> [!NOTE] Real Number Axioms >For simplicity, we assume that real numbers and their operations satisfy the following axioms > > **Addition** > (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 $x,y \in \mathbb{Z}$ then $x+y=y+x$ ([[Commutativity]] of addition). > (A4) if $x \in \mathbb{R}$ there exists $0\in \mathbb{R}$ such that for any $x\in \mathbb{R}$, $x+ 0 = 0+x=x$ (existence of an additive [[Identity element of a binary operation|identity]]). > (A5) if $x \in \mathbb{R}$ there exists $-x \in \mathbb{R}$ such that $x+(-x)=(-x)+x=0$ (existence of an additive [[Inverse under a binary operation|inverse]]). > > **Multiplication** > (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) if $x \in \mathbb{R}/\{ 0 \}$ there exists $x^{-1} \in \mathbb{R}$ such that $x \times x^{-1} = x^{-1} \times x=1$ (existence of multiplicative **inverses**). > > **Distribuitivity** > (D) [[Distributivity]] of multiplication over addition. > > **Order** > (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 $xz<yz$. ^2d1ba9 **Note**: (A1)-(D) can be summarised by noting that the set real numbers forms a [[Field (Algebra)|field]]. # Properties - [[Multiplying inequalities]].