《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§13. Rod calculus multiplication: 81 \times 81

This section gives a fully worked example of the rod calculus multiplication algorithm described in §8.

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
九九八十一、自相乘、得幾何。
答曰、六千五百六十一。
Nine nines [are] eighty-one, [which], multiplied with itself, resulteth in how much?
Answer saith: six thousand five hundred [and] sixty-one.
  • In modern notation:
    \begin{aligned} 9 \times 9 &= 81 \\ 81 \times 81 &= 6561. \end{aligned}
術曰、重置其位。 Method saith: put layered [the] places.
以上八呼下八、八八六十四、即下六千四百於中位。 Calling [the] lower eight with [the] upper eight: eight eights [are] sixty-four, that is, put down six thousand four hundred upon [the] middle places.
以上八呼下一、一八如八、即於中位下八十。 Calling [the] lower one with [the] upper eight: one eight [is] as eight, that is, upon [the] middle places put down eighty.
退下位一等、收上位八十。 Retreat [the] lower places [by] one rank, [and] take back eighty [the] upper place.
以上位一呼下八、一八如八、即於中位下八十。 Calling [the] lower eight with one [the] upper place: one eight [is] as eight, that is, upon [the] middle places put down eighty.
以上一呼下一、一一如一、即於中位下一。 Calling [the] lower one with [the] upper one: one one [is] as one, that is, upon [the] middle places put down one.
上下位俱收、中位即得六千五百六十一。 [The] upper [and the] lower places [now] all taken back, [the] middle places do result in six thousand five hundred [and] sixty-one.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §13". <https://yawnoc.github.io/sun-tzu/i/13> Accessed yyyy-mm-dd.