عدد گویا - ویکی‌پدیا، دانشنامهٔ آزاد

${\displaystyle \mathbb {Q} \cup \mathbb {Q} ^{c}=\mathbb {R} }$
${\displaystyle \mathbb {Q} \cap \mathbb {Q} ^{c}=\emptyset }$

${\displaystyle {\frac {a}{b}}={\frac {a\times d}{b\times d}},{\frac {c}{d}}={\frac {c\times b}{d\times b}}}$

${\displaystyle {\frac {3}{7}}={\frac {3\times 11}{7\times 11}}={\frac {33}{77}}\;,\;{\frac {5}{11}}={\frac {5\times 7}{11\times 7}}={\frac {35}{77}}\Rightarrow 33<35\Rightarrow {\frac {3}{7}}<{\frac {5}{11}}}$

${\displaystyle {\frac {a}{b}}\pm {\frac {c}{d}}={\frac {ad\pm bc}{bd}}}$

${\displaystyle {\frac {4}{3}}+{\frac {2}{5}}={\frac {20+6}{15}}={\frac {26}{15}}}$

${\displaystyle {\frac {5}{6}}-{\frac {5}{4}}={\frac {20-30}{24}}={\frac {-10}{24}}}$

${\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}}$

${\displaystyle {\frac {a}{b}}\div {\frac {c}{d}}={\frac {a}{b}}.{\frac {d}{c}}={\frac {ad}{bc}}}$

${\displaystyle {\frac {5}{7}}\div {\frac {3}{5}}={\frac {5}{7}}\cdot {\frac {5}{3}}={\frac {5\times 5}{7\times 3}}={\frac {25}{21}}}$

${\displaystyle {\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}={\frac {a}{-b}}}}$

${\displaystyle {\displaystyle -\left({\frac {7}{12}}\right)={\frac {-7}{12}}={\frac {7}{-12}}}}$

${\displaystyle {\displaystyle \left({\frac {x}{y}}\right)^{-n}=({\frac {y}{x}})^{n}}}$

${\displaystyle {\displaystyle \left({\frac {3}{5}}\right)^{-2}=({\frac {5}{3}})^{2}}}$

${\displaystyle {\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}}}$

${\displaystyle {\displaystyle \left({\frac {10}{17}}\right)^{3}={\frac {10^{3}}{17^{3}}}}}$

${\displaystyle {\displaystyle \left({\frac {x}{y}}\right)^{0}=1}}$

${\displaystyle {\displaystyle \left({\frac {13}{-5}}\right)^{0}=1}}$