Answer
$3,\dfrac{1}{3} $ or $-3,-\dfrac{1}{3}$
Work Step by Step
Let $x$ be the number. Then its reciprocal is $\dfrac{1}{x}.$
The conditions of the problem translate to the equation
\begin{array}{l}\require{cancel}
x\div\left( 9\cdot\dfrac{1}{x} \right)=1
.\end{array}
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
x\div\dfrac{9}{x}=1
\\\\
x\cdot\dfrac{x}{9}=1
\\\\
\dfrac{x^2}{9}=1
\\\\
9\left( \dfrac{x^2}{9} \right)=(1)9
\\\\
x^2=9
.\end{array}
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{array}{l}\require{cancel}
x=\pm\sqrt{9}
\\\\
x=\pm3
.\end{array}
Hence, the numbers is the pair $
3,\dfrac{1}{3} $ or $-3,-\dfrac{1}{3}$.
