### Теорія:

Два числа, добуток яких дорівнює $$1$$, називають взаємно оберненими.

Взаємно оберненими є числа:
$\begin{array}{l}\frac{5}{9}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}і\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{9}{5}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{5}{9}\cdot \frac{9}{5}=\frac{{}^{1}\overline{)5}\cdot {\overline{)9}}^{1}}{{}_{1}\overline{)9}\cdot {\overline{)5}}_{1}}=\phantom{\rule{0.147em}{0ex}}1\\ \\ \frac{17}{3}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}і\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{3}{17}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{17}{3}\cdot \frac{3}{17}=\frac{{}^{1}\overline{)17}\cdot {\overline{)3}}^{1}}{{}_{1}\overline{)3}\cdot {\overline{)17}}_{1}}=\phantom{\rule{0.147em}{0ex}}1\\ \\ \frac{1}{13}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}і\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}13\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{1}{13}\cdot 13=\frac{1\cdot 13}{13}=\frac{13}{13}=\phantom{\rule{0.147em}{0ex}}1\end{array}$
Числу $\frac{c}{d}$, де $c\ne 0\phantom{\rule{0.147em}{0ex}}$ і $d\ne 0\phantom{\rule{0.147em}{0ex}}$   взаємно оберненим є число $\frac{d}{c}$.
Щоб знайти число, взаємно обернене  мішаному, мішане число записують у вигляді неправильного дробу.

$8\frac{1}{3}=\frac{8\cdot 3+1}{3}=\frac{25}{3}$
Взаємно оберненим $8\frac{1}{3}$ буде число $\frac{3}{25}$.
Число , обернене до 1 , є само число 1.
Числу 0 оберненого числа не існує.