《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§7. Rod calculus: vertical & horizontal forms

This section hints at the way in which numerals are displayed in rod calculus, a means of doing arithmetic computations using counting rods (eventually replaced by the Chinese abacus).

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
凡算之法、先識其位。 [In the] method of all computation, know first [the] places.
一從十橫、百立千僵、千十相望、萬百相當。 [The] ones [are] vertical [and the] tens horizontal; [the] hundreds erect [and the] thousands prostrate. [The] thousands [and] tens look at each other; [the] myriads [and] hundreds face each other.
  • Version A and Version D have 百萬 instead of 萬百. According to the inline note in Version B, this is a transcription error:
    Source text Target text Notes
    案、萬百原本訛作百萬、今據夏侯陽算經改正。 Note: Myriads [and] hundreds [in the] original version [was] erroneously written hundreds [and] myriads. Now corrected according unto Hia‑hou Yang's Computational Classic.
    In my opinion 萬百 vs 百萬 is a minor issue, but I agree that 萬百 is the correct order, since we have 千十 preceding, rather than 十千. Thus I have 萬百 above.

Extended commentary

Since this section is hardly a self-contained explanation of rod calculus, I give a brief introduction here.

For each digit (except zero which is denoted by a blank space) there are two forms, vertical and horizontal:

Digit 1 2 3 4 5 6 7 8 9
Vertical form One vertical rod. Two vertical rods. Three vertical rods. Four vertical rods. Five vertical rods. One horizontal rod with one vertical rod below it. One horizontal rod with two vertical rods below it. One horizontal rod with three vertical rods below it. One horizontal rod with four vertical rods below it.
Horizontal form One horizontal rod. Two horizontal rods. Three horizontal rods. Four horizontal rods. Five horizontal rods. One vertical rod with one horizontal rod below it. One vertical rod with two horizontal rods below it. One vertical rod with three horizontal rods below it. One vertical rod with four horizontal rods below it.

The text tells us to use

For example, the number 345 is displayed as three vertical rods, followed by four horizontal rods, followed by five vertical rods. The alternating between vertical and horizontal forms prevents ambiguities which would arise from placing vertical forms next to each other.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §7". <https://yawnoc.github.io/sun-tzu/i/7> Accessed yyyy-mm-dd.