### Theory:

The reciprocal of a number is one divided by that number. Formally the reciprocal of $$x =$$ $\frac{1}{x}$.
So, for example, the reciprocal of $$5$$ is one quarter: The reciprocal of $$5$$ is $\frac{1}{5}$.
If $\frac{1}{8}$ is number then, the reciprocal of $\frac{1}{8}$ is $$8$$.

Important!
Rational Number: A number can be made by dividing two integers.
• An integer is a number with no fractional part.
Example:
0.5 is a rational number because 0.5 $$=$$  $\frac{3}{2}$  (3 and 2 are both integers.)
When $\frac{15}{8}$ is a rational number then, the reciprocal of $\frac{15}{8}$ is $\frac{8}{15}$, to get the product as $$1$$?

Thus, $\frac{15}{8}$ $$×$$ $\frac{8}{15}$ $$=$$ $$1$$.

Similarly, $-\frac{4}{9}$ must be multiplied by $-\frac{9}{4}$ so as to get the product as $$1$$.

We state that $\frac{15}{8}$ is the reciprocal of $\frac{8}{15}$ and $-\frac{4}{9}$ is the reciprocal of $-\frac{9}{4}$.

Can you guess what is the reciprocal of $$0$$?

Important!
Is there a rational number which when multiplied by $$0$$ give $$1$$? Thus, zero has no reciprocal.
We state that a rational number $\frac{c}{d}$ is called the reciprocal or multiplicative inverse of another non-zero rational number $\frac{a}{b}$ if $\frac{a}{b}$ $$×$$ $\frac{c}{d}$ $$=$$ $$1$$.