### Numbers: Fractions

### Reciprocal of a fraction

Reciprocal of a fraction

If we swap the numerator and the denominator in the fraction #\tfrac{2}{3}#, we get #\tfrac{3}{2}#. We now see that: \[\tfrac{2}{3} \times \tfrac{3}{2} =\tfrac{6}{6} = 1\]

In general it holds that:

*Two numbers are each other's reciprocal (also called inverse) if their product is #1#.*

**Examples**

\begin{array}{rcrcr}\tfrac{3}{5} &\times& \tfrac{5}{3} &=& 1\\\tfrac{1}{10} &\times& 10 &=& 1\\-\tfrac{4}{3} &\times& -\tfrac{3}{4} &=& 1\end{array}

#{{19}\over{5}}#

If we swap the numerator and the denominator of the fraction #{{5}\over{19}}#, we find #{{19}\over{5}}#. To double-check, we multiply the numbers and check if the product equals #1#.

\[{{5}\over{19}} \times {{19}\over{5}}=1\]

Therefore, the reciprocal of #{{5}\over{19}}# equals #{{19}\over{5}}#.

If we swap the numerator and the denominator of the fraction #{{5}\over{19}}#, we find #{{19}\over{5}}#. To double-check, we multiply the numbers and check if the product equals #1#.

\[{{5}\over{19}} \times {{19}\over{5}}=1\]

Therefore, the reciprocal of #{{5}\over{19}}# equals #{{19}\over{5}}#.

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