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| - | ===== Non-negative rational number ===== | ||
| - | ==== Definition ==== | ||
| - | ^ $ \mathbb Q_+ $ ^ | ||
| - | | $ 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 === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Rational_number|Rational number]] | ||
| - | |||
| - | Mizar: [[http://mizar.org/JFM/pdf/arytm_3.pdf|ARYTM_3]] | ||
| - | |||
| - | ==== Context ==== | ||
| - | Set | ||
| - | === Subset of === | ||
| - | [[Rational number]], [[Non-negative real number]] | ||
