Example 1 If you don't have a square root table, nor a calculator, what simple method can be used to find the approximate value of $\sqrt{11}$? Of course, you can use the usual method for finding square roots, but the method below seems more convenient.
Repeating this process yields

One would immediately think of removing the irrational numbers $(\sqrt{11}-3)$ or $(\sqrt{11}-3) / 2$ from equations (2), (3), (4), (5), (6), (7), (8) to obtain the "fractional" values as approximations of $\sqrt{11}$. It is easy to calculate that these "approximations" are sequentially
$$\sqrt{11} \approx 3, \frac{10}{3}, \frac{63}{19}, \frac{199}{60}, \frac{1257}{379}, \frac{3970}{1197}, \frac{25077}{7561}$$

Expressed in decimal form, these "approximations" are (to eight decimal places)
$$\begin{aligned}
\sqrt{11} \approx & 3,3.33333333,3.31578947,3.31666666 \\
& 3.31662269,3.31662489,3.31662478
\end{aligned}$$

These are indeed very precise approximations, as in fact
$$\sqrt{11}=3.31662479 \cdots$$$$\begin{array}{l}
3<\sqrt{11}<4, \\
\sqrt{11}=3+(\sqrt{11}-3) \\
=3+\frac{1}{(\sqrt{11}+3) / 2}=3+\frac{1}{3+(\sqrt{11}-3) / 2} \\
=3+\frac{1}{3+\frac{1}{\sqrt{11}+3}}=3+\frac{1}{3+\frac{1}{6+(\sqrt{11}-3)}} .
\end{array}$$
$$\begin{array}{l}
\sqrt{11}=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+(\sqrt{11}-3) / 2}}} \\
=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+\frac{1}{6+(\sqrt{11}-3)}}}} \\
=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+\frac{1}{6+\frac{1}{3+(\sqrt{11}-3) / 2}}}}} \\
=3+\frac{1}{3+\frac{1}{6+\frac{1}{3+\frac{1}{6+\frac{1}{3+\frac{1}{6+(\sqrt{11}-3)}}}}}}
\end{array}$$