Chapter 14 Section 5 Calculation Multiplication and Division Algorithm

It is very inconvenient to calculate multiplication and division in three lines.After the middle of the Tang Dynasty, to meet the needs of commercial development, people began to simplify the multiplication and division method of calculation.In the fake "Xiahouyang Suanjing", there are many examples of turning multi-digit multiplication into one-bit multiplication.If there are a total of a Ding in a certain area, each Ding should be Nayongchobu 2.45 ends (one end = 5 feet), then the total number of Nab ends should be a × 2.45 = a × 7 × 7÷10÷2, and it is valid to use one with three digits The decimal multiplication of numbers is transformed into two multiplications of one digit and two divisions, and the operation can be completed in one line.One-bit multiplication is called cause, and this method is called refactoring.

The method of turning multiplication and division into addition and subtraction is called external addition and subtraction, which is a multiplication and division shortcut algorithm in which the first digit of the multiplication (division) number is 1.There is a question at the bottom of the volume of the counterfeit "Xia Hou Yang Suan Jing": There are 2454 pieces of silk, each worth 1.7 guan, how much is it worth?Its algorithm is 2454×1.7=2454×17÷10=(24540+2454×7)÷10, and the program of 2454×17 is: Yang Hui systematically summarized the method of turning multiplication and division into addition and subtraction in the Tang and Song Dynasties in "Multiplication and Division Tongbian Basics and Moments", and proposed five techniques of addition instead of multiplication and four techniques of subtraction instead of division.For the multiplication (division) method where the first digit of the multiplication (division) number is not 1, the first digit of the multiplication (division) number can be changed to 1 by doubling, halving, etc., and then the addition (subtraction) method is used to replace the multiplication (division). The method is called "seeking one" technique.In Yang Hui's "Multiplication and Division Tongbian Bianmo", there is a formula of "seeking multiplication and division for one generation". The verse of "seeking a multiplication" is: "Five, six, seven, eight, nine, do not take the multiples. Two or three must be half, and four or two will be folded. Doubling the original follows the law, but in fact it is the opposite. (Self-note: the double method must be folded, and the doubled must be folded). If you use it as a substitute, the number is enough to keep." For example, 237×56=(237÷2)×(56×2)=118.5×112, Multiplication is done by adding two bits. This method became obsolete after the simplification of the 14th century formula.

Return and division is developed on the basis of nine returns and subtraction.Return refers to one-bit division, and the one-bit division from 1 to 9 is called nine return.After summarizing and developing by Yang Hui, Zhu Shijie, etc., the form of the nine-return formula in "Anathematics Enlightenment" is basically the same as that of the current abacus formula: One return is like one advance, nine one advances into ten.Adding two to one makes five, every two makes ten.Three one thirty one, three two sixty two, every three into ten.Four one twenty two, four two add five, four three seven two, every four makes ten.Five returns are doubled, and every five advances becomes ten.Sixty one adds four, six two thirty two, six three adds five, six four sixty-four, six five eighty-two, every six makes ten... nine returns to the body, every nine makes ten.

Zhu Shijie already knows how to return, but he has no fine grass.He Pingzi's "Detailed Algorithm" has return and removal of fine grass, such as 48895÷385, which is three returns and eighty-five divisions, and the fine grass is: column dividend Later, people created the formula of bumping back to solve the problem of how to determine the quotient of the big divisor.So far, the quick calculation algorithm and its formulas have developed to the point where calculation chips and calculations cannot be accommodated, and abacus and abacus calculations have emerged, and the calculation formulas have become abacus formulas, that is, the algorithmic language of abacus calculations.