The Japanese Number System

By Nicholas P. Leveillee
2011, Vol. 3 No. 05 | pg. 1/1

Borrowing heavily from the Chinese, abandoned their own numerals many years ago and used the ones from . When written and spoken, numbers are broken down into their key components, multiples of powers of ten. The Japanese combine the number with ten, hundred, thousand, ten thousand, and so on to create the desired number (Menninger 450). To understand Japanese numbers, one must first understand the variants. Then, there are different symbols for each numeral, depending on the form of the writing (Ifrah 273, 276). Cultural taboos and a desire to avoid ambiguity also mean that, when spoken, people may alternate between two different systems for the names of some numbers (Ifrah 275). There will be emphasis on a mathematician who is called "Japan's Newton" and is recently getting credit for his many innovations hundreds of years ago (Sezi Takakazu). The Japanese number system also has a way to represent decimals, using the same principles as the integers (Japanese Numerals). Japanese is considered one of the most difficult languages to learn, because of the many different symbols that could potentially mean the same thing (Ifrah 273).

Japan's location in close proximity to China influenced its adoption of a Chinese numerical system. It is uncertain exactly when Japan abandoned its own number system and adopted the Chinese system. Certainly, the Chinese numerals have been used for almost one hundred years, as information exists dating back to 1936 (Ifrah 274).

There are four styles of Japanese writing: kanji, hiragana, katakana, and romaji. Kanji are symbols that represent an idea using a Chinese symbol. Kanji used to have many symbols but many are represented by hiragana now. Kanji has 1,945 official symbols and an additional 166 for names. Because of their widespread use, 996 are taught in primary school because of their importance. Hiragana differs from kanji in that these symbols represent syllables. Hiragana has fifty-one symbols and represents inflections and word endings that cannot be expressed through the kanji. Katakana is similar in hiragana as they both represent syllables. Katakana is used instead of hiragana in foreign words, geographic locations, foreign proper names, and several others. The last style is romaji, which uses the Western alphabet. This is used in places that would be inconvenient if another style was used. Dictionaries commonly use romaji. This is just how words are written. With speaking numbers, there are two ways to say them. The Pure Japanese way uses their original words rather than the Sino-Japanese way, using words borrowed from the Chinese (Ifrah 273). For clarity and superstitious reasons, people alternate between the two systems (275). When writing numerals, there are five possible forms. While the most common form is standard form, there are also cursive, calligraphic, commercial, and formal forms. In some of these forms, numerals are the same or similar (276).

Table 1: Common Japanese numbers and numerals (Ifrah 276)

  Number 1 2 3 4 5 6 7 8 9 10
Spoken Pure Japanese hi (to) fu (ta) mi yo itsu mu nana ya kokono
Sino-Japanese ichi ni san shi go roku shichi hachi ku
Written Standard Form

The Japanese people sometimes face ambiguity because of words that sound similar. To deal with this, they alternate between the Pure Japanese and Sino-Japanese words. Ichi.ban could mean 'one evening' or 'first number' so confusion is avoided by saying hito.ban instead. Other words are merely heard more easily when a combination of systems is used. For 17, jû.nana is heard with more clarity than jû.shichi. There is also another reason the Japanese use a different system. Three of the Sino-Japanese words are similar in sound to unpleasant words. Shi might be mistaken for death, shichi for death or loss, and ku for pain. When using these words, many Japanese would use their Pure Japanese form. To the Japanese, invoking such a word would be to suffer the evil raised by the word (Ifrah 275-6). Numbers made using 4 or 9 are very difficult to find in plane seats, hotel and hospital rooms, and even parking bays. The launch of the Renault 4 car had such a terrible reception in Japan because of the 4 (276). As if confusion among the words was not enough, there are different writing forms as well.

Standard form is the most common writing style. Cursive, calligraphic, and commercial are used in situations that are more specialized and are rare as a result. Formal form is used in legal documents to ensure that a single stroke would not change a value. A one could easily be changed into a two or three, a two into a three, a three into a five, and a ten into ten-thousand. There are no different formal numerals used for the other numbers.

Table 2: Formal Numbers (Japanese Numerals)

Number Common Formal

Japan's contributions to mathematics are not as well known as other countries, yet there is one person who is starting to be recognized as "Japan's Newton" (Sezi Takakazu). Sezi Takakaru or Sezi Kōwa lived from 1642 to 1708 in Edo, now Tokyo. He is being called the most important person in wasan, Japanese calculation. He recovered ancient Chinese sources and generalized their problems. It is unsure how he was educated, but his knowledge of contemporary mathematics could not be disputed when he published his first book. His greatest contributions were to algebra, where he created generalizations that could be used to solve any similar problem. He could solve equations with more than one unknown and obtain the roots of an equation. It is hard to pin down exactly what contributions Seki made in some of his books, as he collaborated with some of his disciples (Encyclopedia Britannica). His pupils called him "The Arithmetical Sage" and that phrase is carved on his tombstone as well. Seki studied determinants in 1683, the first person to do so. Leibniz, ten years later, use determinants to solve simultaneous equations without knowing that Seki had already created a general version years earlier. Seki was also ahead of Jacob Bernoulli, discovering Bernoulli numbers first. He studied equations that had positive and negative roots but did not venture into complex numbers. Having seen magic squares used in China, he was the first to bring the topic to Japan in 1683. Two years later, he solved 30 + 14x - 5x2 - x3 = 0 using the method Horner would discover in one hundred years. He also discovered Newton's method for solving equations and created his own version of the Newton interpolation formula. He is also thought to have passed on major discoveries in calculus to his students (O'Connor and Robertson). Seki also calculated π to 10 correct decimal places by using a method called Aitken's delta-squared process, which was rediscovered in the 20th century by Alexander Aitken (Sezi Takakazu). This helps show that Japan had a grasp of transcendental numbers and a decimal system.

An important aspect of the Japanese number system is that it can be written with either Arabic numerals or the Chinese numerals. Because Japan has made a shift from the traditional vertical writing, now used mainly for restoration work, towards western horizontal writing, Arabic numerals are becoming more commonplace. Transcendental numbers are written using their normal form. Fractions are written with Arabic numerals. Decimals can be written with both. There are two decimals systems that are used in Japan. While neither are used very often, they do see use in batting averages and percentages.

Table 3: Traditional Measurement System (Japanese Numerals)

Rank 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10
Reading bu rin shi kotsu bi sen sha jin ai

Table 4: Discount Decimals (Japanese Numerals)

Rank 10−1 10−2 10−3 10−4 10−5
Reading wari bu rin shi

The only decimal that sees real usage is wari, from the second table. When writing with Chinese numerals, they insert a decimal sign '・' and continue writing like normal. There is a zero sign that does not appear to see much use. Written as 零 or 〇 and pronounced zero in Pure Japanese and rei in Sino-Japanese, it can sometimes make the positional system easier to read with symbols. To make the place values easier to determine when using Chinese numerals, sometimes the zero is included. However, this is not necessary because of the multiplier that is used, being ten, hundred, etc. (Japanese Numerals). Another way to ensure there is no confusion is to create a small table and box off the place values, leaving empty boxes when there is a number missing.

Overall, the Japanese numeral system is easy to read, potentially difficult to hear, and time consuming to write. However, being familiar with the symbols could easily speed up writing them without sacrificing clarity. There are virtually no examples of mathematical operations in Japanese. Most numbers are written using Latin numerals, the only Japanese being used for the description of the problem.


Encyclopedia Britannica. "Seki Takakazu." 2011. Britannica Online Encyclopedia. 6 March 2011 .

Ifrah, Georges. The Universal History of Numbers: From Prehistory to the Invention of the Computer. Trans. David Bellos, et al. New York: John Wiley & Sons, Inc., 1998.

"Japanese Numerals." 20 February 2011. Wikipedia. 5 March 2011 .

Menninger, Karl. Number Words and Number Symbols: A Cultural History of Numbers. Trans. Paul Broneer. New York: Dover Publications, Inc., 1969.

O'Connor, J J and E F Robertson. "Seki biography." 1997. 6 March 2011 .

"Sezi Takakazu." 28 February 2011. Wikipedia. 5 March 2011 .

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