This section gives a worked example of simplifying a fraction using the subtraction-only version of the Euclidean algorithm.
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 |
|---|---|---|
| 今有一十八分之一十二。問約之得幾何。 答曰、三分之二。 |
Suppose there be twelve eighteenths.
[We] ask, reducing it resulteth in how much? Answer saith: two thirds. |
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| 術曰、置十八分在下、一十二分在上。副置二位、以少減多、等數得六為法、約之、即得。 | Method saith: put [the] eighteen shares in [the] below, [and the] twelve shares in [the] above. Put subsidiarily [the] two places, [and] subtract of [the] greater by [the] lesser; [for the] equal number [there] resulteth six as [the] divisor, [and] reducing them, [we] are done. |
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In my opinion, this section's description of the subtraction-only Euclidean algorithm is rather lacking in clarity. In particular, there is no explicit instruction to repeat the subtraction until you get the equal number 6.
A much better description of 約分術, "[the] Method of Reducing Fractions", is found in the separate text 《九章算術》, "Nine Chapters [on] Computational Methods", in the chapter 〈方田〉, 'Rectangular Fields', which has the same problem of simplifying 12/18 as well as another, simplifying 49/91. I think it is informative to include a translation of an excerpt here.
The source text for this excerpt is from 《四部叢刊初編》 (ctext.org library), and the bracketed portions are inline two-column annotations by Liu Huei.
Note that this excerpt is not a part of Sun Tzŭ's Computational Classic:
| Source text | Target text | Notes |
|---|---|---|
| 今有十八分之十二。問約之得幾何。 答曰、三分之二。 |
Suppose there be twelve eighteenths.
[We] ask, reducing it resulteth in how much? Answer saith: two thirds. |
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| 又有九十一分之四十九。問約之得幾何。 答曰、十三分之七。 |
And there be forty-nine ninety-firsts.
[We] ask, reducing it resulteth in how much? Answer saith: seven thirteenths. |
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| 約分（按、約分者、物之數量不可悉全、必以分言之。分之為數、繁則難用。設有四分之二者、繁而言之、亦可為八分之四、約而言之、則二分之一也。雖則異辭、至於為數、亦同歸爾。法實相推、動有參差、故為術者、先治諸分。）術曰、 | [The] Method of Reducing Fractions (Note: Reducing Fractions: [when a] number of objects cannot [be] completely whole, [we] must speak of them by fractions. [A] fraction being [a] number, [when] complicated [is] difficult [to] use. Suppose there be two fourths; complicating [it in] speaking of it, [it] also can be four eighths; reducing [it in] speaking of it, one half. Although different [in their] terms, as to [their] being numbers, [they do] return [the] same. Divisors [and] dividends [in] deduction with each other, [their] movements have unevenness; therefore [we] do [the] method, first administering it unto [the] fractions.) saith: |
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| 可半者、半之。 | Those which can [be] halved: halve them. | |
| 不可半者、副置分母子之數、以少減多、更相減損、求其等也、以等數約之。 | Those which cannot [be] halved: put subsidiarily [the] numbers [that are the] denominator [and the] numerator of [the] fraction; subtract of [the] greater by [the] lesser, [and] further subtract [and] diminish of [them] each other, seeking their [being] equal, [and] reduce them using [the] equal number. | |
| （等數約之、即除也。其所以相減者、皆等數之重疊。故以等數約之。） | ([To] reduce them [with the] equal number, is [to] divide [them]. Those of them by which [we have] mutually subtracted, [are] all layered stackings of [the] equal number. Therefore [we] use [the] equal number [to] reduce them.) |
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Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §1". <https://yawnoc.github.io/sun-tzu/ii/1> Accessed yyyy-mm-dd.