Question

What is the opposite and reciprocal of $2\dfrac{1}{2}$ ?

Answer 1

Hint: The opposite of a number is the number which when added to the number gives the sum of zero. And the reciprocal of a number is the number which is obtained by dividing 1 with the number itself. Also, in our problem we will first convert the mixed fraction into an improper fraction as it would make calculation easier.

Complete step-by-step answer:
The number has been given to us in the form of a whole number. So, we will first of all convert it into an improper function. This can be done as follows:
$\begin{align}
  & \Rightarrow 2\dfrac{1}{2}=\dfrac{2\times 2+1}{2} \\
 & \therefore 2\dfrac{1}{2}=\dfrac{5}{2} \\
\end{align}$

Thus, the improper fraction obtained is equal to $\dfrac{5}{2}$. Now, let this improper fraction be denoted by ‘x’. Also, let the opposite (or, the additive inverse) of this improper fraction be ‘y’ and its reciprocal be ‘y’. Then, calculating the additive inverse first of all, we get:
$\begin{align}
  & \Rightarrow y+x=0 \\
 & \Rightarrow y+\dfrac{5}{2}=0 \\
 & \therefore y=-\dfrac{5}{2} \\
\end{align}$

Now, calculating the reciprocal of our improper fraction by diving 1 with the fraction itself, we get the following equation:
$\Rightarrow z=1\div x$
$\Rightarrow z=1\div \dfrac{5}{2}$
$\Rightarrow z=\dfrac{1}{\dfrac{5}{2}}$
$\therefore z=\dfrac{2}{5}$

Thus, the opposite of our improper fraction comes out to be $-\dfrac{5}{2}$. And, the reciprocal of our improper fraction comes out to be $\dfrac{2}{5}$.
Hence, the opposite and reciprocal of $2\dfrac{1}{2}$comes out to be $-\dfrac{5}{2}$ and $\dfrac{2}{5}$ respectively.

Note: Zero is the only number (among real and complex numbers) which is the additive inverse of itself. Also, it is the only number whose reciprocal does not exist as one upon absolute zero has no meaning. These are some of the key facts about opposite and reciprocal numbers that should be known to everyone.