{"id":244,"date":"2011-03-25T06:19:45","date_gmt":"2011-03-25T06:19:45","guid":{"rendered":"http:\/\/avery.morrow.name\/blog\/?p=244"},"modified":"2013-05-15T01:17:30","modified_gmt":"2013-05-15T01:17:30","slug":"on-japanese-asamkhyeyas","status":"publish","type":"post","link":"https:\/\/avery.morrow.name\/blog\/2011\/03\/on-japanese-asamkhyeyas\/","title":{"rendered":"On Japanese Asamkhyeyas"},"content":{"rendered":"

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<\/a>, 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 (\u8f09 sai<\/em>, one hundred tredecillion) to more than a little unusual: take \u6052\u6cb3\u6c99 gougasha<\/em>, 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<\/a>.)<\/p>\n

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 Avata\u1e43saka S\u016btra. When the author of the latter tries to explain how long it took for Buddha to reach enlightenment, he employs the term asa\u1e43khyeya or “innumerable”, which Wikipedia defines <\/a>quite succinctly as follows:<\/p>\n

An asa\u1e43khyeya (Sanskrit: \u0905\u0938\u0902\u0916\u094d\u092f\u0947\u092f) is a Buddhist name for the number 10140<\/sup>, or alternatively for the number \"\" as it is listed in the Avata\u1e43saka S\u016btra 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.<\/p><\/blockquote>\n

An article linked<\/a> by Wikipedia provides another source, giving us the following table of authoritative values:<\/p>\n\n\n\n\n\n\n\n
Date<\/th>\nSource text<\/th>\nValue<\/th>\n<\/tr>\n
300s CE<\/td>\nAbhidharmakosha<\/em> by Vasubandhu<\/td>\nbetween 1051<\/sup> and 1059<\/sup><\/td>\n<\/tr>\n
400s CE<\/td>\nAvata\u1e43saka S\u016btra, tr. Buddhabhadra<\/td>\n105 \u00d7 2^103<\/sup><\/td>\n<\/tr>\n
699 CE<\/td>\nAvata\u1e43saka S\u016btra, tr. Shikshananda<\/td>\n107 \u00d7 2^103<\/sup><\/td>\n<\/tr>\n
768 CE<\/td>\nAvata\u1e43saka S\u016btra, tr. Prajna<\/td>\n107.44 \u00d7 10^37<\/sup> (?)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

A strange person in Amsterdam<\/s> lovely person who has left me a comment on this post<\/a> 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.<\/p>\n

In Japanese, the term for asa\u1e43khyeya is \u963f\u50e7\u7947 asougi<\/em>, which Yoshida defined as 1031<\/sup> and 1064<\/sup> in the various editions of his book, and his disciples redefined as 10104<\/sup>, 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<\/sup> came from, but let us leave that aside.<\/p>\n

Yoshida felt fit to finish his mind-boggling catalog with the limit of ten septillion vigintillion. To this number he assigned the name \u7121\u91cf\u5927\u6570 muryoutaisuu<\/em>, 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<\/em> also translates back to “immeasurable” in the original Sanskrit. I am suspicious that Yoshida was not only unaware of the Avata\u1e43saka S\u016btra, but also the meaning of the Sanskrit words he was pulling out of the Lotus Sutra for convenience.<\/p>\n

When later Japanese researchers did reread the Avata\u1e43saka S\u016btra, 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<\/a> employ Shikshananda’s reckoning to supply values to terms such as these:<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Sanskrit<\/th>\nSino-Japanese<\/th>\nCommon Japanese meaning\n<\/th>\nValue<\/th>\n<\/tr>\n
laksha<\/td>\n\u6d1b\u53c9 rakusha<\/i><\/td>\n–<\/td>\n105<\/sup><\/p>\n
koti<\/td>\n\u5036\u80dd kutei<\/i><\/td>\n–<\/td>\n107<\/sup><\/p>\n
ayuta<\/td>\n\u963f\u5ebe\u591a ayuta<\/i><\/td>\n–<\/td>\n1014<\/sup><\/p>\n
niyuta<\/td>\n\u90a3\u7531\u4ed6 nayuta<\/i><\/td>\n1072<\/sup> (thanks Yoshida)<\/td>\n1028<\/sup><\/p>\n
vivara<\/td>\n\u983b\u6ce2\u7f85 binbara<\/i><\/td>\n–<\/td>\n1056<\/sup><\/p>\n
kshobhya?<\/td>\n\u77dc\u7faf\u7f85 kongara<\/i><\/td>\n–<\/td>\n10112<\/sup><\/p>\n
…<\/p>\n
?<\/td>\n\u8da3 shu<\/i><\/td>\nappearance<\/td>\n107 \u00d7 2^101<\/sup><\/p>\n
?<\/td>\n\u81f3 shi<\/em><\/td>\nlimit\/reach<\/td>\n107 \u00d7 2^102<\/sup><\/p>\n
asa\u1e43khyeya<\/td>\n \u963f\u50e7\u7947 asougi<\/em><\/td>\n10104<\/sup> (thanks Yoshida)<\/td>\n107 \u00d7 2^103<\/sup><\/p>\n
?<\/td>\n\u963f\u50e7\u7947\u8ee2 asougiten<\/em><\/td>\nAn asougifold<\/td>\n107 \u00d7 2^104<\/sup><\/p>\n
?<\/td>\n\u7121\u91cf muryou<\/i><\/td>\nmeasureless<\/td>\n107 \u00d7 2^105<\/sup><\/p>\n
?<\/td>\n\u7121\u91cf\u8ee2 muryouten<\/i><\/td>\nmeasurelessfold<\/td>\n107 \u00d7 2^106<\/sup><\/p>\n
?<\/td>\n\u7121\u8fba muhen<\/i><\/td>\nboundless<\/td>\n107 \u00d7 2^107<\/sup><\/p>\n
?<\/td>\n\u7121\u8fba\u8ee2 muhenten<\/i><\/td>\nboundlessfold<\/td>\n107 \u00d7 2^108<\/sup><\/p>\n
…\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

and so forth.<\/a> 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 \u4e0d\u53ef\u8aac\u4e0d\u53ef\u8aac\u8ee2<\/a> fukasetsufukasetsuten<\/i>, 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.<\/p>\n

n.b. While making this last table I ran across the Lalitavistara Sutra, which gives an ayuta as 109<\/sup>, niyuta as 1011<\/sup>, 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.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[12],"tags":[],"_links":{"self":[{"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/posts\/244"}],"collection":[{"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/comments?post=244"}],"version-history":[{"count":10,"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/posts\/244\/revisions"}],"predecessor-version":[{"id":1482,"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/posts\/244\/revisions\/1482"}],"wp:attachment":[{"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/media?parent=244"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/categories?post=244"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/avery.morrow.name\/blog\/wp-json\/wp\/v2\/tags?post=244"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}