《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§23. Results of multiplication & division: multiples of 1, and all multiples of 9 through 1 combined

§15 through §22 gave 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.

This section gives the multiplication corresponding to m = n = 1, before giving the sum of all single-digit products of the form m \times n, the result of which is used in its own example of multiplication and division.

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
一一如一、自相乘、得一。一乘不長。 One one [is] as one, [which], multiplied with itself, resulteth in one. Multiplying [by] one groweth not.
  • 長： groweth

    長、上聲、 Cantonese: chœng2, Mandarin: zhǎng

右從九九至一一、總成一千一百五十五、自相乘、得一百三十三萬四千二十五。九人分之、人得一十四萬八千二百二十五。 From nine nines unto one one above, [the] total becometh one thousand one hundred [and] fifty-five, [which], multiplied with itself, resulteth in one hundred [and] thirty-three myriad four thousand [and] twenty-five. [With] nine people sharing it, [each] person getteth fourteen myriad eight thousand two hundred [and] twenty-five.
  • 右： above; lit. aright
  • Version C erroneously has 一百十三萬 for 一百三十三萬.
  • In modern notation:
    \begin{gathered} \begin{alignedat}{2} 9 \times 9 & + 8 \times 9 +{} & \dots & + 1 \times 9 \\ & + 8 \times 8 +{} & \dots & + 1 \times 8 \\[1ex] & & \ddots \\ & & & + 1 \times 1 = 1155 \end{alignedat} \\ \begin{aligned} 1155 \times 1155 &= 1334025 \\ \quad 1334025 \div 9 &= 148225. \end{aligned} \end{gathered}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §23". <https://yawnoc.github.io/sun-tzu/i/23> Accessed yyyy-mm-dd.