0.1 Real Number Axioms
The real numbers \(\mathbb{R}\) are any decimal i.e. the set of both [[rational-numbers|rational numbers]] and [[rational-numbers|irrational numbers]]. The algebraic properties of \(\mathbb{R}\) are summarized in the following axioms, which state that \(\mathbb{R}\) is a field.
There exists binary operations \[ a,m:\mathbb{R} \times \mathbb{R} \to \mathbb{R}, \] written \(a(x,y)=x+y\) and \(m(x,y)=x \cdot y = xy\) and elements \(0,1 \in \mathbb{R}\) such that \(\forall x,y,z \in \mathbb{R}\): 1. \(x+0=x\) (existence of additive identity 0) 2. \(\forall x \in \mathbb{R}, \exists y \in R:x+y=0\) (existence of additive inverse) 3. \(x+(y+z)=(x+y)+z\) (addition is associative) 4. \(x+y=y+x\) (addition is commutative) 5. \(x 1=x\) (existence of multiplicative identity) 6. \(\forall x \in \mathbb{R} \setminus \{ 0 \}, \exists y \in \mathbb{R}:xy=1\) (existence of multiplicative inverse) 7. \(x(yz)=(xy)z\) (multiplication if associative) 8. \(xy=yx\) (multiplication is commutative) 9. \((x+y)z=xz+yz\) (multiplication is distributive over addition)
Axioms 1 \(\to\) 4 state that \(\mathbb{R}\) is a commutative group with respect to addition, 5 \(\to\) 8 state that \(\mathbb{R} \setminus \{ 0 \}\) is a commutative group with resect to multiplication and 9 that addition and multiplication are compatible.
0.2 Ordering Properties
The real numbers have a [[order|natural order relation]] that is compatible with their algebraic structure. We visualize the real numbers as the real line, with smaller numbers to the left and larger numbers to the right.
There is a strict linear order \(<\) on \(\mathbb{R}\) such that for all \(x,y,z \in \mathbb{R}\): 1. either \(x<y\), \(x=y\) or \(x>y\); 2. if \(x<y\) then \(x+z<y+z\); 3. if \(x<y\) and \(z>0\), then \(xz<yz\).
0.3 Open & Closed Intervals
For any \(a,b \in \mathbb{R}\) with \(a \leq b\) we define the open intervals: \[ \begin{align} (-\infty, b) & :=\{ x \in \mathbb{R} : x < b \} \\ (a,b) & := \{ x \in \mathbb{R}: a<x<b \} \\ (a,\infty) & := \{ x \in \mathbb{R}: a<x \}, \end{align} \] the closed intervals: \[ \begin{align} (-\infty,b] & := \{ x \in \mathbb{R}:x \leq b \} \\ [a,b] & := \{ x \in \mathbb{R}:a \leq x \leq b \} \\ [a,\infty) & := \{ x \in \mathbb{R} : a \leq x \}, \end{align} \] and the half-open intervals: \[ \begin{align} (a,b] & := \{ x \in \mathbb{R}:a<x\leq b\} \\ [a,b) & := \{ x \in \mathbb{R}:a \leq x < b \}. \\ \end{align} \]
All standard properties of inequalities follow from the previous axioms. [[archimedean-property]]
The set \(\mathbb{R}\) of real numbers is uncountable.
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