Calculators and computers use Newton's Method to compute square roots. On this slide, we'll see how to compute $\sqrt{2}$.
Finding the square root of 2 is the same thing as solving $x^2 - 2 = 0$. So we set $f(x)=x^2-2$, $f'(x)=2x$, and apply the recursive formula $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{x_n^2-2}{2x_n}.$$ It only takes a few steps to get 10 or 20 decimal places.
| $x_n$ | $x$ | $f(x) = x^2-2$ | $f'(x)=2x$ | $x-\dfrac{f(x)}{f'(x)}$ | 1.4142135623731 |
| $x_1$ | 1 | -1 | 2 | $1 - \tfrac{-1}{2} = 3/2$ | $\underline{1}.5000000000000$ |
| $x_2$ | $\tfrac{3}{2}$ | $\tfrac{1}{4}$ | 3 | $\tfrac{3}{2} - \tfrac{1/4}{3}=\tfrac{17}{2}$ | $\underline{1.41}66666666667$ |
| $x_3$ | $\tfrac{17}{12}$ | $\tfrac{1}{144}$ | $\tfrac{17}{6}$ | $\tfrac{17}{12} - \tfrac{1/144}{17/6} = \tfrac{577}{408}$ | $\underline{1.41421}56862745$ |
| $x_4$ | $\tfrac{577}{408}$ | $\tfrac{1}{166464}$ | $\tfrac{577}{204}$ | $\tfrac{665857}{470832}$ | $\underline{1.41421356237}47$ |