Real number
Set
| definiendum | $ r \in \mathbb R $ |
| for all | $x,y\in \mathbb Q$ |
| for all | $ r\subset \mathbb Q, r\neq \emptyset $ |
| postulate | $ y\in r\implies x\in r $ |
| postulate | $ \neg\ \exists (b\in r).\ \forall (a\in r).\ a<_{\mathbb Q}b $ |
Discussion
Remark: We distinguish between “$\subset$” and “$\subseteq$”, i.e. the above definition implies $ r\neq \mathbb Q $.
Each real number is modeled as a Dedekind cut of $ \mathbb Q $ into two pieces. The axioms above say that $r$ is a $<_{\mathbb Q}$-complete set with no upper bound. We define the total order of the reals via $s<r\equiv s\subset r$, see Order structure of real numbers.
If the complement of such a subset in $\mathbb Q$ has a lowest element, it corresponds to just that rational number.
One can define the real numbers in several other different ways, e.g. via Cauchy sequences w.r.t. to rational numbers, and it's worth stating that it's not a priori clear that the Dedekind cuts enable us to capture the full wealth of the reals - but that do.
As an example, consider that there is no solution $x\in \mathbb Q$ for equation $x^2=2$, but each such formula can be used to define a set which is taken to be it's solution. More explicitly, we represent the solution, denoted $\sqrt{2}$, by the set of rational numbers, which fulfill $x^2<2$:
$\sqrt{2}\equiv\{x\ |\ x\in\mathbb Q\ \land\ (x<0\ \lor\ x^2_{\mathbb Q}2)\}$.
The arithmetic structure of real numbers is defined in terms of the Arithmetic structure of rational numbers and it's worth noting that it's not a priori obvious that e.g. the above $\sqrt{2}$ fulfills $\sqrt{2}\cdot \sqrt{2}=2$.
Theorems
Any natural number $n$ can be written as a unique finite sequence of digits. Any integer $k$ can be written as $k=\mathrm{sign}(k)\cdot|k|$, where $|k|$ is a unique natural number. Any rational number $q$ can be written as $q=k/n$, where $k$ is an integer and $n$ is a natural number. For $q\neq 0$ we can determine a unique $k$ and $n$ by demanding |k| to be minimal (namely be canceling common factors of given $k$ and $n$). Any real number can be written as $r=k\cdot s$, where $k$ is an integer and $s\in(0,1]$ can be written as an infinite sequence $(a_i)_{i\in\mathbb N}$ of digits. For $r\neq 0$ we can determine a unique $k$ and $s$ by demanding |k| to be minimal (namely $k=\mathrm{sign}(r)\bigl\lceil \strut|r|\bigr\rceil$ and $s=|r|/\bigl\lceil \strut|r|\bigr\rceil$). Then, also, each number is $r=k+\sum_{i\in\mathbb N}a_i\cdot \mathrm{b}^i$, where $\mathrm{b}$ is the base.
Reference
Wikipedia: Real number, Construction of the real numbers, Dedekind cut
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