《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§28. Gaussian elimination

This section gives a worked example of solving a system of linear equations by Gaussian elimination.

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
今有甲乙二人持錢、各不知數。 Suppose there be two people A [and] B holding coins, each of [which we] know not [the] number of.
甲得乙中半、可滿四十八、乙得甲太半、亦滿四十八。 A getting B's half, indeed reacheth forty-eight; B getting A's two thirds, also reacheth forty-eight.
  • 中半： half; lit. middle half
  • 可： indeed; lit. can
  • 太半： two thirds; lit. grand half
問甲乙二人元持錢各幾何。 [We] ask, how many coins each held [the] two people A [and] B originally?
  • In modern notation, we have the system
    \begin{aligned} a + \frac{b}{2} &= 48 \\[\tallspace] b + \frac{2a}{3} &= 48 \end{aligned}
    in a and b the number of coins held by each person.
答曰、甲持錢三十六、乙持錢二十四。 Answer saith: A held coins thirty-six; B held coins twenty-four.
術曰、如方程求之。 Method saith: seek them according unto the rectangular system.
  • 方程： the rectangular system

    This is the classical name for Gaussian elimination; today it is the term used to translate English "equation".

置二甲一乙錢九十六於右方、置二甲三乙錢一百四十四於左方。 Put two A, one B, [and] coins ninety-six upon [the] right, [and] put two A, three B, [and] coins one hundred [and] forty-four upon [the] left.
  • Unwritten in the text is the doubling and tripling of the two equations to obtain a system with integer coefficients,
    \begin{aligned} 2a + 1b &= 96 \\ 2a + 3b &= 144. \end{aligned}
    Classical Chinese is written top-to-bottom, right-to-left, so the Gaussian elimination actually has the equations laid out in columns (with elementary column operations rather than row operations):
    \roundbr{ \begin{array}{cc} 2a & 2a \\ 3b & 1b \\ \hline 144 & 96 \end{array} }.
    While it is modern practice to include only the coefficients, I have retained a and b in the augmented matrix for readability.
以右方二乘左方、上得四、中得六、下得二百八十八錢。 Multiplying [the] left by [the] right's two: above [there] resulteth four, [in the] middle [there] resulteth six, [and] below [there] resulteth two hundred [and] eighty-eight coins.
  • In modern notation, double the left column to get
    \roundbr{ \begin{array}{cc} 4a & 2a \\ 6b & 1b \\ \hline 288 & 96 \end{array} }.
    This step and the next are redundant, as both equations already had the same coefficient for a. Nevertheless, it demonstrates that it was known how to deal with the case of unequal coefficients.
以左方二乘右方、上得四、中得二、下得一百九十二。 Multiplying [the] right by [the] left's two: above [there] resulteth four, [in the] middle [there] resulteth two, [and] below [there] resulteth one hundred [and] ninety-two.
  • Version A and Version C are missing this sentence entirely.
  • Version B erroneously has 九十六 for 一百九十二.
  • In modern notation, double the right column to get
    \roundbr{ \begin{array}{cc} 4a & 4a \\ 6b & 2b \\ \hline 288 & 192 \end{array} }.
以右行再減左行、左上空、中餘四乙為法、下餘九十六錢為實。 Furthermore subtracting of [the] left column by [the] right column: [the] upper left [becometh] empty, [in the] middle [there] remaineth four B as [the] divisor, [and] below [there] remaineth ninety-six coins as [the] dividend.
  • In modern notation, this gives
    \roundbr{ \begin{array}{cc} & 4a \\ 4b & 2b \\ \hline 96 & 192 \end{array} }.
上法下實、得二十四錢為乙錢。 [The] upper divisor [and the] lower dividend, result in twenty-four coins as B's coins.
  • In modern notation, divide the left column by 4 to get
    \roundbr{ \begin{array}{cc} & 4a \\ 1b & 2b \\ \hline 24 & 192 \end{array} }.
以減右下九十六、餘七十二為實、以右上二甲為法。 With [which], subtracting of [the] lower right's ninety-six, [there] remaineth seventy-two as [the] dividend; [and] use [the] upper right's two A as [the] divisor.
  • Here the text reverts the right column to before it was doubled,
    \roundbr{ \begin{array}{cc} & 2a \\ 1b & 1b \\ \hline 24 & 96 \end{array} },
    then does the subtraction to obtain
    \roundbr{ \begin{array}{cc} & 2a \\ 1b & \\ \hline 24 & 72. \end{array} }.
上法下實、得三十六為甲錢也。 [The] upper divisor [and the] lower dividend, result in thirty-six as A's coins.
  • In modern notation, divide the right column by 2 to get
    \roundbr{ \begin{array}{cc} & 1a \\ 1b & \\ \hline 24 & 36 \end{array} }.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §28". <https://yawnoc.github.io/sun-tzu/iii/28> Accessed yyyy-mm-dd.