Rod Calculus - Extraction of Square Root

Extraction of Square Root

Algorithm for extraction of square root was described in Jiuzhang suanshu and with minor difference in terminology in The Mathematical Classic of Sun Zi.

The animation shows the algorithm for rod calculus extraction of an approximation of the square root from the algorithm in chap 2 problem 19 of The Mathematical Classic of Sun Zi:

Now there is a square area 234567, find one side of the square.

The algorithm is as follows:

• Set up 234567 on the counting board, on the second row from top, named shi
• Set up a marker 1 at 10000 position at the 4th row named xia fa
• Estimate the first digit of square root to be counting rod numeral 4, put on the top row (shang) hundreds position,
• Multiply the shang 4 with xiafa 1, put the product 4 on 3rd row named fang fa
• Multiply shang with fang fa deduct the product 4x4=16 from shi: 23-16=7, remain numeral 7.
• double up the fang fa 4 to become 8, shift one position right, and change the vertical 8 into horizontal 8 after moved right.
• Move xia fa two position right.
• Estimate second digit of shang as 8: put numeral 8 at tenth position on top row.
• Multiply xia fa with the new digit of shang, add to fang fa

.

• 8 calls 8 =64, subtract 64 from top row numeral "74", leaving one rod at the most significant digit.
• double the last digit of fang fa 8, add to 80 =96
• Move fang fa96 one position right, change convention;move xia fa "1" two position right.
• Estimate 3rd digit of shang to be 4.
• Multiply new digit of shang 4 with xia fa 1, combined with fang fa to make 964.
• subtract successively 4*9=36,4*6=24,4*4=16 from the shi, leaving 311
• double the last digit 4 of fang fa into 8 and merge with fang fa
• result

North Song dynasty mathematician Jia Xian developed an additive multiplicative algorithm for square root extraction, in which he replaced the traditional "doubling" of "fang fa" by adding shang digit to fang fa digit, with same effect.