Does the following approximate equality hold accurately? Is the approximate value of $\sqrt{7}$ obtained from this related to any of the approximate values in the continued fraction expansion of $\sqrt{7}$?

$$
84 \sqrt{7} \approx 222 \frac{2}{2+7}
$$[^0]


[^0]:    ${ }^{1}$ The necessary knowledge about continued fractions can be found in problem 1283, K. M. L. 29 (October 1964), p. 67.
The right-hand side fraction is a pure repeating decimal: $222,222 \ldots$ The left side, however, is $\sqrt{7 \cdot 84^{2}}=\sqrt{49392}=$ $222,243 \ldots$.., the approximate equality (1) holds to 1 decimal place, that is, to the fourth significant digit.

The right side can also be expressed as $2000 / 9$, thus

$$
\sqrt{7} \approx \frac{2000}{9 \cdot 84}=\frac{500}{189}
$$

The continued fraction expansion, following the pattern of the recommended examples:

$$
\begin{aligned}
& \text{For } 21 \text{, it is possible to form so-called side fractions by substituting the integers } 1,2, \ldots, q_{k}-1 \text{ for } q_{k}. \text{ These, according to (3), give approximate fractions with numerators smaller than } S_{k} \text{ and denominators smaller than } N_{k}, \text{ but - as the following examples show - they approximate the root less well, as substituting 1, 2, 3 for the second 4 yields:} \\
& \frac{127 \cdot 1+82}{48 \cdot 1+31}=\frac{209}{79}=2,64556 \ldots \\
& \frac{127 \cdot 2+82}{48 \cdot 2+31}=\frac{336}{127}=2,64566 \ldots \\
& \frac{127 \cdot 3+82}{48 \cdot 3+31}=\frac{463}{175}=2,64571 \ldots
\end{aligned}
$$

The right-hand side of (2) does not belong to any of these - and even the first one approximates better than the others.