《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§19. Results of multiplication & division: multiples of 5

§15 through §22 give the results of multiplications and divisions of the form

\begin{aligned} (m n) \times (m n) &= m^2 n^2 \\ (m^2 n^2) \div m &= m n^2, \end{aligned}

along with the result of the division

\Bigl[ n \times n + (n - 1) \times n + \dots + 1 \times n \Bigr]^2 \div n.

Since this gets rather repetitive and boring, you may wish to skip to §23.

This section corresponds to n = 5.

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
五五二十五、自相乘、得六百二十五。五人分之、人得一百二十五。 Five fives [are] twenty-five, [which], multiplied with itself, resulteth in six hundred [and] twenty-five. [With] five people sharing it, [each] person getteth one hundred [and] twenty-five.
四五二十、自相乘、得四百。四人分之、人得一百。 Four fives [are] twenty, [which], multiplied with itself, resulteth in four hundred. [With] four people sharing it, [each] person getteth one hundred.
三五一十五、自相乘、得二百二十五。三人分之、人得七十五。 Three fives [are] fifteen, [which], multiplied with itself, resulteth in two hundred [and] twenty-five. [With] three people sharing it, [each] person getteth seventy-five.
二五一十、自相乘、得一百。二人分之、人得五十。 Two fives [are] ten, [which], multiplied with itself, resulteth in one hundred. [With] two people sharing it, [each] person getteth fifty.
一五如五、自相乘、得二十五。一人得二十五。 One five [is] as five, [which], multiplied with itself, resulteth in twenty-five. One person getteth twenty-five.
右五五一條、得七十五、自相乘、得五千六百二十五。五人分之、人得一千一百二十五。 [The] five fives above [as] one strand, result in seventy-five, [which], multiplied with itself, resulteth in five thousand six hundred [and] twenty-five. [With] five people sharing it, [each] person getteth one thousand one hundred [and] twenty-five.
  • 右： above; lit. aright
  • In modern notation:
    \begin{gathered} 5 \times 5 + 4 \times 5 + \dots + 1 \times 5 = 75; \\ 75 \times 75 = 5625; \quad 5625 \div 5 = 1125. \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §19". <https://yawnoc.github.io/sun-tzu/i/19> Accessed yyyy-mm-dd.