For this problem, it is useful to know the following square root algorithm that allows for digit-by-digit extraction of \sqrt{x}$and gives one decimal place of \sqrt{x}$ for each two decimal places of $x$. We will illustrate how to extract the second digit after the decimal point of \sqrt{\pi}$, knowing that \pi=3.1415 \cdots$ and \sqrt{\pi}=1.7 \cdots$. Let$d$be the next decimal digit. Then$d$should be the largest digit such that$(1.7+0.01 d)^{2}<\pi$, which in this case we will treat as$(1.7+0.01 d)^{2}<3.1415$. Expanding this, we get$2.89+0.034 d+0.0001 d^{2}<$3.1415, from which we get the value of$d$to be approximately \left\lfloor\frac{3.1415-2.89}{0.034}\right\rfloor=\left\lfloor\frac{0.2515}{0.034}\right\rfloor=7$, since the $0.0001 d^{2}$ term is negligible. Indeed, 7 is the largest such digit, and so $d=7$ is the second digit of \sqrt{\pi}$. Because we are constantly subtracting the square of our extracted answer so far, we can record the difference in a manner similar to long division, which yields a quick method of extracting square roots by hand.