Square Root Extraction by the Return-and-Divide Method

Original: 歸除開平方 (Guichu kai pingfang). This method uses specific abacus division rhymes to determine the digits of the root.

Suppose we have a square area of 54,756 paces. Question: What is the length of one side?

The rules for the Return-and-Divide method for square roots are as follows: Place the total area 54,756 paces as the Dividend Shi: the number being operated upon in the center of the board the suanpan or abacus and examine it.

Estimate the first quotient as 200 and place it to the left of the Dividend. ◯ Also place 200 at the lower right. Calling out from left to right: "Two times two [is four]," subtract this from the Dividend.

40,000 is subtracted, leaving a remainder of 14,756 paces in the Dividend. Double the 200 at the lower right to get 400, which becomes the Divisor Fa: the number used to divide the remainder.

Using the Return-and-Divide rhymes, call out: "Four-one: twenty; when meeting four, advance one." This mnemonic (si-yi er-shi) is for dividing by 4. It indicates that 1 divided by 4 is 0.2, with a remainder. Here it is used to find the next digit, 3. This yields a quotient of 30 paces. Immediately place 30 paces below the 400 on the right. Calling out: "Three times three [is nine]," subtract 900 paces from the Dividend. This leaves a remainder of 1,856 paces. ◯ Immediately double the 30 paces at the lower right to get 60 paces, which combined with the previous 400 makes 460 paces as the Divisor.

Using the Return-and-Divide rhymes again, call out: "Four-one: twenty; when meeting eight, advance two." 4 into 8 goes 2 times; this helps determine the units digit of 4. This yields a quotient of 4 paces. ◯ Also place 4 paces below the 60 on the right. Calling out: "Four..."