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Non-negative rational number
Definition
| $ n,m\in \mathbb N $ |
| $ m>1 $ |
| $ \mathbb N \cup \{r\ |\ \exists n,m\ (\ r=\langle n,m\rangle \land \mathrm{gcd}(n,m)\neq 1\ )\} $ |
Ramifications
Definitions
| $ a,b,c,x\in \mathbb N $ |
| $b\neq 0$ |
| $ r,s,t \in \mathbb Q_+ $ |
| $ a\div b:=x,\hspace{1cm}(c<b)\land (a=x\cdot b+c)$ |
| $ \mathrm{red}(a,b):=a\div\mathrm{gcd}(a,b)$ |
| $ \frac{a}{b} \equiv \begin{cases} \mathrm{red}(a,b) & \mathrm{if}\ \mathrm{red}(b,a)=1\\\\ \langle\mathrm{red}(a,b),\mathrm{red}(b,a)\rangle & \mathrm{else} \end{cases}$ |
| $ \mathrm{num}\frac{a}{b}\equiv a $ |
| $ \mathrm{den}\frac{a}{b}\equiv b $ |
| $ r+s=\frac{\mathrm{num} r\ \cdot\ \mathrm{den} s\ +\ \mathrm{num} s\ \cdot\ \mathrm{den} r}{\mathrm{den} r\ \cdot\ \mathrm{den} s}$ |
| $ r\cdot s=\frac{\mathrm{num} r\ \cdot\ \mathrm{num} s}{\mathrm{den} r\ \cdot\ \mathrm{den} s}$ |
Predicates
| $r\leq s\equiv \exists t\ (s=r+t)$ |
| $r < s\equiv s\leq r$ |
Reference
Context
Subset of
