Suppose we want to find the square root of 3. The square root of 3 is a number whose square is 3. The most obvious way to find square roots is by trial and error.
12 is too small, and 22 is too big, so the square root of 3 is between 1 ad 2. By trial and error we see that
We continue using trial and error to discover that
Using trial and error to find the digit in the next decimal place, we find.
An unfortunate feature of this process is that the work that the algorithm we use for computing squares is a right to left algorithm. As a result, we do not get to use the work that was done in previous steps does not appear in the subsequent steps. We have to start from scratch at every step.
If we had a left to right algorithm, we could perhaps use the work from previous computations when we try to extend the accuracy of our approximation to the next place.