Chinese numerals

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Chinese numerals are words and characters used to denote numbers in Chinese.

Today speakers of Chinese use three written numeral systems: the system of Arabic numerals used world-wide, and two indigenous systems. The more familiar indigenous system is based on Chinese characters that correspond to numerals in the spoken language. These are shared with other languages of the Chinese cultural sphere such as Japanese, Korean and Vietnamese. Most people and institutions in China primarily use the Arabic system for convenience, with traditional Chinese numerals used in finance, mainly for writing amounts on checks, banknotes, some ceremonial occasions, some boxes, and on commercials.[citation needed]

The other indigenous system is the Suzhou numerals, or huama, a positional system, the only surviving form of the rod numerals. These were once used by Chinese mathematicians, and later in Chinese markets, such as those in Hong Kong before the 1990s, but have been gradually supplanted by Arabic (and also Roman) numerals.

Written numbers[edit]

Chinese and Arabic numerals may coexist, as on this kilometer marker: 1620 km on Hwy G209 (G二〇九)

The Chinese character numeral system consists of the Chinese characters used by the Chinese written language to write spoken numerals. Similar to spelling-out numbers in English (e.g., "one thousand nine hundred forty-five"), it is not an independent system per se. Since it reflects spoken language, it does not use the positional system as in Arabic numerals, in the same way that spelling out numbers in English does not.

Characters used to represent numbers[edit]

Standard numbers[edit]

There are characters representing the numbers zero through nine, and other characters representing larger numbers such as tens, hundreds, thousands and so on. There are two sets of characters for Chinese numerals: one for everyday writing and one for use in commercial or financial contexts known as dàxiě (simplified Chinese: 大写; traditional Chinese: 大寫; literally: "big writing"). The latter arose because the characters used for writing numerals are geometrically simple, so simply using those numerals cannot prevent forgeries in the same way spelling numbers out in English would.[1] A forger could easily change the everyday characters 三十 (30) to 五千 (5000) just by adding a few strokes. That would not be possible when writing using the financial characters 參拾 (30) and 伍仟 (5000). They are also referred to as "banker's numerals", "anti-fraud numerals", or "banker's anti-fraud numerals". For the same reason, rod numerals were never used in commercial records.

T denotes Traditional Chinese characters, S denotes Simplified Chinese characters.

Financial Normal Value Pīnyīn
(Mandarin)
Jyutping
(Cantonese)
Pe̍h-ōe-jī
(Hokkien)
Notes
Character (T) Character (S) Character (T) Character (S)
/ 0 líng ling4 lîng Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero. The traditional 零 is more often used in schools. 〇 is not a standard Chinese character, because Chinese characters never contain ovals or circles. In Unicode, 〇 is treated as a Chinese symbol or punctuation, rather than a Chinese ideograph.
1 jat1 it/chit Also (obsolete financial), can be easily manipulated into 弍 (two) or 弎 (three).
2 èr ji6 jī/nn̄g Also (obsolete financial), can be easily manipulated into 弌 (one) or 弎 (three).
Also (T) or (S), see Characters with regional usage section.
3 sān saam1 sam/saⁿ Also (obsolete financial), can be easily manipulated into 弌 (one) or 弍 (two).
Also (T) or (S) sān. Vulgar variants include , .
4 sei3 sù/sì Also (obsolete financial)[2]
5 ng5 ngō/gōo  
6 liù luk6 lio̍k/la̍k  
7 cat1 chhit  
8 baat3 pat/peh  
9 jiǔ gau2 kiú/káu  
10 shí sap6 si̍p/tsa̍p Although some people use as financial[citation needed], it is not ideal because it can be easily manipulated into 伍 (five) or 仟 (thousand).
100 bǎi baak3 pah  
1,000 qiān cin1 chhian
104 wàn maan6 bān Chinese numbers group by ten-thousands; see Reading and transcribing numbers below.
亿 108 jik1 ik For variant meanings and words for higher values, see Large numbers below and ja:大字 (数字).

Characters with regional usage[edit]

Financial Normal Value Pinyin (Mandarin) Standard alternative Notes
1 yāo Literally means "the smallest". It is used in Mandarin to unambiguously pronounce "#1" in a series of one (一) such as phone numbers and ID numbers, because reading them together in a row is not distinguishable (e.g. 一一一 would be pronounced as "yao-yao-yao" instead of sounding like "YEEEEEE"). In Taiwan, it is only used by soldiers, police, and emergency services. This usage is not observed in Cantonese except for 十三幺 (a special winning hand) in Mahjong.
(T) or
(S)
2 liǎng A very common alternative way of saying "two". Its usage varies from dialect to dialect, even person to person. For example "2222" can read as "二千二百二十二", "兩千二百二十二" or even "兩千兩百二十二" in Mandarin. See Reading and transcribing numbers section below.
3 In regional dialects of Northeastern Mandarin, 仨 represents a "lazy" pronunciation of three within the local dialect. It can be used as a general number to represent "three" (e.g.第仨号 dì sā hào, "number three"; 星期仨 xīngqīsā, "Wednesday"), or as an alternative for 三个 "three of" (e.g. 我们仨 Wǒmen sā, "the three of us", as opposed to 我们三个 Wǒmen sān gè). Regardless of usage, a measure word (such as 个) never follows after 仨.
10 In Cantonese speech, when 十 is used in the middle of a number, preceded by a multiplier and followed by a ones digit, 十 becomes 呀 (aa6), e.g. 六呀三, 63. This usage is not observed in Mandarin.
廿 20 niàn 二十 The written form is still used to refer to dates, especially Chinese calendar dates.
Spoken form is still used in various dialects of Chinese. See Reading and transcribing numbers section below.
In Cantonese, 廿 (jaa6) must be followed by another digit 1-9 (e.g. 廿三, 23), or in a phrase like 廿幾 ("twenty-something"); it is not used by itself to mean 20.
is a rare variant.
30 三十 The written form is still used to abbreviate date references in Chinese. For example, May 30 Movement (五卅運動).
Spoken form is still used in various dialects of Chinese. See Reading and transcribing numbers section below. As with 廿, 卅 must be used with another number to mean 卅幾 ("thirty-something") in Cantonese. Used in other dialects too, as well as historical writings.
40 四十 Spoken form is still used in various dialects of Chinese, albeit very rare, as well as historical writings written in Classical Chinese.
As with 廿, 卌 must be used with another number to mean 卌幾 ("forty-something") in Cantonese. The usage of the word 卌 is done in the following matter, "sei(四) ah ##", or "4 ah ##". Thus 41 would be pronounced "sei ah yat", i.e. "four ah one".
200 二百 Very rarely used, one common example is the literature 《皕宋樓》.

Characters with military usage[edit]

In the PLA, some numbers will have altered names when used for clearer radio communications. They are:

  • 0: renamed 洞 (dòng) lit. hole
  • 1: renamed 幺 (yāo) lit. small
  • 2: renamed 两 (liǎng) lit. double
  • 4: renamed 刀 (dāo) lit. knife
  • 7: renamed 拐 (guǎi) lit. turn
  • 9: renamed 勾 (gōu) lit. hook

Large numbers[edit]

For numbers larger than 10,000, similarly to the long and short scales in the West, there have been four systems in ancient and modern usage. The original one, with unique names for all powers of ten up to the 14th, is ascribed to the Yellow Emperor in the 6th century book by Zhen Luan, Wujing suanshu (Arithmetic in Five Classics). In modern Chinese only the second system is used, in which the same ancient names are used, but each represents a number 10,000 (myriad, wàn) times the previous:

Character (T) Factor of increase
Character (S) 亿
Pinyin zhào jīng gāi ráng gōu jiàn zhēng zài
Jyutping jik1 siu6 ging1 goi1 zi2 joeng4 kau1 gaan3 zing3 zoi3
Hokkien POJ ik tiāu kiann gai tsí liōng kau kán tsiànn tsài
Alternative /
1 105 106 107 108 109 1010 1011 1012 1013 1014 Each numeral is 10 (十 shí) times the previous.
2
(current usage)
108 1012 1016 1020 1024 1028 1032 1036 1040 1044 Each numeral is 10,000 (萬 (T) or 万 (S) wàn) times the previous.
3 108 1016 1024 1032 1040 1048 1056 1064 1072 1080 Each numeral is 108 (萬乘以萬 (T) or 万乘以万 (S) wàn chéng yǐ wàn, 10000 times 10000) times the previous.
4 108 1016 1032 1064 10128 10256 10512 101024 102048 104096 Each numeral is the square of the previous.

In practice, this situation does not lead to ambiguity, with the exception of 兆 (zhào), which means 1012 according to the system in common usage throughout the Chinese communities as well as in Japan and Korea, but has also been used for 106 in recent years (especially in mainland China for megabyte). To avoid problems arising from the ambiguity, the PRC government never uses this character in official documents, but uses 万亿 (wànyì) instead. The ROC government in Taiwan uses 兆 (zhào) to mean 1012 in official documents.

Numbers from Buddhism[edit]

Numerals beyond 載 zài come from Buddhist texts in Sanskrit, but are mostly found in ancient texts.

Character (T) Character (S) Pinyin Jyutping Hokkien POJ Value Notes
gik1 ki̍k 1048 Literally means "Extreme"
恆河沙 恒河沙 héng hé shā hang4 ho4 sa1 hîng-hô-sua 1052[citation needed] Literally means "Sands of the Ganges"; a metaphor used in a number of Buddhist texts referring to the grains of sand in the Ganges River.
阿僧祇 ā sēng qí aa1 zang1 zi2 a-sing-kî 1056 From Sanskrit Asaṃkhyeya
那由他 nà yóu tā naa5 jau4 taa1 ná-iû-thann 1060 From Sanskrit Nayuta
不可思議 不可思议 bùkě sīyì bat1 ho2 si1 ji3 put-khó-su-gī 1064 Literally translated as "unfathomable".
無量 无量 wú liàng mou4 loeng6 bô-liōng 1068 Literally translated as "without measure"
大數 大数 dà shù daai6 sou3 ta-siàu 1072 Literally translated as "a large number"

Small numbers[edit]

The following are characters used to denote small order of magnitude in Chinese historically. With the introduction of SI units, some of them have been incorporated as SI prefixes, while the rest have fallen into disuse.

Character(s) (T) Character(s) (S) Pinyin Value Notes
涅槃寂靜 涅槃寂静 niè pán jì jìng 10−24 Literally, "Nirvana's Tranquility"

(T) or (S) corresponds to the SI prefix yocto-.

阿摩羅 阿摩罗 ā mó luó 10−23 (Ancient Chinese, from Sanskrit अमल amala)
阿頼耶 阿赖耶 ā lài yē 10−22 (Ancient Chinese, from Sanskrit आलय ālaya)
清靜 清净 qīng jìng 10−21 Literally, "Quiet"

(T) or (S) corresponds to the SI prefix zepto-.

虛空 虚空 xū kōng 10−20 Literally, "Void"
六德 liù dé 10−19 (Ancient Chinese)
剎那 刹那 chà nà 10−18 Literally, "Brevity", from Sanskrit क्षण ksaṇa

corresponds to the SI prefix atto-.

彈指 弹指 tán zhǐ 10−17 Literally, "Flick of a finger"
瞬息 shùn xī 10−16 Literally, "Moment of Breath"
須臾 须臾 xū yú 10−15 (Ancient Chinese)

(T) or (S) corresponds to the SI prefix femto-.

逡巡 qūn xún 10−14 (Ancient Chinese)
模糊 mó hu 10−13 Literally, "Blurred"
10−12 (Ancient Chinese)

corresponds to the SI prefix pico-.

miǎo 10−11 (Ancient Chinese)
āi 10−10 (Ancient Chinese)
chén 10−9 Literally, "Dust"

(T) or (S) corresponds to the SI prefix nano-.

shā 10−8 Literally, "Sand"
xiān 10−7 Literally, "Fiber"
wēi 10−6 still in use, corresponds to the SI prefix micro-.
10−5 (Ancient Chinese)
10−4 "Thread"
háo 10−3 also .

still in use, corresponds to the SI prefix milli-.

10−2 also .

still in use, corresponds to the SI prefix centi-.

fēn 10−1 still in use, corresponds to the SI prefix deci-.

SI prefixes[edit]

In the People's Republic of China, the translations for the SI prefixes in 1981 were different from those used today. The larger (兆, 京, 垓, 秭, 穰) and smaller Chinese numerals (微, 纖, 沙, 塵, 渺) were defined as translations for the SI prefixes as mega, giga, tera, peta, exa, micro, nano, pico, femto, atto, resulting in the creation of more values for each numeral.[3]

Republic of China (Taiwan) defined 百萬 as the translation for mega. This translation is widely used in official documents, academic communities, informational industries, etc. However, the civil broadcasting industries sometimes use 兆赫 to represent "megahertz".

Today, both the governments of the People's Republic of China (Mainland China, Hong Kong and Macau) and Republic of China (Taiwan) use phonetic transliterations for the SI prefixes. However, the governments have each chosen different Chinese characters for certain prefixes. The following table lists the two different standards together with the early translation.

SI Prefixes
Value Symbol English Early translation PRC standard ROC standard
1024 Y yotta-   yáo yòu
1021 Z zetta-   jiē
1018 E exa- [3] ráng ài ài
1015 P peta- [3] pāi pāi
1012 T tera- [3] gāi tài zhào
109 G giga- [3] jīng
106 M mega- [3] zhào zhào 百萬 bǎiwàn
103 k kilo- qiān qiān qiān
102 h hecto- bǎi bǎi bǎi
101 da deca- shí shí shí
100 (None) one  
10−1 d deci- fēn fēn fēn
10−2 c centi-
10−3 m milli- háo háo háo
10−6 µ micro- [3] wēi wēi wēi
10−9 n nano- [3] xiān nài
10−12 p pico- [3] shā
10−15 f femto- [3] chén fēi fēi
10−18 a atto- [3] miǎo à à
10−21 z zepto-   jiè
10−24 y yocto-   yāo yōu

Reading and transcribing numbers[edit]

Whole numbers[edit]

Multiple-digit numbers are constructed using a multiplicative principle; first the digit itself (from 1 to 9), then the place (such as 10 or 100); then the next digit.

In Mandarin, the multiplier (liǎng) is often used rather than (èr) for all numbers greater than 200 with the "2" numeral (although as noted earlier this varies from dialect to dialect and person to person). Use of both 兩 (liǎng) or 二 (èr) are acceptable for the number 200. When writing in the Cantonese dialect, 二 (yi6) is used to represent the "2" numeral for all numbers. In the southern Min dialect of Chaozhou (Teochew), 兩 (no6) is used to represent the "2" numeral in all numbers from 200 onwards. Thus:

Number Structure Characters
Mandarin Cantonese Chaozhou Shanghainese
60 [6] [10] 六十 六十 六十 六十
20 [2] [10] or [20] 二十 二十 or 廿 二十 廿
200 [2] (èr) or (liǎng) [100] 二百 or 兩百 二百 or 兩百 兩百 兩百
2000 [2] (èr) or (liǎng) [1000] 二千 or 兩千 二千 or 兩千 兩千 兩千
45 [4] [10] [5] 四十五 四十五 or 卌五 四十五 四十五
2,362 [2] [1,000] [3] [100] [6] [10] [2] 兩千三百六十二 二千三百六十二 兩千三百六十二 兩千三百六十二

For the numbers 11 through 19, the leading "one" (一) is usually omitted. In some dialects, like Shanghainese, when there are only two significant digits in the number, the leading "one" and the trailing zeroes are omitted. Sometimes, the one before "ten" in the middle of a number, such as 213, is omitted. Thus:

Number Strict Putonghua Colloquial or dialect usage
Structure Characters Structure Characters
14 [10] [4] 十四    
12000 [1] [10000] [2] [1000] 一萬兩千 [1] [10000] [2] 一萬二 or 萬二
114 [1] [100] [1] [10] [4] 一百一十四 [1] [100] [10] [4] 一百十四
1158 [1] [1000] [1] [100] [5] [10] [8] 一千一百五十八 See note 1 below

Notes:

  1. Nothing is ever omitted in large and more complicated numbers such as this.

In certain older texts like the Protestant Bible or in poetic usage, numbers such as 114 may be written as [100] [10] [4] (百十四).

For numbers larger than a myriad, the same grouping system used in English applies, except in groups of four places (myriads) rather than in groups of three (thousands). Hence it is more convenient to think of numbers here as in groups of four, thus 1,234,567,890 is regrouped here as 12,3456,7890. Larger than a myriad, each number is therefore four zeroes longer than the one before it, thus 10000 × wàn (萬) = yì (億). If one of the numbers is between 10 and 19, the leading "one" is omitted as per the above point. Hence (numbers in parentheses indicate that the number has been written as one number rather than expanded):

Number Structure Characters
12,345,678,902,345
(12,3456,7890,2345)
(12) [1,0000,0000,0000] (3456) [1,0000,0000] (7890) [1,0000] (2345) 十二兆三千四百五十六億七千八百九十萬兩千三百四十五

Interior zeroes before the unit position (as in 1002) must be spelt explicitly. The reason for this is that trailing zeroes (as in 1200) are often omitted as shorthand, so ambiguity occurs. One zero is sufficient to resolve the ambiguity. Where the zero is before a digit other than the units digit, the explicit zero is not ambiguous and is therefore optional, but preferred. Thus:

Number Structure Characters
205 [2] [100] [0] [5] 二百〇五
100,004
(10,0004)
[10] [10,000] [0] [4] 十萬〇四
10,050,026
(1005,0026)
(1005) [10,000] (26) or
(1005) [10,000] (026)
一千〇五萬〇二十六 or
一千〇五萬二十六

Fractional values[edit]

To construct a fraction, the denominator is written first, followed by 分之 ("parts of") and then the numerator. This is the opposite of how fractions are read in English, which is numerator first. Each half of the fraction is written the same as a whole number. Mixed numbers are written with the whole-number part first, followed by ("and"), then the fractional part.

Fraction Structure Characters
2/3 [3] [parts of] [2] 三分之二
15/32 [3] [10] [2] [parts of] [10] [5] 三十二分之十五
1/3000 [3] [1000] [parts of] [1] 三千分之一
3 5/6 [3] [and] [6] [parts of] [5] 三又六分之五

Percentages are constructed similarly, using 百 (100) as the denominator. The 一 (one) before 百 is omitted.

Percentage Structure Characters
25% [100] [parts of] [2] [10] [5] 百分之二十五
110% [100] [parts of] [1] [100] [1] [10] 百分之一百一十

Decimal numbers are constructed by first writing the whole number part, then inserting a point (simplified Chinese: ; traditional Chinese: ; pinyin: diǎn), and finally the decimal expression. The decimal expression is written using only the digits for 0 to 9, without multiplicative words.

Decimal expression Structure Characters
16.98 [10] [6] [point] [9] [8] 十六點九八
12345.6789 [1] [10000] [2] [1000] [3] [100] [4] [10] [5] [point] [6] [7] [8] [9] 一萬兩千三百四十五點六七八九
75.4025 [7] [10] [5] [point] [4] [0] [2] [5] 七十五點四〇二五 or 七十五點四零二五
0.1 [0] [point] [1] 零點一

Ordinal numbers[edit]

Ordinal numbers are formed by adding ("sequence") before the number.

Ordinal Structure Characters
1st [sequence] [1] 第一
2nd [sequence] [2] 第二
82nd [sequence] [8] [10] [2] 第八十二

Negative numbers[edit]

Negative numbers are formed by adding (simplified Chinese: ; traditional Chinese: , Pinyin: fù, Jyuping: fu6) before the number.

Number Structure Characters
-1158 [negative] [1] [1000] [1] [100] [5] [10] [8] 負一千一百五十八
-3 5/6 [negative] [3] [and] [6] [parts of] [5] 負三又六分之五
-75.4025 [negative] [7] [10] [5] [point] [4] [0] [2] [5] 負七十五點四零二五

Usage[edit]

Chinese grammar requires the use of classifiers (measure words) when a numeral is used together with a noun to express a quantity. For example, "three people" is expressed as 三个人 sān ge rén, "three GE person", where 个 ge is a classifier. There exist many different classifiers, for use with different sets of nouns, although 个 is the most common, and may be used informally in place of other classifiers. For details see the article Chinese classifier.

Chinese uses cardinal numbers in certain situations in which English would use ordinals. For example, 三楼 sān lóu (literally "three story") means "third floor" ("second floor" in British numbering). Likewise, 二十一世纪 èrshí yī shìjì (literally "twenty-one century") is used for "21st century".[4]

Numbers of years are commonly spoken as a sequence of digits, as in 二零零一 èr líng líng yī ("two zero zero one") for the year 2001.[5] Names of months and days (in the Western system) are also expressed using numbers: 一月 yīyuè ("one month") for January, etc.; and 星期一 xīngqīyī ("week one") for Monday, etc. (although Sunday is 星期日 xīngqīrì, or informally 星期天 xīngqītiān, "week day"). Full dates are usually written in the format 2001年1月20日 for January 20, 2001 (using 年 nián "year", 月 yuè "month", and 日 "day") – all the numbers are read as cardinals, not ordinals, and the year is read as a sequence of digits as just mentioned. (See also Chinese calendar.)

Counting rod and Suzhou numerals[edit]

Counting rod numerals
Main article: Suzhou numerals

In the same way that Roman numerals were standard in ancient and medieval Europe for mathematics and commerce, the Chinese formerly used the rod numerals, which is a positional system. The Suzhou numerals (simplified Chinese: 苏州花码; traditional Chinese: 蘇州花碼; pinyin: Sūzhōu huāmǎ) system is a variation of the Southern Song rod numerals. Nowadays, the huāmǎ system is only used for displaying prices in Chinese markets or on traditional handwritten invoices.

Hand gestures[edit]

There is a common method of using of one hand to signify the numbers one to ten. While the five digits on one hand can express the numbers one to five, six to ten have special signs that can be used in commerce or day-to-day communication.

Historical use of numerals in China[edit]

Shang oracle bone numerals of 14th century B.C.[6]
West Zhou dynasty bronze script
rod numeral place value from Yongle Encyclopedia
Japanese counting board with grids

Most Chinese numerals of later periods were descendants of the Shang dynasty oracle numerals of the 14th century BC. The oracle bone script numerals were found on tortoise shell and animal bones. In early civilizations, the Shang were able to express any numbers, however large, with only nine symbols and a counting board.[7]

Some of the bronze script numerals such as 1, 2, 3, 4, 10, 11, 12, and 13 became part of the system of rod numerals.

In this system, horizontal rod numbers are used for the tens, thousands, hundred thousands etc. Sun Tzu wrote that "one is vertical, ten is horizontal".[8]

Counting rod v7.png Counting rod h1.png Counting rod v8.png Counting rod h2.png Counting rod v4.png
7 1 8 2 4

The counting rod numerals system has place value and decimal numerals for computation, and was used widely by Chinese merchants, mathematicians and astronomers from the Han dynasty to the 16th century.

Alexander Wylie, Christian missionary to China, in 1853 already refuted the notion that "the Chinese numbers were written in words at length", and stated that in ancient China, calculation was carried out by means of counting rods, and "the written character is evidently a rude presentation of these". After being introduced to the rod numerals, he said "Having thus obtained a simple but effective system of figures, we find the Chinese in actual use of a method of notation depending on the theory of local value [i.e. place-value], several centuries before such theory was understood in Europe, and while yet the science of numbers had scarcely dawned among the Arabs."[9]

During the Ming and Qing dynasties (when Arabic numerals were first introduced into China), some Chinese mathematicians used Chinese numeral characters as positional system digits. After the Qing period, both the Chinese numeral characters and the Suzhou numerals were replaced by Arabic numerals in mathematical writings.

Cultural influences[edit]

Traditional Chinese numeric characters are also used in Japan and Korea and were used in Vietnam before the 20th century. In vertical text (that is, read top to bottom), using characters for numbers is the norm, while in horizontal text, Arabic numerals are most common. Chinese numeric characters are also used in much the same formal or decorative fashion that Roman numerals are in Western cultures. Chinese numerals may appear together with Arabic numbers on the same sign or document.

See also[edit]

References[edit]

  1. ^ 大寫數字『
  2. ^ Note: Variant Chinese character of 肆, with a 镸 radical next to a 四 character. Not all browsers may be able to display this character, which forms a part of the Unicode CJK Unified Ideographs Extension A group.
  3. ^ a b c d e f g h i j k (Chinese) 1981 Gazette of the State Council of the People's Republic of China, No. 365, page 575, Table 7: SI prefixes
  4. ^ Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 12.
  5. ^ Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 13.
  6. ^ The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, Table 20, p. 6, Cambridge University Press ISBN 0-521-23582-0
  7. ^ The Shorter Science & Civilisation in China Vol 2, An abridgement by Colin Ronan of Joseph Needham's original text, p5,Cambridge University Press ISBN 0-521-23582-0
  8. ^ Chinese Wikisource 孫子算經: 先識其位,一從十橫,百立千僵,千十相望,萬百相當。
  9. ^ Alexander Wylie, Jottings on the Sciences of the Chinese, North Chinese Herald, 1853, Shanghai

External links[edit]