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The Chinese numeral system is a system of words used to describe numbers in the Chinese language, as well as other languages that use Chinese characters, primarily Japanese and Korean.

Basic numbers Edit

The Chinese numeral system is a positional decimal system. There are characters for the numbers from zero through ten, and then characters for the powers of 10 thereafter.

Numeral

Value

Chinese pronunciation

(Mandarin, pinyin)

Japanese pronunciation

0 líng rei, zero
1 ichi
2 èr ni
3 sān san
4 shi, yon
5 go
6 liù roku
7 shichi, nana
8 hachi
9 jiǔ kyū
10 shí
100 bǎi hyaku
1000 qiān sen
万/萬 10,000 wàn man
亿/億 108 oku

In Chinese, the power-of-ten numbers are used as suffixes to the basic numerals. For example, 二十 is 20, because 十 marks the place value of the 二. 一百二十三 is 123, because 百 is prefixed with a 一, 十 is prefixed with a 二, and 三 is at the end of the number. If the place value is 一十, then the 一 is usually omitted, so 13 is 十三, not 一十三. Any zeros are expressed using a single 零 without any place value markers. So 103 is 一百零三, and 10003 is 一万零三.

In Japanese, a 一 can be omitted before any place-value marker, not just 十, and 零 is not used as a positional placeholder, so 100 is just 百, 103 is simply 百三, and 123 is just 百二十三.

The symbols 十, 百, and 千 are themselves repeated after 万, so 100,000 is 十万, and 1,230,000 is 一百二十三万.

In certain cases, the word 二 can be replaced with 兩 (两 in simplified Chinese, 両 in shinjitai kanji) to mean "two", usually when talking about counts of objects (e.g. 两个人 for "two people", or 两只猫 for "two cats"). However, using 二 for standalone numbers is always correct. However, when using two as an prefix before a number larger than hundred, 两 should be used. For example, 两千两白二十 is correct for 2,220. The use of 两 is sometimes very important, because for example 二百五 means 'idiot' or 'very stupid person', while the correct version 两百五 is just 205 as it should be.

Large numbers Edit

Numbers larger than 亿 are not generally used in daily life. The Wujing Suanshu[1] defines a series of numbers larger than 亿, and assigns three different scales to them:

  1. The "lower numbers" (下數 (Traditional), 下数 (Simplified)), in which each character after 万 is valued 10 times the previous one. In this system, 亿 is equal to 105.
  2. The "middle numbers" (中數 (Traditional), 中数 (Simplified)), in which each character after 亿 is valued 108 (一亿) times the previous one.
  3. The "upper numbers" (上數 (Traditional), 上数 (Simplified)), in which each character after 万 is the square of the previous one.

In modern times, these numbers have been redefined such that each character after 万 is valued 104 (一万) times the previous one, giving a fourth system.

The higher numerals are as follows:

Numeral Non-numeric

meaning/definition

Chinese pinyin Japanese Lower

numbers

Modern

usage

Middle

numbers

Upper

numbers

omen zhào chō 106 1012 1016 1016
capital jīng kei 107 1016 1024 1032
border, frontier gāi gai 108 1020 1032 1064
shi/jo 109 1024 1040 10128
穣/穰 abundant, lush ráng 1010 1028 1048 10256
沟/溝 ditch, canal gōu 1011 1032 1056 10512
涧/澗 brook, stream jiàn kan 1012 1036 1064 101024
upright, righteous zhèng sei 1013 1040 1072 102048
载/ load, debt zài sai 1014 1044 1080 104096

The upper numbers are the same as Knuth's -yllion system, with 亿 being one myllion, 兆 being one byllion, and so on.

Examples of usage Edit

In the above examples, each way of counting has its own grouping systems. Below are some examples of the usage of these systems.

Lower numbers Edit

In the lower numbers, each power of 10 has its own positional marker.

Number Representation
647,890 六亿四万七千八百九十
123,456,789,012 一溝二穰三秭四垓五京六兆七亿八万九千零一十二

The lower numbers can express numbers with up to 15 digits. After that, the symbols are undefined.

Modern usage Edit

In modern usage, 十, 百, and 千 repeat after 万, and each of them again once after each symbol afterward.

From here onwards, a group of four digits will be implied to have 千, 百, and 十 in between them.

Number Representation
647,890

六十四万七千八百九十 (64万7890)

123,456,789,012 一千二百三十四亿五千六百七十八万九千零一十二 (1234亿5678万9012)
264 - 1 1844京6744兆0737亿0955万1615
30! + 116 2溝6525穰2859秭8121垓9105京8636兆3084亿8177万1561

This system can express numbers with up to 48 digits.

Middle numbers Edit

In the middle numbers, 十, 百, and 千 repeat after 万, and 万 repeats once after each symbol afterward.

Number Representation
647,890 64万7890
123,456,789,012 1234亿5678万9012
264 - 1 1844兆6744万0737亿0955万1615
30! + 116 2垓6525万2859京8121万9105兆8636万3084亿8177万1561
ceil(e5π√163) 1247万2571載5601万9637正3048万5610澗7520万0180溝7455万2566穰8245万

8586秭2995万2721垓7336万8815京7940万8549兆5792万2996亿2109万3744

The middle numbers can express numbers with up to 88 digits.

Upper numbers Edit

In the upper numbers, each symbol 万 and above repeats once between occurrences of the next higher symbol, in a fractal-like pattern.

Number Representation
264 - 1 1844兆6744万0737亿0955万1615
30! + 116 2京6525万2859亿8121万9105兆8636万3084亿8177万1561
RSA-100 1522京6050万2792亿2533万3605兆3561万8378亿1326万3742垓9718万0681亿1496万1380兆6886万5790亿8494万5801京2296万3258亿9528万 9765兆4000万3506亿9200万6139
108191 1千万亿兆京垓秭穰溝澗正載

The upper numbers can express numbers with up to 8192 digits.

Larger numbers Edit

Numbers larger than appear in some Buddhist texts, which are not defined in the Wujing Suanshu. In Chinese, they are not used; however, some of them have been standardized in Japanese usage.[2] A list is given below with the hypothetical values that the upper numbers would have if the system in thet Wujing Suanshu were extended to include these numbers.

Numeral Non-numeric

meaning/definition

Chinese

pinyin

Japanese Modern

usage

Upper

numbers

极/極 extreme, the limit goku 1048 108192
恒河沙/恆河沙 The sands of the Ganges River héng hé shā gōgasha 1052 1016,384
阿僧祇 Asamkhyeya ā sēng qí asōgi 1056 1032,768
那由他 Nayuta, figuratively "myriad" in Sanskrit nà yóu tā nayuta 1060 1065,536
不可思议/不可思議 unimaginable, inconceivable bù kě sī yì fukashigi 1064 10131,072
无量大数/無量大数 immeasurably large number wú liàng dà shù muryōtaisū 1068 10262,144

References Edit

  1. 大数系统, a section in article 中文数字, on 维基百科 (Chinese Wikipedia). http://zh.wikipedia.org/wiki/%E4%B8%AD%E6%96%87%E6%95%B0%E5%AD%97#.E5.A4.A7.E6.95.B8.E7.B3.BB.E7.B5.B1
  2. 大数系统, a section in article 日语数字, on 维基百科 (Chinese Wikipedia). http://zh.wikipedia.org/wiki/%E6%97%A5%E8%AA%9E%E6%95%B8%E5%AD%97#.E5.A4.A7.E6.95.B8.E7.B3.BB.E7.B5.B1
Googology in Asia

Fish numbers: Fish number 1 · Fish number 2 · Fish number 3 · Fish number 4 · Fish number 5 · Fish number 6 · Fish number 7
Mapping functions: S map · SS map · S(n) map · M(n) map · M(m,n) map
By Aeton: Okojo numbers · N-growing hierarchy
By BashicuHyudora: Pair sequence number · Bashicu matrix system
Indian counting system: Lakh · Crore
Chinese and Japanese counting system: Wan · Yi · Zhao · Jing · Gai · Zi · Rang · Gou · Jian · Zheng · Zai · Ji · Gougasha · Asougi · Nayuta · Fukashigi · Muryoutaisuu
Buddhist text: Tallakshana · Dvajagravati · Mahakathana · Asankhyeya · Dvajagranisamani · Vahanaprajnapti · Inga · Kuruta · Sarvanikshepa · Agrasara · Uttaraparamanurajahpravesa · Avatamsaka Sutra
Other: Taro's multivariable Ackermann function · Sushi Kokuuhen

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