Question

What is the opposite and reciprocal of 9?

Answer 1

Hint: We first take the general forms and numbers to express the concept of opposite and reciprocal numbers. We then use mathematical forms to find the equations and relations. We place the value of 9 and get its opposite and reciprocal value.

Complete step by step solution:
Let us take an arbitrary number $x$. The opposite of the number $x$ is $y$ then we have $x+y=0$ which gives $y=-x$.
Similarly, the reciprocal of the number $x$ is $z$ then we have $xz=1$ which gives $z={\displaystyle \frac{1}{x}}$.
The condition for the reciprocal to exist is that $x\ne 0$.
Using the previously discussed theorems, we now find the opposite and reciprocal of 9.
Let the opposite and reciprocal numbers of 9 are $a,b$ respectively.
Therefore, the condition gives us $a+9=0$ which gives the simplified from. We subtract 9 from both sides and get
$\begin{array}{rl}& a+9=0\\ & \Rightarrow a+9-9=0-9\\ & \Rightarrow a=-9\end{array}$
Similarly, the reciprocal condition gives us $9\times b=9b=1$ which gives the simplified from. We divide 9 from both sides and get
$$\begin{array}{rl}& 9b=1\\ & \Rightarrow {\displaystyle \frac{9b}{9}}={\displaystyle \frac{1}{9}}\\ & \Rightarrow b={\displaystyle \frac{1}{9}}\end{array}$$

Therefore, the opposite and reciprocal of 9 are $$-9,{\displaystyle \frac{1}{9}}$$ respectively.

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.