On Japanese Asamkhyeyas

Quadrillion, quintillion… decillion…? Anyway, in English, we can use our high school Latin to construct names for ever increasing powers of 10. In Japanese, though, this system was never used, because enough kanji were available for any human need. As described by Matt a few years ago, in 1627 a Japanese guy named Mitsuyoshi Yoshida decided to venture forth and invent numbers with names that went from an acceptable extension of kanji (載 sai, one hundred tredecillion) to more than a little unusual: take 恒河沙 gougasha, one hundred septendecillion, but literally “the number of grains of sand on the Ganges”, the Ganges being more of a figurative concept in Japan at the time. (Yoshida himself technically changed the definitions of these numbers in the various editions of his book; these were revised later by the scientific community at large. The numbers I give here are the standard form. Japanese Wikipedia has more details.)

Yoshida grabs various terms from the Renge-kyo, or Lotus Sutra, but these terms were actually more precisely defined in the Gandavyuha appendix to the Kegon-kyou, or Flower Garland Sutra, or Avataṃsaka SÅ«tra. When the author of the latter tries to explain how long it took for Buddha to reach enlightenment, he employs the term asaṃkhyeya or “innumerable”, which Wikipedia defines quite succinctly as follows:

An asaṃkhyeya (Sanskrit: असंख्येय) is a Buddhist name for the number 10140, or alternatively for the number as it is listed in the Avataṃsaka Sūtra where the values are a=5, b=103 in the translation of Buddhabhadra, a=7, b=103 in that of Shikshananda and a=10, b=104 in that of Thomas Cleary who makes errors in the calculation.

An article linked by Wikipedia provides another source, giving us the following table of authoritative values:

Date Source text Value
300s CE Abhidharmakosha by Vasubandhu between 1051 and 1059
400s CE Avataṃsaka Sūtra, tr. Buddhabhadra 105 × 2^103
699 CE Avataṃsaka Sūtra, tr. Shikshananda 107 × 2^103
768 CE Avataṃsaka Sūtra, tr. Prajna 107.44 × 10^37 (?)

A strange person in Amsterdam lovely person who has left me a comment on this post also has things to say about these numbers, including the claim that all the translators have some sort of error. I think she wrote the Wikipedia article.

In Japanese, the term for asaṃkhyeya is 阿僧祇 asougi, which Yoshida defined as 1031 and 1064 in the various editions of his book, and his disciples redefined as 10104, simply because they wanted words for large numbers. We can already see some trouble here. It is not clear to me why Yoshida strayed from the calculations of Buddhabhadra and Shikshananda who predated him by a very long time. It is also not clear where Wikipedia’s figure 10140 came from, but let us leave that aside.

Yoshida felt fit to finish his mind-boggling catalog with the limit of ten septillion vigintillion. To this number he assigned the name 無量大数 muryoutaisuu, which means “an immeasurably large number”. Certainly it is beyond the needs of any ordinary human. But it is obviously not beyond measurement. Also, if you will scroll up a little, you will find that the mysterious word asougi also translates back to “immeasurable” in the original Sanskrit. I am suspicious that Yoshida was not only unaware of the Avataṃsaka SÅ«tra, but also the meaning of the Sanskrit words he was pulling out of the Lotus Sutra for convenience.

When later Japanese researchers did reread the Avataṃsaka SÅ«tra, they discovered a bountiful supply of Buddhist words to express numbers far beyond the septillion vigintillion level. Both Japanese Wikipedia and a website linked from there employ Shikshananda’s reckoning to supply values to terms such as these:

Sanskrit Sino-Japanese Common Japanese meaning Value
laksha 洛叉 rakusha 105

koti 倶胝 kutei 107

ayuta 阿庾多 ayuta 1014

niyuta 那由他 nayuta 1072 (thanks Yoshida) 1028

vivara 頻波羅 binbara 1056

kshobhya? 矜羯羅 kongara 10112

? 趣 shu appearance 107 × 2^101

? 至 shi limit/reach 107 × 2^102

asaṃkhyeya 阿僧祇 asougi 10104 (thanks Yoshida) 107 × 2^103

? 阿僧祇転 asougiten An asougifold 107 × 2^104

? 無量 muryou measureless 107 × 2^105

? 無量転 muryouten measurelessfold 107 × 2^106

? 無辺 muhen boundless 107 × 2^107

? 無辺転 muhenten boundlessfold 107 × 2^108

and so forth. Being based on the same source, both Cleary’s translation and the Japanese table end up with the same number, which Cleary calls a “square untold”, but in Japanese is given the fantastic name 不可説不可説転 fukasetsufukasetsuten, that is “untheorizable-untheorizable–fold”. This must truly be the largest number nameable in Japanese without scientific notation, although due to the multiple translations, its value is a little shaky: between 10^10^36 and 10^10^37. Unfortunately it is dwarfed by the googolplex, 10^10^100. But it is a sufficiently large number that to write it in regular notation as 10 followed by 0s, you would have to have an intergalactic amount of empty space to write all the 0s.

n.b. While making this last table I ran across the Lalitavistara Sutra, which gives an ayuta as 109, niyuta as 1011, etc. The age of the Lalitavistara Sutra is unknown. However, comparing it to the enormous numbers of the Avatamsaka Sutra and the smaller numbers of the earlier Abhidharmakosha, we can see that the Lalitavistara Sutra seems to use the same numbers as the Abhidharmakosha and therefore predate the Avatamsaka Sutra, or else the author had an unusually small imagination.

Posted: March 25th, 2011 | Excerpts 4 Comments »