Question

What is the reciprocal of $\dfrac{7}{9}$?

Answer 1

Hint: We first take the general forms and numbers to express the concept of reciprocal numbers. We then use mathematical forms to find the equations and relations. We place the value of $\dfrac{7}{9}$ and get its reciprocal value.

Complete step by step solution:
Let us take an arbitrary number $x$. The reciprocal of the number $x$ is $z$ then we have $xz=1$ which gives $z=\dfrac{1}{x}$.
The condition for the reciprocal to exist is that $x\ne 0$.
Using the previously discussed theorems, we now find the reciprocal of $\dfrac{7}{9}$.
Let the reciprocal number of $\dfrac{7}{9}$ be $a$.
The reciprocal condition gives us $\dfrac{7}{9}\times a=\dfrac{7a}{9}=1$ which gives the simplified form.
\[\begin{align}
  & \dfrac{7a}{9}=1 \\
 & \Rightarrow 7a=9 \\
 & \Rightarrow a=\dfrac{9}{7} \\
\end{align}\]

Therefore, the reciprocal of $\dfrac{7}{9}$ is \[\dfrac{9}{7}\].

Note: The opposite number is free of conditions. It is defined for every real number. But the reciprocal number has only one condition where the number is non-zero. 0 has no reciprocal number. The relation between a fraction and its reciprocal is the interchange of the denominator and the numerator value with each other.