Jiuzhang Suanshu Chapter 1 - Fang Tian

**Translation and explanatory notes by Tao Steven Zheng (鄭濤)**

**劉徽注**
[1] 以御田疇界域。

**Liu Hui's Commentary**
[1] To determine the areas and boundaries of fields.

## **Rectangular field method 方田術 (Problems 1.01 - 1.02)**

荅曰：一畝。
〔二〕又有田廣十二步，從十四步。問：為田幾何？

荅曰：一百六十八步。[1]
方田術曰：廣從步數相乘得積步。[2] 以畝法二百四十步除之，即畝數。百畝為一頃。[3]

**劉徽注**
[1] 圖：從十四，廣十二。

[2] 此積謂田冪。凡廣從相乘謂之冪。

**李淳風注**
[2] 臣淳風等謹按：經云“廣從相乘得積步”，注云“廣從相乘謂之冪”。觀斯注意，積冪義同。以理推之，固當不爾。何則？冪是方面單布之名，積乃眾數聚居之稱。循名責實，二者全殊。雖欲同之，竊恐不可。今以凡言冪者據廣從之一方；其言積者舉眾步之都數。經云相乘得積步，即是都數之明文。注云謂之為冪，全乖積步之本意。此註前云積為田冪，於理得通。復云謂之為冪，繁而不當。今者註釋存善去非，略為料簡，遺（wèi）諸後學。

[3] 臣淳風等謹按：此為篇端，故特舉頃、畝二法。餘術不復言者，從此可知。按：一 畝田，廣十五步，從而疏之，令為十五行，則每行廣一步而從十六步。又橫而截之，令為十六行，即每行廣一步而從十五步。此即從疏橫截之步，各自為方。凡有二百四十步，為一畝之地，步數正同。以此言之，即廣從相乘得積步，驗矣。二百四十步者，畝法也；百畝者，頃法也。故以除之，即得。

Answer: 1 *mu*.

[1.02] Suppose there is a field (田 *tian*) of width 12 *bu* wide and length 14 *bu*. Question: How large [in area] is the field?

Answer: 168 [square] *bu*. [1]

Rectangular field method (方田術 *fang tian shu*): Multiply the number of *bu* (步) of the width with the length to obtain the product (積 *ji*) in [square] *bu*. Divide by 240 [square] *bu* to obtain the number of *mu* (畝). 100 *mu* is 1 *qing* (頃).

**Liu Hui's Commentary**
[1] Diagram: the length is 14, and the width is 12.

[2] The product (積 *ji*) is called the area (冪 *mi*) of the field. Whenever width and length are multiplied, it is called the area (冪 *mi*).

**Li Chunfeng's Commentary**
[2] Li Chunfeng and his associates comment: The classic text says: "multiply the width with the length to obtain the product (積 *ji*) in *bu* (步)", and the commentary [of Liu Hui] says: "width and length are multiplied, it is called the area (冪 *mi*)". By examining the meaning of this commentary, the product (積 *ji*) and the area (冪 *mi*) have the same meaning. From this principle, this cannot be the case. Why? The area (冪 *mi*) is the name of the surface of the rectangle, and the product (積 *ji*) is the name of the product of numbers. Comparing the two words in terms and what they refer to, the two are completely different. Although one may wish to treat them as if they were the same, I fear this cannot be so. Nowadays it is common to call the surface across the width and length as the area (冪 *mi*), whereas the product (積 *ji*) refers to the total number of *bu*. When the classic text says: "multiply the width with the length to obtain the product (積 *ji*) in *bu* (步)", it clearly explains a total number. When the commentary says: "it is called the area (冪 *mi*)", it completely defies the original meaning of the product (積 *ji*) in *bu* (步). The commentary first mentions "the product (積 *ji*) is called the area (冪 *mi*) of the field", which is reasonable in principle, but it then refers to it again as "an area (冪 *mi*)", which is unnecessary and incorrect. Our commentary retains what is correct and rejects what is incorrect, for the sake of simplifying things for all future learners.
[3] Li Chunfeng and his associates comments: This is the start of the chapter; therefore, the two rules for *qing* (頃) and *mu* (畝) are specially mentioned. The other methods in this chapter do not mention them again, but they can be understood from here. Comment: Given a field of 1 *mu* and width 15 *bu*, cutting it vertically results in 15 columns, with each column being 1 *bu* wide and 16 *bu* long. Alternatively, cutting horizontally results in 16 rows, with each row being 1 *bu* wide and 15 *bu* long. Cutting horizontally and vertically in unit *bu*, each subdivision being a unit square, gives a total of 240 [square] *bu*. For a field of 1 *mu*, the number of *bu* is the same. According to this explanation, multiplying the width and the length together to obtain the product in [square] *bu* is verified. Thus, 240 [square] bu is the *mu* [conversion] standard; 100 *mu* is the *qing* [conversion] standard. Therefore, using these two rules for the purpose of dividing the product gives the desired result.

### **Explanatory Notes**

**Terminology**

(1) 田 tián: This word literally means "field". In traditional Chinese mathematics, all areas are called fields, with the rectangular field being the default field. In ancient China, squares and rectangles are also called "upright fields" (直田 *zhi tian*) or "width fields" (廣田 *guang tian*).

(2) 廣 guǎng: The horizontal width.

(3) 從 zòng or zōng: The vertical length.

(4) 相乘 xiāng chéng: This term literally translates to "mutually multiply", which is equivalent to multiplication.

(5) 積 jī: The product of two or more numbers under multiplication.

(6) 冪 mì: The area of a surface.

(7) 步 bù: This term literally translates to "step" or "pace". In traditional Chinese mathematics, *bu* is a unit of length that approximates the double pace of man

(8) 積步 jī bù: This term literally translates to "product in *bu*". In traditional Chinese mathematics, the "product in *bu*" can mean a different unit depending on the context: for areas, it is "square *bu*"; for volumes, it is "cubic *bu*".

(9) 畝 mǔ: A unit of area equal to 240 square *bu*.

(10) 頃 qǐng: A unit of area equal to 100 *mu*.

(11) 都數 dōu shù: The total number.

(12）臣淳風等謹按 chén chún fēng děng jǐn àn: Literally translates to "Your subject Chunfeng and his associates carefully comment".

**Principles**

(1) Let

A
,
a
,
b

{\displaystyle A, a, b }

denote the area, width, and length of the rectangle. The area of a rectangle is equal to the product of its width and length:

# A
a
b
.

{\displaystyle A = ab.}

**Solutions**

[1.01] Substitute the given width and length into the area formula:

A

=
15

b
u
×
16

b
u

=
240

b

u
2

.

{\displaystyle \begin{align}
A &= 15 \; bu \times 16 \; bu \\
&= 240 \; bu^2.
\end{align}}

Since

1

m
# u
240

b

u
2

{\displaystyle 1 \; mu = 240 \; bu^2 }

, the area of the rectangular field is

1

m
u

{\displaystyle 1 \; mu }

.

[1.02] Substitute the given width and length into the area formula:

A

=
12

b
u
×
14

b
u

=
168

b

u
2

.

{\displaystyle
\begin{align} A &= 12 \; bu \times 14 \; bu \\
&= 168 \; bu^2.
\end{align} }

## **Rectangular field method [measured] in *li* 里田術 (Problems 1.03 - 1.04)**

荅曰：三頃七十五畝。

〔四〕又有田廣二里，從三里。問：為田幾何？

荅曰：二十二頃五十畝。

里田術曰：廣從里數相乘得積里。以三百七十五乘之，即畝數。[1]

**劉徽注**
[1] 按：此術廣從里數相乘得積里。故方里之中有三頃七十五畝，故以乘之，即得畝數也

Answer: 3 *qing* 75 *mu*.

[1.04] Suppose there is a field (田 *tian*) of width 2 *li* and length 3 *li*. Question: How large [in area] is the field?

Answer: 22 *qing* 50 *mu*.
Rectangular field method [measured] in *li* (里田術 *li tian shu*): Multiply the number of *li* (步) of the width with the length to obtain the product (積 *ji*) in [square] *li*. Multiply by 375 [square] to obtain the number of *mu* (畝).
**Liu Hui's Commentary**
[1] Comment: This method prescribes the multiplication of the number of *li* of the width and the length, thus giving the product (積 *ji*) in [square] *li*. Since 1 square *li* has 3 *qing* 75 *mu*, multiplying by this gives the numeber of *mu*.

### **Explanatory Notes**

**Terminology**

(1) 里 lǐ: A unit of length that is equal to 300 *bu*.

**Principles**

(1) Since 1 *li* is equal to 300 *bu*, one square of width 1 *li* and length 1 *li* has an area of 90000 square *bu*. Because 1 *mu* is equal to 240 square *bu*, dividing 90000 square *bu* by 240 square *bu* per *mu* gives 375 *mu*, or 3 *qing* 75 square *bu*.

**Solutions**

[1.03] Substitute the given width and length into the area formula (including the conversion factor):

A

=
1

l
i
×
1

l
i
×
375

=
375

m
u
.

{\displaystyle \begin{align}
A &= 1 \; li \times 1 \; li \times 375 \\
&= 375 \; mu .
\end{align}}

Since

1

q
i
n
# g
100

m
u

{\displaystyle 1 \; qing = 100 \; mu }

, the area of the rectangular field is

3

q
i
n
g

75

m
u

{\displaystyle 3 \; qing \; 75 \; mu }

.

[1.04] Substitute the given width and length into the area formula:

A

=
2

l
i
×
3

l
i
×
375

=
2250

m
u
.

{\displaystyle \begin{align}
A &= 2 \; li \times 3 \; li \times 375 \\
&= 2250 \; mu .
\end{align}}

Since

1

q
i
n
# g
100

m
u

{\displaystyle 1 \; qing = 100 \; mu }

, the area of the rectangular field is

22

q
i
n
g

50

m
u

{\displaystyle 22 \; qing \; 50 \; mu }

.

## **Reduction of fractions method 約分術 (Problems 1.05 - 1.06)**

荅曰：三分之二。

〔六〕又有九十一分之四十九。問：約之得幾何？

荅曰：十三分之七。

約分術曰[1]：可半者半之；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。[2]

**劉徽注**
[1] 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（cēn cī），故為術者先治諸分。

[2] 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。

Answer: 2 parts of 3.

[1.06] Again, suppose there are 49 parts of 91. Question: Reducing [the fraction] gives how much?

Answer: 7 parts of 13.
Reduction of fractions method (約分術 *yue fen shu*)[1]: If both quantities [i.e. the numerator and denominator] can be halved, halve them. If they cannot be halved, set the numbers separately, subtract the smaller from the larger, and continue subtracting until equality, to obtain the equal number (等數 deng shu). Use the equal number to reduce the fraction. [2]

**Liu Hui's Commentary**
[1] Comment: As to reduce of fractions, note that the quantities of things may not always be expressed as whole [numbers], and it is necessary to express them as fractions. Fractions are difficult to use if they are not reduced. Suppose there are 2 parts of 4; these are the same as 4 parts of 8 when unreduced. However, in reduced form, this is 1 parts of 2. Although the expressions might be different, but as numbers, they are the same [quantity]. When comparing the divisor with the dividend, one often finds that they are uneven (參差 *cen ci*) [i.e. larger or smaller]. Hence, one must first learn how to deal with fractions calculations.

[2] Use the equal number (等數 *deng shu*) to reduce fractions via division. All the numbers involved in the subtractions are all multiples of the equal number. This is why we can reduce the given quantities by it.

### **Explanatory Notes**

**Terminology**

(1) 分之 fēn zhī: Literally translates to "parts of". This is the standard way in which fractions are expressed in Chinese.

(2) 母 mǔ: The denominator; literally translates to "mother".

(3) 子 zi: The numerator; literally translates to "child".

**Principles**

(1) The reduction of fractions method (約分術 *yue fen shu*) prescribes a generalized procedure for reducing fractions to lowest terms. The process is equivalent to the Euclidean algorithm. This process begins by making two numbers undergo a series of successive subtractions until both numbers are equal. This result, referred to as the "equal number" (等數 *deng shu*), is equivalent to the greatest common divisor (GCD) or highest common factor (HCF).

**Solutions**

[1.05] Both the numerator 12 and the denominator 18 are even numbers, so halve them.

12
÷
2

18
÷
2

=

6
9

{\displaystyle \frac{12 \div 2}{18 \div 2} = \frac{6}{9} }

Although the numerator 6 is an even number, the denominator 9 is an odd number, so they cannot both be halved. Use the Euclidean algorithm to find the equal number (等數 *deng shu*).

9
−
6

=
3

6
−
3

=
3

3

=
3

{\displaystyle
\begin{align}
9 - 6 &= 3 \\
6 - 3 &= 3 \\
3 &= 3
\end{align}
}

Thus, the equal number (等數 *deng shu*) is 3. Divide the numerator and denominator by 3 to obtain the fully reduced fraction.

6
÷
3

9
÷
3

=

2
3

{\displaystyle \frac{6 \div 3}{9 \div 3} = \frac{2}{3} }

[1.06]

The numerator 49 and the denominator 91 cannot be halved because both are odd numbers. Use the Euclidean algorithm to find the equal number (等數 *deng shu*).

91
−
49

=
42

49
−
42

=
7

42
−
7

=
35

35
−
7

=
28

28
−
7

=
21

21
−
7

=
14

14
−
7

=
7

7

=
7

{\displaystyle
\begin{align}
91 - 49 &= 42 \\
49 - 42 &= 7 \\
42 - 7 &= 35 \\
35 - 7 &= 28 \\
28 - 7 &= 21 \\
21 - 7 &= 14 \\
14 - 7 &= 7 \\
7 &= 7
\end{align}
}

Thus, the equal number (等數 *deng shu*) is 7. Divide the numerator and denominator by 7 to obtain the fully reduced fraction.

49
÷
7

91
÷
7

=

7
13

{\displaystyle \frac{49 \div 7}{91 \div 7} = \frac{7}{13} }

## **Addition of fractions method 合分術 (Problems 1.07 - 1.09)**

荅曰：十五分之十一。

〔八〕又有三分之二，七分之四，九分之五。問：合之得幾何？

荅曰：得一、六十三分之五十。

〔九〕又有二分之一，三分之二，四分之三，五分之四。問：合之得幾何？

荅曰：得二、六十分之四十三。

合分術曰[1]：母互乘子，並以為實，母相乘為法。[2] 實如法而一。不滿法者，以法命之。[3] 其母同者，直相從之。
**劉徽注**
[2] 母互乘子：約而言之者，其分麄（cū）；繁而言之者，其分細。雖則麄細有殊，然其實一也。眾分錯難，非細不會。乘而散之，所以通之。通之則可並也。凡母互乘子謂之齊，羣母相乘謂之同。同者，相與通同共一母也；齊者，子與母齊，勢不可失本數也。方以類聚，物以羣分。數同類者無遠；數異類者無近。遠而通體知，雖異位而相從也；近而殊形知，雖同列而相違也。然則齊同之術要矣：錯綜度數，動之斯諧，其猶佩觿解結，無往而不理焉。乘以散之，約以聚之，齊同以通之，此其筭之綱紀乎。其一術者，可令母除為率，率乘子為齊。
[3] 今欲求其實，故齊其子，又同其母，令如母而一。其餘以等數約之，即得知，所謂同法為母，實餘為子，皆從此例。

**李淳風注**
[1] 臣淳風等謹按︰合分知，數非一端，分無定準，諸分子雜互，羣母參差，麄細既殊，理難從一。故齊其眾分，同其羣母，令可相併，故曰合分。

Answer: 11 parts of 15.

[1.08] Again, suppose there are 2 parts of 3, 4 parts of 7, and 5 parts of 9. Question: Adding [the fractions] gives how much?

Answer: The result is 1 and 50 parts of 63.

[1.09] Again, suppose there are 1 parts of 2, 2 parts of 3, 3 parts of 4, and 4 parts of 5. Question: What is the sum?

Answer: The result is 2 and 43 parts of 60 .

Addition of fractions method (合分術 *he fen shu*) [1]: Cross-multiply (互乘 *hu cheng*) the numerators and denominators, and combine them as the dividend (實 *shi*). Multiply all the denominators together as the divisor (法 *fa*). [2] Divide the dividend by the divisor; if it does not divide completely, then use the divisor to name (命 *ming*) the fraction. [3] If their denominators are the same, then directly add them together.

**Liu Hui's Commentary**
[2] Cross-multiply (互乘 *hu cheng*) the numerators and denominators: a fraction in its reduced form has a coarse subdivision (麄 *cu*), whereas with an unreduced form, such a subdivision is fine (細 *x*i). Although the denominators may be coarse or fine, they represent the same fraction. A set of fractions are often difficult to handle [in that their denominators are different], one cannot combine (會 *hui*) them without finely subdividing their denominators. Multiply to refine (散 *san*) the denominators, so the fractions can interconnect (通 *tong*). Once interconnected [with a common denominator], [the numerators] can be added together. Cross-multiplying every numerator and denominator is called homogenization (齊 *qi*); multiplying the denominators together is called equalizing (同 *tong)*. To equalize is to make each each fraction have the same common denominator; to homogenize is to align the numerator and the denominator, so that the ratios (勢 *shi*) do not alter the original value of the fractions. The principles are classified into types; things are divided into groups. Numbers of the same types are not far apart (無遠 *wu yuan*); whereas numbers of different types are not close together (無近 *wu jin*). Numbers that are far apart but interconnected [with the same denominator] can be added even if they occupy different positions [on the counting board]. On the other hand, numbers that are close together but in different forms [with different denominators], even if they are positioned in the same rank [on the counting board], they are still remain separated. Thus, the methods of homogenizing and equalizing (齊同之術 *qi tong zhi shu*) are very important: the fractions may be in disaccord, but by such methods can bring them in harmony. This is like using an awl for untying knots; there is no case where it will not work. Multiplying to refine the fractions, dividing to reduce them, homogenizing and equalizing to interconnect them, is this not the guiding principle of computation? Another method: we may let the rate (率 *lü*) be the product of all denominators divided by each denominator, and the rate multiplied by the numerators serves to homogenize the numerators.

[3] Now to find the dividend, first homogenize the numerators and make the denominators equal. Let the dividend be divided by the common denominator. Reduce any remainder with the divisor by the equal number, thereby obtaining the result. This is called the equalized divisor as denominator, and the remainder from the dividend as numerator. Henceforth, all cases of this situation follow this rule.
**Li Chunfeng's Commentary**
[1] Li Chunfeng and his associates comment: When adding fractions, due to the numbers not being the same, the divisions not being standard, where all the numerators being mixed with each other, and the denominators not the same (being coarse and fine), it is in principle difficult to combine them into a single fraction. Thus, it is necessary to homogenize the numerators and equalize the denominators before the fractions can be added. This is why we called it the addition of fractons.

### **Explanatory Notes**

**Terminology**

(1) 合分 hé fēn: Literally translates to "combining fractional parts".

(2) 方 fāng: The Dao (the way) or the principle. This usage of 方 and the quote 方以類聚，物以羣分。comes from the *Yi Jing* (*I Ching*): 方以類聚，物以羣分，吉凶生矣。(Affairs are arranged together according to their tendencies, and things are divided according to their classes. Hence were produced what is good and evil.)

**Principles**

(1) The addition of fractions method (合分術 *he fen shu*) gives a general algorithm for computing the sums of fractions.

Step 1: Given an arbitrary number of fractions

a
1

b
1

,

a
2

b
2

,

a
3

b
3

,
.
.
.
,

a
n

b
n

{\displaystyle \frac{a\_1}{b\_1}, \frac{a\_2}{b\_2}, \frac{a\_3}{b\_3}, ..., \frac{a\_n}{b\_n} }

use the numerators and denominators to calculate the adjusted numerators for each fraction via a cross-multiplication procedure called *hu cheng* (互乘). This process is calculated by multiplying each numerator by the denominators of the other fractions. Chinese mathematicians would later call this operation "homogenizing" (齊 *qi*).

A
1

=

a
1

×

b
2

×

b
3

×
⋯
×

b
n

A
2

=

a
2

×

b
1

×

b
3

×
⋯
×

b
n

A
3

=

a
3

×

b
1

×

b
2

×
⋯
×

b
n

⋮

A
n

=

a
n

×

b
1

×

b
2

×
⋯
×

b

n
−
1

{\displaystyle \begin{align}
A\_1 = a\_1 \times b\_2 \times \;&\;b\_3 \times \cdots \times b\_n \\[5 pt]
A\_2 = a\_2 \times b\_1 \times \;&\;b\_3 \times \cdots \times b\_n \\[5 pt]
A\_3 = a\_3 \times b\_1 \times \;&\;b\_2 \times \cdots \times b\_n \\[5 pt]
&\vdots \\
A\_n = a\_n \times b\_1 \times \;&\; b\_2 \times \cdots \times {b}\_{n-1} \\
\end{align} }

Step 2: Add all the *hu cheng* (互乘) products together to obtain the dividend

A

{\displaystyle A}

.

# A

A
1

A
2

A
3

⋯

A
n

{\displaystyle A = A\_1 + A\_2 + A\_3 + \cdots + A\_n }

Step 3: Calculate the divisor

B

{\displaystyle B}

 by multiplying all the given denominators.

# B

b
1

×

b
2

×

b
3

×
⋯
×

b
n

{\displaystyle B = b\_1 \times b\_2 \times b\_3 \times \cdots \times b\_n }

Chinese mathematicians would later call this operation "equalizing" (同 *tong*). Together with "homogenizing " (齊 *qi*) , both operations forms the "homogenizing and equalization method" (齊同術 *qi tong shu*).

Step 4: Divide the dividend (實 *shi*) by the divisor (法 *fa*).

A
B

{\displaystyle \frac{A}{B}}

Step 5: Lastly, reduce the fraction to lowest terms if needed

A
B

→

a
b

{\displaystyle \frac{A}{B} \to \frac{a}{b} }

where

g
c
d
(
a
,
b
# )
1

{\displaystyle gcd(a,b) = 1}

. If

a
>
b

{\displaystyle a > b}

, express the fraction as a mixed number

a
b

=
q

r
b

{\displaystyle \frac{a}{b} = q + \frac{r}{b} }

, where

# a
b
q
r

{\displaystyle a=bq+r}

 and

0
<
r
<
b

{\displaystyle 0 < r < b }

.

**Solutions**

[1.07]

Calculate the dividend (實 *shi*).

A

=
1
×
5
2
×
3

=
5
6

=
11

{\displaystyle
\begin{align}
A &= 1 \times 5 + 2 \times 3 \\
&= 5 + 6 \\
&= 11
\end{align}}

Calculate the divisor (法 *fa*):

# B
3
×
5

{\displaystyle B = 3 \times 5 }

Divide the dividend (實 *shi*) by the divisor (法 *fa*):

A
B

=

11
15

{\displaystyle \frac{A}{B} = \frac{11}{15}}

[1.08]

Calculate the dividend (實 *shi*):

A

=
2
×
7
×
9
4
×
3
×
9
5
×
3
×
7

=
126
108
105

=
339

{\displaystyle
\begin{align}
A &= 2 \times 7 \times 9 + 4 \times 3 \times 9 + 5 \times 3 \times 7 \\
&= 126 + 108 + 105 \\
&= 339
\end{align}}

Calculate the divisor (法 *fa*):

# B
3
×
7
×
# 9
189

{\displaystyle B = 3 \times 7 \times 9 = 189 }

Divide the dividend (實 *shi*) by the divisor (法 *fa*):

A
B

=

339
189

=

113
63

=
1

50
63

{\displaystyle \frac{A}{B} = \frac{339}{189} = \frac{113}{63} = 1\frac{50}{63} }

[1.09]

Calculate the dividend (實 *shi*):

A

=
1
×
3
×
4
×
5
2
×
2
×
4
×
5
3
×
2
×
3
×
5
4
×
2
×
3
×
4

=
60
80
90
96

=
326

{\displaystyle
\begin{align}
A &= 1 \times 3 \times 4 \times 5 + 2 \times 2 \times 4 \times 5 + 3 \times 2 \times 3 \times 5 + 4 \times 2 \times 3 \times 4 \\
&= 60 + 80 + 90 + 96 \\
&= 326
\end{align}}

Calculate the divisor (法 *fa*):

# B
2
×
3
×
4
×
# 5
120

{\displaystyle B = 2 \times 3 \times 4 \times 5 = 120 }

Divide the dividend (實 *shi*) by the divisor (法 *fa*):

A
B

=

326
120

=

163
60

=
2

43
60

{\displaystyle \frac{A}{B} = \frac{326}{120} = \frac{163}{60} = 2\frac{43}{60} }

## **Subtraction of fractions method 減分術 (Problems 1.10 - 1.11)**

荅曰：四十五分之三十一。

〔一一〕又有四分之三，減其三分之一。問：餘幾何？

荅曰：十二分之五。

減分術曰[1]：母互乘子，以少減多，餘為實。母相乘為法。實如法而一。[2]
**劉徽注**
[2] 「母互乘子」知，以齊其子也，「以少減多」知，齊故可相減也。「母相乘為法」者，同其母。母同子齊，故如母而一，即得。

**李淳風注**
[1] 臣淳風等謹按︰諸分子、母數各不同，以少減多，欲知餘幾，減餘為實，故曰減分。

Answer: 31 parts of 45.

[1.11] Again, suppose there are 3 parts 4; subtract 1 parts 3 from it. Question: What is the remainder?

Answer: 5 parts 12.

Subtraction of fractions method (減分術 *jian fen shu*) [1]: Cross-multiply (互乘 *hu cheng*) the numerators and denominators, subtract the smaller from the larger, and take the remainder (餘 *yu*) as the dividend (實 *shi*). Multiply all the denominators together as the divisor (法 *fa*). Divide the dividend by the divisor. [2]

**Liu Hui's Commentary**
[2] "Cross-multiply (互乘 *hu cheng*) the numerators and denominators" means to homogenize the numerators; "subtract the smaller from the larger" means the fractions can be subtracted after the numerators are homogenized. "Multiply all of the denominators together as the divisor" means to equalize the denominators. Once the denominators are equalized and the numerators are homogenized, divide the dividend by the divisor to obtain the result.

**Li Chunfeng's Commentary**
[1] Li Chunfeng and his associates comment: The numerators and denominators of fractions are not all the same; subtract the smaller from the larger, one wishes to know the remainder. Taking the remainder from the subtraction as the dividend is why this is called "subtraction of fractions."

### **Explanatory Notes**

**Terminology**

(1) 減分 hé fēn: Literally translates to "subtracting fractional parts".

**Principles**

(1) The subtraction of fractions method (減分術 *jian fen shu*) is very similar to the addition of fractions method (合分術 *he fen shu*); the difference being only subtacting all the cross-multiplication products instead of adding.

**Solutions**

[1.10]

Calculate the dividend (實 *shi*).

A

=
8
×
5
−
1
×
9

=
40
−
9

=
31

{\displaystyle
\begin{align}
A &= 8\times 5 - 1 \times 9 \\
&= 40 -9 \\
&= 31
\end{align}}

Divisor (法 *fa*):

# B
9
×
# 5
45

{\displaystyle B = 9 \times 5 = 45 }

Divide the dividend (實 *shi*) by the divisor (法 *fa*):

A
B

=

31
45

{\displaystyle \frac{A}{B} = \frac{31}{45}}

[1.11]

Calculate the dividend (實 *shi*).

A

=
3
×
3
−
1
×
4

=
9
−
4

=
5

{\displaystyle
\begin{align}
A &= 3\times 3 - 1 \times 4 \\
&= 9 -4 \\
&= 5
\end{align}}

Calculate the divisor (法 *fa*):

# B
4
×
# 3
12

{\displaystyle B = 4 \times 3 = 12 }

Divide the dividend (實 *shi*) by the divisor (法 *fa*):

A
B

=

5
12

{\displaystyle \frac{A}{B} = \frac{5}{12}}

## **Comparison of fractions method 課分術 (Problems 1.12 - 1.14)**

荅曰：二十五分之十六多，多二百分之三。
〔一三〕又有九分之八，七分之六。問：孰多？多幾何？
荅曰：九分之八多，多六十三分之二。
〔一四〕又有二十一分之八，五十分之十七。問：孰多？多幾何？
荅曰：二十一分之八多，多一千五十分之四十三。
課分術曰[1]：母互乘子，以少減多，餘為實。母相乘為法。實如法而一，即相多也。[2]

**劉徽注**
[2] 按：此術多與減分義同。唯相多之數，意共減分有異：減分知，求其餘數有幾；課分知，以其餘數相多也。

**李淳風注**
[1] 臣淳風等謹按︰分各異名，理不齊一，校其相多之數，故曰課分也。
[2] 臣淳風等謹按︰此術母互乘子，以少分減多分。

Answer: 16 parts of 25 is larger, and larger by 3 parts of 200.

[1.13] Again, suppose there are 8 parts of 9, and 6 parts of 7. Question: Which is larger? By how much?

Answer: 8 parts of 9 is larger, and larger by 2 parts of 63.

[1.14] Again, suppose there are 8 parts of 21, and 17 parts of 50. Question: Which is larger? By how much?

Answer: 8 parts 21 is larger, and larger by 43 parts of 1050.

Comparison of fractions method (課分術 *ke fen shu*) [1]: Cross-multiply (互乘 *hu cheng*) the numerators and denominators, subtract the smaller from the larger, and take the remainder (餘 *yu*) as the dividend (實 *shi*). Multiply all the denominators together as the divisor (法 *fa*). Divide the dividend by the divisor, the result is the amount one is larger than the other. [2]

**Liu Hui's Commentary**
[2] The meaning of this method is largely the same as subtracting fractions. Only the sense (意 *yi*) of how much one fraction is larger than the other differs from that of subtracting fractions: subtracting fractions means finding out how much their difference is; comparing fractions means determining how much one fraction is larger than the other by the remainder.
**Li Chunfeng's Commentary**
[1] Li Chunfeng and his associates comment: The fractions all have different denominators (名 *ming*), and in principle are not homogenized. Comparing them and the amounts of their differences is why this is called "comparison of fractions".

[2] Li Chunfeng and his associates comment: This method instructs to cross-multiply the numerators and denominators, then subtract the smaller from the larger.

### **Explanatory Notes**

**Terminology**

(1) 課分 kè fēn: Literally translates to "comparing fractional parts".

**Principles**

(1) The algorithm for the comparison of fractions method (課分術 *ke fen shu*) is identical to the subtraction of fractions method (減分術 *jian fen shu*). In addition to performing the arithmetic calculation, one has to judge which fraction is larger.

**Solutions**

[1.12]

Compare each fraction:

5
8

=

5
×
25

8
×
25

=

125
200

16
25

=

16
×
8

25
×
8

=

128
200

{\displaystyle
\begin{align}
\frac{5}{8} &= \frac{5 \times 25}{8 \times 25} = \frac{125}{200} \\[5 pt]
\frac{16}{25} &= \frac{16 \times 8}{25 \times 8} = \frac{128}{200}
\end{align}}

Therefore,

16
25

{\displaystyle \frac{16}{25} }

 is larger than

5
8

{\displaystyle \frac{5}{8}}

. Subtract the larger fraction by the smaller fraction to obtain the difference:

128
200

−

125
200

=

3
200

.

{\displaystyle
\frac{128}{200} - \frac{125}{200} = \frac{3}{200}. }

[1.13]

Compare each fraction:

8
9

=

8
×
7

9
×
7

=

56
63

6
7

=

6
×
9

7
×
9

=

54
63

{\displaystyle
\begin{align}
\frac{8}{9} &= \frac{8 \times 7}{9 \times 7} = \frac{56}{63} \\[5 pt]
\frac{6}{7} &= \frac{6 \times 9}{7 \times 9} = \frac{54}{63}
\end{align}}

Therefore,

8
9

{\displaystyle \frac{8}{9} }

 is larger than

6
7

{\displaystyle \frac{6}{7} }

. Subtract the larger fraction by the smaller fraction to obtain the difference:

56
63

−

54
63

=

2
63

.

{\displaystyle
\frac{56}{63} - \frac{54}{63} = \frac{2}{63}. }

[1.14]

Compare each fraction:

8
21

=

8
×
50

21
×
50

=

400
1050

17
50

=

17
×
21

50
×
21

=

357
1050

{\displaystyle
\begin{align}
\frac{8}{21} &= \frac{8 \times 50}{21 \times 50} = \frac{400}{1050} \\[5 pt]
\frac{17}{50} &= \frac{17 \times 21}{50 \times 21} = \frac{357}{1050}
\end{align}}

Therefore,

8
21

{\displaystyle \frac{8}{21} }

 is larger than

17
50

{\displaystyle \frac{17}{50} }

. Subtract the larger fraction by the smaller fraction to obtain the difference:

400
1050

−

357
1050

=

43
1050

.

{\displaystyle
\frac{400}{1050} - \frac{357}{1050} = \frac{43}{1050}. }

## **Averaging fractions method 平分術 (Problems 1.15 - 1.16)**

荅曰：減四分之三者二，三分之二者一，並，以益三分之一，而各平於十二分之七。
〔一六〕又有二分之一，三分之二，四分之三。問：減多益少，各幾何而平？
荅曰：減三分之二者一，四分之三者四，並以益二分之一，而各平於三十六分之二十三。
**平分〈臣淳風等謹按︰平分知，諸分參差，欲令齊等，減彼之多，增此之少，故曰平分也。〉術曰**：母互乘子，〈齊其子也。〉副並為平實。〈臣淳風等謹按︰母互乘子，副並為平實知，定此平實主限，眾子所當損益知，限為平。〉母相乘為法。〈「母相乘為法」知，亦齊其子，又同其母。〉以列數乘未並者各自為列實。亦以列數乘法。〈此當副置列數除平實，若然則重有分，故反以列數乘同齊。臣淳風等謹又按︰問︰雲所平之分多少不定，或三或二，列位無常。平三知，置位三重；平二知，置位二重。凡此之例，一準平分不可預定多少，故直雲列數而已。〉以平實減列實，餘，約之為所減。並所減以益於少。以法命平實，各得其平。

**劉徽注**
[2] 按：此術多與減分義同。唯相多之數，意共減分有異：減分知，求其餘數有幾；課分知，以其餘數相多也。

**李淳風注**
[1] 臣淳風等謹按︰分各異名，理不齊一，校其相多之數，故曰課分也。
[2] 臣淳風等謹按︰此術母互乘子，以少分減多分。

Answer: From 3 parts of 4, subtract 2 [parts of 12]; from 2 parts 3, subtract 1 [parts 12]. Then add them together [3 parts 12], and add it to 1 parts 3, which gives the average for all [fractions] being 7 parts of 12.

[1.16] **Again, suppose there are 8 parts of 9, and 6 parts of 7. Question: Which is larger? By how much?**

Answer: 8 parts of 9 is larger, and larger by 2 parts of 63.

Comparison of fractions method (課分術 *ke fen shu*) [1]: Cross-multiply (互乘 *hu cheng*) the numerators and denominators, subtract the smaller from the larger, and take the remainder (餘 *yu*) as the dividend (實 *shi*). Multiply all the denominators together as the divisor (法 *fa*). Divide the dividend by the divisor, the result is the amount one is larger than the other. [2]

**Liu Hui's Commentary**
[2] The meaning of this method is largely the same as subtracting fractions. Only the sense (意 *yi*) of how much one fraction is larger than the other differs from that of subtracting fractions: subtracting fractions means finding out how much their difference is; comparing fractions means determining how much one fraction is larger than the other by the remainder.
**Li Chunfeng's Commentary**
[1] Li Chunfeng and his associates comment: The fractions all have different denominators (名 *ming*), and in principle are not homogenized. Comparing them and the amounts of their differences is why this is called "comparison of fractions".

[2] Li Chunfeng and his associates comment: This method instructs to cross-multiply the numerators and denominators, then subtract the smaller from the larger.

### **Explanatory Notes**

**Terminology**

(1) 平分 kè fēn: Literally translates to "comparing fractional parts".

**Principles**

(1) The algorithm for the comparison of fractions method (課分術 *ke fen shu*) is identical to the subtraction of fractions method (減分術 *jian fen shu*). In addition to performing the arithmetic calculation, one has to judge which fraction is larger.

**Solutions**

[1.15]

Compare each fraction:

5
8

=

5
×
25

8
×
25

=

125
200

16
25

=

16
×
8

25
×
8

=

128
200

{\displaystyle
\begin{align}
\frac{5}{8} &= \frac{5 \times 25}{8 \times 25} = \frac{125}{200} \\[5 pt]
\frac{16}{25} &= \frac{16 \times 8}{25 \times 8} = \frac{128}{200}
\end{align}}

Therefore,

16
25

{\displaystyle \frac{16}{25} }

 is larger than

5
8

{\displaystyle \frac{5}{8}}

. Subtract the larger fraction by the smaller fraction to obtain the difference:

128
200

−

125
200

=

3
200

.

{\displaystyle
\frac{128}{200} - \frac{125}{200} = \frac{3}{200}. }

[1.16]

Compare each fraction:

8
9

=

8
×
7

9
×
7

=

56
63

6
7

=

6
×
9

7
×
9

=

54
63

{\displaystyle
\begin{align}
\frac{8}{9} &= \frac{8 \times 7}{9 \times 7} = \frac{56}{63} \\[5 pt]
\frac{6}{7} &= \frac{6 \times 9}{7 \times 9} = \frac{54}{63}
\end{align}}

Therefore,

8
9

{\displaystyle \frac{8}{9} }

 is larger than

6
7

{\displaystyle \frac{6}{7} }

. Subtract the larger fraction by the smaller fraction to obtain the difference:

56
63

−

54
63

=

2
63

.

{\displaystyle
\frac{56}{63} - \frac{54}{63} = \frac{2}{63}. }