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Today, speakers of Chinese use three written numeral systems: the system of Hindu-Arabic numerals used worldwide, 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 and Taiwan primarily use the Hindu-Arabic or mixed Arabic-Chinese systems for convenience, with traditional Chinese numerals used in finance, mainly for writing amounts on checks, banknotes, some ceremonial occasions, some boxes, and on commercials.
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 Hindu-Arabic (and also Roman) numerals.
Characters used to represent numbersEdit
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.
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, known as xiǎoxiě (simplified Chinese: 小写; traditional Chinese: 小寫; literally: 'small writing, i.e. lower case'), and one for use in commercial or financial contexts, known as dàxiě (simplified Chinese: 大写; traditional Chinese: 大寫; literally: 'big writing, i.e. upper case'). 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. 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 叁拾 (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.
|Character (T)||Character (S)||Character (T)||Character (S)|
|零||零/〇||0||líng||ling4||khòng/lêng||Usually 零 is preferred, but in some areas, 〇 may be a more common informal way to represent zero. The original Chinese character is 空 or 〇, 零 is referred as remainder something less than 1 yet not nil [說文] referred. The traditional 零 is more often used in schools. In Unicode, 〇 is treated as a Chinese symbol or punctuation, rather than a Chinese ideograph.|
|壹||一||1||yī||jat1||it/chi̍t||Also 弌 (obsolete financial), can be easily manipulated into 弍 (two) or 弎 (three).|
|貳||贰||二||2||èr||ji6||jī/nn̄g||Also 弍 (obsolete financial), can be easily manipulated into 弌 (one) or 弎 (three).|
Also 兩 (T) or 两 (S), see Characters with regional usage section.
|叄||叁||三||3||sān||saam1||sam/saⁿ||Also 弎 (obsolete financial), can be easily manipulated into 弌 (one) or 弍 (two).|
Also 參 (T) or 参 (S) sān.
|肆||四||4||sì||sei3||sù/sì||Also 䦉 (obsolete financial)|
|拾||十||10||shí||sap6||si̍p/cha̍p||Although some people use 什 as financial, it is not ideal because it can be easily manipulated into 伍 (five) or 仟 (thousand).|
|萬||萬||万||104||wàn||maan6||bān||Chinese numbers group by ten-thousands; see Reading and transcribing numbers below.|
|億||億||亿||108||yì||jik1||ek||For variant meanings and words for higher values, see Large numbers below and ja:大字 (数字).|
Characters with regional usageEdit
|Financial||Normal||Value||Pinyin (Mandarin)||Standard alternative||Notes|
|幺||1||yāo||一||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.|
|2||liǎng||二||Replaces 二 before a classifier. For example, "two people" is "两个人", not "二个人". It appears where "a pair of" would in English, but 两 is always used in such cases. It is also used for numbers, with usage varying from dialect to dialect, even person to person. For example, "2222" can be 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.|
|倆||2||liǎ||兩||In regional dialects of Northeastern Mandarin, 倆 represents a "lazy" pronunciation of 兩 within the local dialect. It can be used as an alternative for 兩个 "two of" (e.g. 我们倆 Wǒmen liǎ, "the two of us", as opposed to 我们兩个 Wǒmen liǎng gè). A measure word (such as 个) never follows after 倆.|
|仨||3||sā||三||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.|
|呀||10||yà||十||In spoken Cantonese, 呀 (aa6) can be used in place of 十 when it is used in the middle of a number, preceded by a multiplier and followed by a ones digit, e.g. 六呀三, 63; it is not used by itself to mean 10. This usage is not observed in Mandarin.|
|念||廿||20||niàn||二十||A contraction of 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 spoken Cantonese, 廿 (jaa6) can be used in place of 二十 when followed by another digit such as in numbers 21-29 (e.g. 廿三, 23), a measure word (e.g. 廿個), a noun, or in a phrase like 廿幾 ("twenty-something"); it is not used by itself to mean 20.
卄 is a rare variant.
|卅||30||sà||三十||A contraction of 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.
In spoken Cantonese, 卅 (saa1) can be used in place of 三十 when followed by another digit such as in numbers 31-39, a measure word (e.g. 卅個), a noun, or in phrases like 卅幾 ("thirty-something"); it is not used by itself to mean 30. When spoken 卅 is pronounced as 卅呀 (saa1 aa6). Thus 卅一 (31), is pronounced as saa1 aa6 jat1.
|卌||40||xì||四十||A contraction of 四十. Found in historical writings written in Classical Chinese.|
Spoken form is still used in various dialects of Chinese, albeit very rare. See Reading and transcribing numbers section below.
In spoken Cantonese 卌 (sei3) can be used in place of 四十 when followed by another digit such as in numbers 41-49, a measure word (e.g. 卌個), a noun, or in phrases like 卌幾 ("forty-something"); it is not used by itself to mean 40. When spoken, 卌 is pronounced as 卌呀 (sei3 aa6). Thus 卌一 (41), is pronounced as sei3 aa6 jat1.
|皕||200||bì||二百||Very rarely used; one example is in the name of a library in Huzhou, 皕宋樓 (Bìsòng Lóu).|
Characters with military usageEdit
- 0: renamed 洞 (dòng) lit. hole
- 1: renamed 幺 (yāo) lit. small
- 2: renamed 两 (liǎng) lit. double
- 7: renamed 拐 (guǎi) lit. cane, turn
- 9: renamed 勾 (gōu) lit. hook
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, 萬 wàn) times the previous:
|Character (T)||萬||億||兆||京||垓||秭||穰||溝||澗||正||載||Factor of increase|
Each numeral is 10 (十 shí) times the previous.
(萬進, current usage)
Each numeral is 10,000 (萬 (T) or 万 (S) wàn) times the previous.
Starting with 亿, 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.
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) or 太 (tài, as the translation for tera) instead. Partly due to this, combinations of 万 and 亿 are often used instead of the larger units of the traditional system as well, for example 亿亿 (yìyì) instead of 京. The ROC government in Taiwan uses 兆 (zhào) to mean 1012 in official documents.
Large numbers from BuddhismEdit
|Character (T)||Character (S)||Pinyin||Jyutping||Hokkien POJ||Value||Notes|
|極||极||jí||gik1||ke̍k||1048||Literally means "Extreme".|
|恆河沙||恒河沙||héng hé shā||hang4 ho4 sa1||hêng-hô-soa||1052||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 kei4||a-seng-kî||1056||From Sanskrit Asaṃkhyeya असंख्येय, meaning "incalculable, innumerable, infinite".|
|那由他||nà yóu tā||naa5 jau4 taa1||ná-iû-thaⁿ||1060||From Sanskrit Nayuta नयुत, meaning "myriad".|
|不可思議||不可思议||bùkě sīyì||bat1 ho2 si1 ji3||put-khó-su-gī||1064||Literally translated as "unfathomable". This word is commonly used in Chinese as a chengyu, meaning "unimaginable", instead of its original meaning of the number 1064.|
|無量大數||无量大数||wú liàng dà shù||mou4 loeng6 daai6 sou3||bû-liōng tāi-siàu||1068||"无量" literally translated as "without measure", and can mean 1068. This word is also commonly used in Chinese as a commendatory term, means "no upper limit". E.g.: 前途无量 lit. front journey no limit, which means "a great future". "大数" literally translated as "a large number; the great number", and can mean 1072.|
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|
|微||wēi||10−6||still in use, corresponds to the SI prefix micro-.|
|絲||丝||sī||10−4||also 秒. |
|毫||háo||10−3||also 毛. |
|厘||lí||10−2||also 釐. |
|分||fēn||10−1||still in use, corresponds to the SI prefix deci-.|
Small numbers from BuddhismEdit
|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"|
|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"|
|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|
|彈指||弹指||tán zhǐ||10−17||Literally, "Flick of a finger". Still commonly used in the phrase "弹指一瞬间" (A very short time)|
|瞬息||shùn xī||10−16||Literally, "Moment of Breath". Still commonly used in Chengyu "瞬息万变" (Many things changed in a very short time)|
|須臾||须臾||xū yú||10−15||(Ancient Chinese, rarely used in Modern Chinese as "a very short time")|
|逡巡||qūn xún||10−14||(Ancient Chinese)|
|模糊||mó hu||10−13||Literally, "Blurred"|
In the People's Republic of China, the early translation for the SI prefixes in 1981 was different from those used today. The larger (兆, 京, 垓, 秭, 穰) and smaller Chinese numerals (微, 纖, 沙, 塵, 渺) were defined as translation for the SI prefixes as mega, giga, tera, peta, exa, micro, nano, pico, femto, atto, resulting in the creation of yet more values for each numeral.
The Republic of China (Taiwan) defined 百萬 as the translation for mega and 兆 as the translation for tera. 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, the governments of both China and 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.
|Value||Symbol||English||Early translation||PRC standard||ROC standard|
Reading and transcribing numbersEdit
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 兩 (liǎng) is often used rather than 二 (èr) for all numbers 200 and greater 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:
|20||  or ||二十||二十 or 廿||二十||廿|
|200|| (èr) or (liǎng) ||二百 or 兩百||二百 or 兩百||兩百||兩百|
|2000|| (èr) or (liǎng) ||二千 or 兩千||二千 or 兩千||兩千||兩千|
|45||  ||四十五||四十五 or 卌五||四十五||四十五|
|2,362|| [1,000]     ||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; yī) 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|
|12000||   ||一萬兩千||  ||一萬二 or 萬二|
|114||    ||一百一十四||   ||一百十四|
|1158||      ||一千一百五十八||See note 1 below|
- Nothing is ever omitted in large and more complicated numbers such as this.
Outside of Taiwan, digits are sometimes grouped by myriads instead of 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):
|(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||十二万三千四百五十六亿七千八百九十万二千三百四十五|
In Taiwan, pure Arabic numerals are officially always and only grouped by thousands. Unofficially, they are often not grouped, particularly for numbers below 100,000. Mixed Arabic-Chinese numerals are often used in order to denote myriads. This is used both officially and unofficially, and come in a variety of styles:
|12,345,000||(1234) [1,0000] (5) [1,000]||1,234萬5千|
|123,450,000||(1) [1,0000,0000] (2345) [1,0000]||1億2345萬|
|12,345||(1) [1,0000] (2345)||1萬2345|
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:
|205||   ||二百零五|
| [10,000]  ||十萬零四|
|(1005) [10,000] (26) or
(1005) [10,000] (026)
To construct a fraction, the denominator is written first, followed by 分之; fēn zhī ("parts of/dividing") 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 又; yòu ("and"), then the fractional part.
|2/3|| [parts of] ||三分之二||sān fēnzhī èr|
|15/32||   [parts of]  ||三十二分之十五||sān-shí èr fēnzhī shí wǔ|
|1/3000||  [parts of] ||三千分之一||sān qiān fēnzhī yī|
|3 5/6|| [and]  [parts of] ||三又六分之五||sān yòu liù fēnzhī wǔ|
Percentages are constructed similarly, using 百; bǎi (100) as the denominator. The 一; yī (one) before 百 is omitted. (Like the English "one hundred" or "a hundred", typically the quantity 100 is denoted 一百; yībǎi; 'one hundred' in Chinese.)
|25%|| [parts of]   ||百分之二十五||bǎi fēnzhī èr-shí wǔ|
|110%|| [parts of]    ||百分之一百一十||bǎi fēnzhī yībǎi yīshí|
Decimal numbers are constructed by first writing the whole number part, then inserting a point (simplified Chinese: 点; traditional Chinese: 點; pinyin: diǎn), and finally the decimal expression. The decimal expression is written using only the digits for 0 to 9, without multiplicative words.
|16.98||  [point]  ||十六點九八|
|12345.6789||         [point]    ||一萬兩千三百四十五點六七八九|
|75.4025||   [point]    ||七十五點四〇二五 or 七十五點四零二五|
|0.1|| [point] ||零點一|
Ordinal numbers are formed by adding 第; dì ("sequence") before the number.
|82nd||[sequence]   ||第八十二||dì-bāshí-èr|
|−1158||[negative]       ||負一千一百五十八|
|−3 5/6||[negative]  [and]  [parts of] ||負三又六分之五|
|−75.4025||[negative]    [point]    ||負七十五點四零二五|
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; 三個人; sān ge rén , "three (ge particle) person", where 个/個 ge is a classifier. There exist many different classifiers, for use with different sets of nouns, although 个/個 is the most common, and may be used informally in place of other classifiers.
Chinese uses cardinal numbers in certain situations in which English would use ordinals. For example, 三楼/三樓; sān lóu (literally "three story/storey") means "third floor" ("second floor" in British § Numbering). Likewise, 二十一世纪/二十一世紀; èrshí yī shìjì (literally "twenty-one century") is used for "21st century".
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. 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. Only one exception, Sunday is 星期日; xīngqīrì, or informally 星期天; xīngqītiān, both literally "week day". When meaning "week", "星期" xīngqī and "禮拜; 礼拜" lǐbài are interchangeable. "禮拜天" lǐbàitiān or "禮拜日" lǐbàirì means "day of worship". Chinese Catholics call Sunday "主日" zhǔrì, "Lord's 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; rì "day") – all the numbers are read as cardinals, not ordinals, with no leading zeroes, and the year is read as a sequence of digits. For brevity the nián, yuè and rì may be dropped to give a date composed of just numbers. For example "6-4" in Chinese is "six-four", short for "month six, day four" i.e. June Fourth, a common Chinese shorthand for the 1989 Tiananmen Square protests (because of the violence that occurred on June 4th). For another example 67, in Chinese is sixty seven, short for year nineteen sixty seven, a common Chinese shorthand for the Hong Kong 1967 leftist riots.
Counting rod and Suzhou numeralsEdit
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.
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 ChinaEdit
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.
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.
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."
During the Ming and Qing dynasties (after Arabic numerals were 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.
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.
|Wikimedia Commons has media related to Chinese numerals.|
- 大寫數字『 Archived 2011-07-22 at the Wayback Machine
- 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.
- “军语”里的那些秘密 http://news.xinhuanet.com/mil/2016-03/21/c_128818839.htm
- (in Chinese) 1981 Gazette of the State Council of the People's Republic of China, No. 365, page 575, Table 7: SI prefixes
- "中華民國統計資訊網（專業人士）". 中華民國統計資訊網 (in Chinese). Retrieved 31 July 2016.
- "中華民國統計資訊網（專業人士）" (in Chinese). 中華民國統計資訊網. Retrieved 31 July 2016.
- "石化氣爆 高市府代位求償訴訟中". 中央社即時新聞 CNA NEWS. 中央社即時新聞 CNA NEWS. Retrieved 31 July 2016.
- "陳子豪雙響砲 兄弟連2天轟猿動紫趴". 中央社即時新聞 CNA NEWS. 中央社即時新聞 CNA NEWS. Retrieved 31 July 2016.
- Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 12.
- Yip, Po-Ching; Rimmington, Don, Chinese: A Comprehensive Grammar, Routledge, 2004, p. 13.
- Days of the Week in Chinese: Three Different Words for 'Week' http://www.cjvlang.com/Dow/dowchin.html
- 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
- 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
- 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。
- Note: The code for the lowercase 〇 (IDEOGRAPHIC NUMBER ZERO) is U+3007, not to be confused with the O mark (CIRCLE).
- Alexander Wylie, Jottings on the Sciences of the Chinese, North Chinese Herald, 1853, Shanghai