See [[Set-theoretic construction of the natural numbers]], See [[Set-theoretic construction of the integers]], See [[Constructing Reals]]. # Definitions ###### Real number axioms 1. $(\mathbb{R},+, \times)$ is a [[Field (Algebra)|field]], i.e. satisfies - (A1) if $x,y\in \mathbb{R}$ then $x+y \in \mathbb{R}$ (Closure under addition). - (A2) if $x,y,z \in \mathbb{R}$ then $(x+y)+z=x+(y+z)$ (Associativity of addition). - (A3) if $x,y \in \mathbb{R}$ then $x+y=y+x$ (Commutativity of addition). - (A4) there exists $0\in \mathbb{R}$, such that for any $x\in \mathbb{R}$, $x+ 0 = 0+x=x$ (existence of an additive identity). - (A5) if $x \in \mathbb{R}$ there exists $-x \in \mathbb{R}$ such that $x+(-x)=(-x)+x=0$ (existence of an additive inverses). - (M1) if $x,y\in \mathbb{R}$ then $xy\in \mathbb{R}$ (**closure** under multiplication). - (M2) if $x,y,z \in \mathbb{R}$ then $(xy)z=x(yz)$ (**associativity** of multiplication). - (M3) if $x,y\in \mathbb{R}$ then $xy=yx$ (**commutativity** of multiplication). - (M4) there exists $1 \in \mathbb{R}$, such that for any $x \in \mathbb{R}$, $1\times x= x\times 1=x$ (existence of a multiplicative **identity**). - (M5) for $x \in \mathbb{R}\setminus \{ 0 \}$ there exists $x^{-1} \in \mathbb{R}$ such that $x \times x^{-1} = x^{-1} \times x=1$ (existence of multiplicative **inverses**). - (D) for all $x, y, z \in \mathbb{R}$, $- x(y + z) = xy + xz$ (distributivity of multiplication over addition). 2. that is totally ordered by $\leq$: - (O1) if $x,y\in\mathbb{R}$ exactly one of the statements $x<y,x=y,x>y$ is true ([[Trichotomy]]). - (O2) if $x,y,z \in \mathbb{R}$, $x<y$ and $y<z$ then $y<z,x<z$ ([[Transitive Relation|transitivity]]). - (O3) if $x,y,z\in \mathbb{R}$ and $x<y$, then $x+z<y+z$. - (O4) if $x,y,z \in \mathbb{R}$, and $x<y$, and $z>0$, then $xz<yz$. 3. and satisfies the least upper bound axiom: any non-empty subset of $\mathbb{R}$ that is bounded above has a [[Infimum and supremum|least upper bound]]. Note that the least upper bound axiom is the only listed property that differentiates the reals distinct from the rationals. ###### Cantor's construction See [[Cantor's construction of real numbers]]. Some authors have also suggested that it suffices to use conditionally convergent series of rational numbers since the [[Riemann rearrangement theorem]] means that such a series can be rearranged to converge to any real number. # Properties **Algebra** See [[Multiplying inequalities]]. (2) [[Real Power of Real Number]]. ###### Completeness The [[Archimedean Property of Real Numbers|Archimedean property]] of the reals is that for any real number, there is a [[Natural Numbers|natural number]] greater than it. The set of rational numbers is a [[Existence of a rational in any closed real interval|dense subset]] of $\mathbb{R}$ and it follows that [[Between two Different Real Numbers exists an Irrational Number|a real number lies between any two non-equal reals]]. See [[Bolzano-Weierstrass theorem]]. It follows from this that closed and bounded subsets of $\mathbb{R}$ are sequentially compact. We can use BW to show Cauchy sequences of real numbers converge (see [[Completeness of real numbers]]). The proof use the fact that Cauchy sequences are bounded, i.e. lie in a sequentially compact subset of $\mathbb{R}$. More generally, Cauchy sequences in compact metric spaces converge (see [[Compact metric spaces are complete]]). ###### Convergence See [[Real sequences]].