522. By what factor should the side of a square be increased to double its area?
$\triangle$ When the side of a square is increased by $k$ times, its area increases by $k^{2}$ times: if the side was $a$ and the area $a^{2}$ before, now the side is $k a$ and the area $(k a)^{2}=k^{2} a^{2}$. Therefore, we need to find such a $k$ that $k^{2}$ equals 2. This number (positive) is called the square root of 2 and is denoted by $\sqrt{2}$. (On many calculators, there is a corresponding key; by pressing it, one can verify that $\sqrt{2}=1,4142 \ldots) \triangleleft$

![](https://cdn.mathpix.com/cropped/2024_05_21_90703b5d5e76e3b5cd3dg-159.jpg?height=129&width=120&top_left_y=2037&top_left_x=477)

On the other hand, doubling the area of a square is easily done geometrically. Let's draw a square on graph paper along the diagonal (see figure).