The Japanese use two sets of numerals for counting, naming quantities, etc. The main set is Chinese in origin, and it is used most of the time. The second set is Japanese in origin, but only counts from 1 to 10 and is only used occasionally as described below.

The Chinese-origin numbers are as follows:

  1. ichi
  2. ni
  3. san
  4. shi (yo/yon)
  5. go
  6. roku
  7. shichi (nana)
  8. hachi
  9. kyuu
  10. juu
  11. ju-ichi
  12. ju-ni
  13. ju-san
  14. ju-yon
  15. ju-go
  16. ...
20. ni-ju
21. ni-ju-ichi
...

100. hyaku
101. hyaku-ichi
102. hyaku-ni
...

999. kyuu-hyaku-kyuu-juu-kyuu
1000. sen
1001. sen-ichi
...

You get the idea. For even larger quantities, man indicates 10,000; ju-man indicates 100,000; hyaku-man indicates 1,000,000; and sen-man indicates 10,000,000. Oku indicates one hundred million and combines with ju-, hyaku-, and sen- to form 1 billion, ten billion, and one hundred billion. 1 trillion is denoted by choo, which is the highest number word I can find in my textbooks -- I can't think of too many situations where you would need to specifically refer to more than one thousand trillion things anyways.

The numbers often change slightly (but predictably) in pronunciation as they combine to represent larger quantities. For example, 300 is sanbyaku rather than "sanhyaku". Likewise, 800 is happyaku rather than "hachihyaku". 8000 is pronounced hassen, 600 is pronounced roppyaku, and so on. It's not as complicated as it seems, once you get used to it.

These Chinese-origin numbers are used for the vast majority of counting and denoting quantities, including times of day and ordinals (1st, 2nd, 3rd, ...). For counting objects or describing a quantity of things, such as three pieces of paper, a counter is suffixed to the number. This is like saying "three head of cattle" or "seven slices of bread" in English -- the difference is, the counter cannot be omitted in Japanese. (See "Japanese Counters" for a short list of common Japanese counters, or individual counters by name).

To generate ordinals, simply suffix "ban" to the Chinese number:

Ichiban, niban, sanban.
Number one, number two, number three.

To express "first", "second", etc, use "banme":

Yonbanme, gobanme, rokubanme.
Fourth, fifth, sixth.

The Japanese-origin numbers are as follows:

  1. hito
  2. futa
  3. mi
  4. yo
  5. itsu
  6. mu
  7. nana
  8. ya
  9. kokono
  10. too
These numbers are used in very certain circumstances that are difficult to catalogue. Of particular interest to foreigners is that these numerals are used with the counter "tsu", which denotes miscellaneous things or is used when the proper counter is unknown. For example:

Piiza o futatsu o-negai shimasu.
I/We'll have two pizzas, please.

Ringo wa kokonotsu arimasu.
I have nine apples.

The Japanese numerals are also used for some calendar dates, specifially the 2nd through the 10th, the 14th, the 20th, and the 24th. Why only those dates is a matter of history, I imagine (not to mention a matter of confusion for students of Japanese).

You've read the other writeups and now think you can count in Japanese? <amused chuckle> Think again.

One of the more annoying features of the Japanese language is that they use variable counting suffixes for different things. While based on the above Chinese origin numbers, the suffix changes according to what you are counting. Some examples:

To count the floors of a building, you use kai:

ikkai
nikai
sankai
yonkai
gokai

To count people, you use nin, but the first two are irregular:

hitori
futari
sannin
yonin
gonin

To count flat things, such as pieces of paper, you use mai:

ichimai
nimai
sanmai
yonmai
gomai

To count large machines, you use dai:

ichidai
nidai
sandai
yondai
godai

To count steps, you use ho:

ippo
niho
sanbo
yonho
goho

There are dozens of these suffixes. Students of Japanese have a lot of fun memorizing them...

Japanese numbers and counting

Here are all the numbers in Japanese.  Note that the higher-up units have conflicting definitions among modern dictionaries, but the table below is the more accepted version.  In real life, units kei and above are rarely used, if ever, and neither is it for units bun and below.

Exponent
 of 10     Kanji     Pronounciation
-------------------------------------
      68   無量大数    muryoutaisuu
      64   不可思議    fukashigi
      60   那由他     nayuta
      56   阿曽僧祇    asougi
      52   恒河沙     kougasha
      48   極       koku
      44   載       sai
      40   正       sei
      36   澗       kan
      32   溝       kou
      28   穣       jou
      24   *       jo
      20   該       gai
      16   京       kei
      12   兆       chou
       8   億       oku
       4   万       man
       3   千       sen
       2   百       hyaku
       1   十       juu
       0   一       ichi
      -1   分       bun
      -2   厘       rin
      -3   毛       mou
      -4   糸       shi
      -5   忽       kotsu
      -6   微       bi
      -7   繊       sen
      -8   沙       sha
      -9   塵       jin
     -10   埃       ai
     -11   渺       byou
     -12   漠       baku
     -13   模糊      moko
     -14   逡巡      shunjun
     -15   須臾      shuyu
     -16   瞬息      shunsoku
     -17   弾指      danshi
     -18   刹那      setsuna
     -19   六徳      rittoku
     -20   空虚      kuukyo
     -21   清浄      seijou

       WA-ON      KAN-ON   MANDARIN
 1 一   hito       ichi     yi
 2 二   futa       ni       ar, liang *
 3 三   mi         san      san
 4 四   yon        shi *    si
 5 五   itsutsu    go       wu
 6 六   mu         roku     liu
 7 七   nana       shichi * qi
 8 八   ya         hachi    ba
 9 九   kokonotsu  kyuu     jiu
10 十   tou        jyuu     shi

Also:
20 廿 (archaic kanji) ni jyuu
30 卅 (archaic kanji) san jyuu
 0 零   rei, zero
For comparison, Mandarin is added to the last column for 1-10. Note that "r" and "f" are not pronounced as they are in English, but close to it. Chinese has two character representations for the number 2, with unique prounciations for each.

Each kanji can have multiple pronounciations. A pronounciation of a kanji is either wa-on (Japanese native) or kan-on (imported from China). For numbers, kan-on pronounciation is more common than wa-on pronounciation with the exceptions of 4 and 7. Several reasons can be thought of why kan-on is preferred. For one, wa-on counting stops at ten and then skips to 20 (hata) and ends there. wa-on counting is limited maybe because it never evolved or got lost in time. 4 is most often pronounced as "yon" instead of "shi," because "shi" is impolite: it is how the word for death is pronounced in Japanese. Many other countries with strong Chinese cultural influence also have this peculiar homophone situation. Numbers from 10 through 99 are made from naturally combining words without special exception rules like European languages have. (e.g. not "thirty", "twelve", but san-juu, juu-ni)

There are many special cases where kan-on is not the appropriate pronounciation choice. Here are a few examples where wa-on are used:

hitori - 1 person
futari - 2 people
Kokonoka - 9th day of a month
Touka - 10th day of a month
Hatsuka - 20th day of a month
Hatachi - Age 20, when one is considered to have entered adulthood. Also the drinking/smoking age. All other ages end with the "sai" counter.

Some examples of numbers beyond ten:

12 - jyuu ni
30 - san jyuu

Note that just like the units thousand, million, billion, etc. multiply by 1,000 each time, Japanese counting units multiply by 10,000 after "man". Even so, Japanese put their commas every three digits, not four. Don't ask me why. Chou is about the largest unit used in real life, like it may be used when talking about the Japanese national defecit.

The character for "jo" is not included in the Shift-JIS or EUC characterset, most likely because nobody ever uses it. A different character for 1 0000 is used in Taiwan (character originates from a serpent pictograph rather than that of a swastika), where a more traditional one is used. (Also I'm sure there are more differences in Chinese character usage amongst mainland China, Taiwan, Korea, or Vietnam.) Kougasha and beyond were made by Chinese monks. Here are the etymologies of those words:

kougasha - The sands of Ganghis river.
asougi - From the Hindu word "Asamkhya", meaning uncountably large number.
nayuta - Ancient Hindu word "Nayuta", a very large number.
fukashigi - (Word origin not recorded)
muryou taisuu - An unmeasurably large number.

In Buddhist literature, Kougasha is a word used to describe something that is uncountably many. In Taoism, for example in the book Tao Te Ching, 1 0000 often is supposed to mean a count of everything. The Great Wall of China may be called "wan li chang zhang" in Mandarin, which means 1 0000 li long wall. Even though it probably isn't precisely 1 0000 li (li is close to miles or kilometers), it has an implied meaning of very long.

Writing numbers in kanji:

Kanji can be written in vertical or horizontal style. There are two methods for representing numbers in kanji. It may contain units (十 百 千 万) or it may be written with just the kanjis for 1-9 (一 二 三 四 五 六 七 八 九) plus a circle for a place holder, in the same way 0 is a place holder in Hindu-Arabic Numerals. (The circle is not considered as a kanji but merely a symbol.)  Kanji is never used in math.

Decimal expansion:

Commas are not pronounced, and periods are pronounced as "ten", which means "dot." The convention for numbers between 1 and zero is to not omit the one's digit zero, followed by a decimal dot, and then a decimal expansion. Commonly, each written decimal is read out loud without mentioning the unit, although there are unit names for many decimal places. For example, "one tenth" would be written as "0.1" and pronounced as "rei ten ichi" or "zero ten ichi" and "one hundredth" can be read as "rei ten rei ichi."

Fractions:

Written as it would be in Europe. When pronounced, the denominator is pronounced first. (denominator "bunno" numerator)

Percentages and negatives:

Negative numbers are pronounced starting with "mainasu" (minus), and there is no kanji substitute for this. Percentage was used before Western math influence, but the only one still commonly used is the "wari" unit, which is equivalent to 10 percent. Percentage is pronounced "paasento" with no kanji equivalent.


Sources:
http://village.infoweb.ne.jp/~fxba0016/misc/suumei/shiryou.html
http://www.usiwakamaru.or.jp/~doraemon/jyuku/suuji.htm

Here are some old names for numbers in Japanese.

20 - hatachi
30 - misoji
40 - yosoji
50 - isoji
60 - musoji
70 - nanasoji
80 - yasoji
90 - kokonosoji
100 - momo (not sure how to make the other hundreds)
1000 - chi (I think I've seen "michi" for 3000)

Nowadays, these numbers are obsolete for the most part. The tens, I think, are still used to talk about ages.

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