I discovered that we can use Newton's method to conveniently compute the square root of a positive integer.
Newton's method stipulates that we keep iterating as such:
$$ x_{n + 1} = x_n - \frac{f(x_n)}{f'(x_n)} $$
However, apparently, we can just compute the square root by keep iterating on
$$ x = \frac{1}{2} \left( x + \frac{a}{x} \right) $$
where $a$ is the number we are finding the square root of and $x$ is our eventual answer.
I don't understand how this method for computing square roots is a manifestation of Newton's method though.
Let $f(x) = x^2 - a$. Then
$$x - \frac{f(x)}{f'(x)} = x - \frac{x^2-a}{2x} = \frac{2x^2 - (x^2-a)}{2x} = \frac{x^2+a}{2x} = \frac{1}{2}\left(x + \frac{a}{x}\right).$$