbb R using the least upper bound (supremum)

Let $X\subseteq \mathbb{R}$ be a non-empty subset that is bounded above, meaning there exists $z\in \mathbb{R}$ such that $x\le z$ for all $x\in X$. Then the supremum of $X$ exists.

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Review §

• Prove the theorem about completeness of $\mathbb{R}$. (Hint: set the supremum to be $c$.)