Let's determine - without using decimal approximations of fractions - which of the following numbers is larger:

$$
\sqrt{11}-\sqrt{5}, \quad \sqrt{19}-\sqrt{11}
$$
I. solution. Determine the sign of the difference $d$ between the first and second numbers, $2 \sqrt{11}-(\sqrt{5}+\sqrt{19})$. Of two positive numbers, the larger is the one whose square is larger. The square of the minuend is 44, and the square of the subtrahend is $24+\sqrt{380}$. Here $\sqrt{380}\frac{43+33}{44+30}=1+\frac{1}{37}>1
\end{gathered}
$

thus the first number is larger. In the fourth transformation, we replaced the square root in the numerator with the immediately smaller integer, and the square root in the denominator with the immediately larger integer, reducing the value of the fraction both times, and still obtained a number greater than 1.

Béla Vizvári (Budapest, Berzsenyi D. g. II. o. t.)