Example 11. If the decimal part of $\sqrt{11}$ is $a$, and the decimal part of $\frac{1}{a}$ is $b$, then $b=(\quad$.
(1994, Hope Cup Mathematics Competition)
Since $9<11<16$, then $3<\sqrt{11}<4$.
Let $a=\sqrt{11}-3$,
$$
\begin{array}{l}
\frac{1}{a}=\frac{1}{\sqrt{11}-3}=\frac{\sqrt{11}+3}{2} . \\
\text { Also } b<\sqrt{11}+3<7,3<\frac{\sqrt{11}+3}{2}<3.5,
\end{array}
$$

Therefore, $b=\frac{\sqrt{11}+3}{2}-3=\frac{\sqrt{11}-3}{2}$.