The standard square root algorithm is based on this left to right guess and check method. The steps are just reversed. Consider
First notice that the number of places in the square is twice the number of places in the number being squared. As a result, starting at the decimal point and going both ways, group the digits in pairs. In this case, there is only one digit to the left of the decimal point, but the 0s go on forever to the right.
The biggest digit whose square is less than 3 is 1. Instead of adding the square of 1 as we did in the guess and check processes, we reverse the procedure and subtract
We put the 1, which is the first digit in the answer on top of the radical, and when we subtract, we bring down the next two digits.
Double the number on top, and write the result off to the side.
The next digit in the answer is the largest digit whose product with the number one gets when one multiplies it times the number one gets when one puts it after the number on the left is smaller than the remainder. In this case it is 7 because
Notice that if we add 1 to these numbers we get
which are the same numbers as one gets in the guess and check procedure.
Subtract the .7 times 2.7 from the remainder
Repeat the process. Double the number on top, and write it off the the left.
The next digit will be a 3. Put a 3 after the 1.7 on top and after the 3.4 on the left,
and subtract .03 x 3.43 from the remainder
Repeat the process. Double the number on top, and write it off to the left.
By dividing the number on the left into the remainder we can tell that the next digit would be a 2. Put the 2 after the 1.73 on top and after the 3.46 on the left.
Note that if we add the numbers which were subtracted, we get
and we get the same numbers we see in the guess and check procedure.
Usually when one uses this algorithm, since the decimal places will take care of themselves, one does not copy down the decimal points or the 0s between the decimal points and the first nonzero digit on the left in the subtraction steps or their remainders. The procedure looks like
Notice that the numbers in green on the left are the same as the digits in the answer, and that the numbers in red are obtained by doubling the number to the left of the green digit in the answer which corresponds to the green digit on the left.
Standard Guess and Check Method
Left to Right Guess and Check Method