Example with Perfect Square
Example with Imperfect Square

Calculate the square root of 2.

Step 1: Divide the number into pairs of digits, starting from the right. If there is an odd number of digits, the leftmost digit will be by itself.

Step 2: Find the largest digit that will go into the leftmost group of digits that many times, and put that as the quotient. In this case, 1 will go into 2 one time. One is the largest digit, because 2 will not go into 2 two times.

Step 3: Multiply the divisor by the largest digit, and subtract the product from the leftmost group of digits. In this case, multiply 1 by 1, and subtract 1 from 2 for a remainder of 1.

Step 4: Bring down the next group of digits to form the new dividend, and double the quotient to form the new divisor. If there are no more digits, but the remainder is not 0, bring down two zeros to form the new dividend, and add a decimal point to the quotient. In this case, bring down 00 to form 100 as the new dividend, add a decimal point to the 1 in the quotient, and double 1 to form 2 as the new divisor.

Repeat step 2: Find the largest digit that can be appended to the new divisor such that the resulting number will go into the new dividend that many times, and append that to the quotient. In this case, 24 will go into 100 four times. Four is the largest digit, because 25 will not go into 100 five times.

Repeat step 3: Multiply the divisor by the largest digit, and subtract the product from the dividend. In this case, multiply 24 by 4, and subtract 96 from 100 for a remainder of 4.

Repeat step 4: Bring down two more zeros to form the new dividend, and double the quotient (without the decimal point) to form the new divisor. In this case, bring down 00 to form 400 as the new dividend, and double 14 to form 28 as the new divisor.

Repeat step 2: Find the largest digit that can be appended to the new divisor such that the resulting number will go into the new dividend that many times, and append that to the quotient. In this case, 281 will go into 400 one time. One is the largest digit, because 282 will not go into 400 two times.

Repeat step 3: Multiply the divisor by the largest digit, and subtract the product from the dividend. In this case, multiply 281 by 1, and subtract 281 from 400 for a remainder of 119.

Repeat step 4: Bring down two more zeros to form the new dividend, and double the quotient (without the decimal point) to form the new divisor. In this case, bring down 00 to form 11900 as the new dividend, and double 141 to form 282 as the new divisor. Repeat steps 2-4 to produce more digits after the decimal point.