How does that method work? Say you have a number which can be written as , where the are positive integers less than 100, that is, they are pairs of digits in . Start with and find the largest digit whose square is no greater than , that is, .
Now consider , that is, the number formed from the first two pairs of digits in . Its square root is a little more than (or equal to) where is the largest digit which, appended to , gives you a number whose square is no greater than . That is, . But and so . The number on the left is just the remainder from the previous step with the next pair of digits from appended, and the first term on the right is twice the result from the previous step with a new digit appended, multiplied by that new digit. Their difference gives our new remainder, .
Iterate that process and you get each digit of the square root.