It's All About the 不可説不可説転円札 (or, Archive of Sino-Japanese Buddhist Hyper-Astronomical Numbers)

Despite the fact that even historically, I don't know of anyone who has read this blog other than myself, perhaps it is because of that that I have a reason to make a post after 5 years for an expected audience of one.

For context, one of the main focuses of my early education was in mathematics, and though my career has departed into the semi-related field of computer science, not much has changed about my familiarity and interest in numerical curiosities. As well as that, my university studies led me down the paths of linguistical intrigue, Japanese didacticism, and East Asian studies. So anything that is at the intersection of the Venn diagram of all four topics would certainly call my attention.

I don't know exactly when I first encountered this particular topic, but perhaps it was semi-contemporaneous with my long-ago post on Japanese ten-power numbers in relation to intuition about currency-conversion and various culturo-economic touchstones for how much 一兆円 can get you (answer: probably still a 新幹線 line but I haven't bothered doing any additional research to account for inflation or the like). But in any case, there are certainly much bigger numbers than (一兆 = 一(万万万) = (10 000)³ = 1 0000 0000 0000 = 1,000,000,000,000). For the sake of convenience going forward, I will probably adopt a neutral convention of using direct 10^x notation to avoid the need to list both thousand-scale and myriad-scale (fun fact: though used generically to refer to a large quantity and often an assortment, the term 'myriad' specifically indicates 一万 or 10⁴) versions of every large number, especially since we have a long ways to go.

Since I am not wikipedia, I won't go into the full details which, assuming that the ravages of time leave the page unscathed, ought to remain on the relevant page for long enough that future-me doesn't have to reiterate the basics.

This post will be a partial translation, partial exploration, and partial archival post for a resource that has, at some point, become inaccessible due to the impermanence of all things (which is oddly on-topic, and though there probably isn't a notion of the internet in historical Buddhism, is true especially on the internet). The resource in question was hosted at this now-dead URL: http://www.sf.airnet.ne.jp/~ts/language/largenumber.html, but which, thanks to the Internet Archive's Wayback Machine, can still be found here. Even though 100% of my known audience can read Japanese, that same 100% is a native English-speaker and so might prefer that I (viz. they) encapsulate some of the details in a more direct fashion to avoid a language barrier.

I don't know much about the original author 高杉親知 (which, based on Rikai-tan's assessment, is probably read as Chikatomo TAKASUGI), and my interest is not so much in the writer as it is in the post itself, which introduced me to a list of etymologically Buddho-centric Japanese astronomical numbers I had not previously encountered. Since my ability to read through large troves of Japanese is limited severely due to my inexposure and inexperience to such tasks, and my generally stagnant Japanese skills not having been necessarily strong enough even at their peak to read through archaic sources on esoteric topics, I won't really try to claim that I can be sure that the information I am re-presenting is authentic and unbiased, and because the names that we call any given numeral quantity are as much a social construct as most other linguistic conventions (though I do not wish to denigrate any historical significance or importance of the influential origins on words in a languge or from a culture I have no direct personal grounding in), even a misrepresentation of what sequence of characters, what reading thereof, and what etymology they derive from, ought to be relatively innocuous and uncontentious when applied solely to numbers, which are rarely ever victims (except in the case of the famous incident where nine was mercilessly eaten by seven).

Of course, even without the blog-post I am parroting, everyone probably knows by now that the Japanese counting and number-system uses explicit place-markers in the form of fixed-ten-powers as post-positional unit suffixes for any digital scalar (being 1-9, leaving 0 out). In other words, the notation for a number greater than 9 is always the concatenation of each of its significant place-values in highest-to-lowest sequence, additionally omitting the leading 一 (1*) for explicit (i.e. not the unrendered 10⁰) tens-power units up to 千=10³. Though there is no direct way of adapting this notion to the standard western conventions in using arabic numerals with purely positional implicit tens-power units, using 0 to fill in the gaps between non-consecutive tens-powers, it is somewhat close to the way that spoken English treats large numbers. "Ten thousand and four" being the way we convey the idea that is rendered orthographically as 10004, rather than reading each digit by itself in sequence like 'one zero zero zero four' (though for pure number-sequences not meant to have place-value associations, like phone-numbers and other pure digit-strings, both Japanese and English fall back on reading the numbers verbatim in sequence without attaching place-value to them, in either respective convention; with some exceptions, of course, most notably that of the emergency phone number in Japan, 110, being referred to in common speech as １１０番 (ひゃくとおばん) which literally means 'hundred-and-ten (phone) number').

I would also like to note for those who either lack the patience or ability (due either to issues relating to the Internet or the task of reading Japanese being difficult or impossible) to read the original post, that the relevant 'values' of large tens-powers are by no means a diachronic immutable, and have had different attributions either between their modern and historical usage, or between multiple 'historical' sources using the same effective kanji-sequence to denote differing tens-power-values. As with many linguistic conventions, especially those that are notational and highly abstract, how big a 'stride' one takes to get from one place-value to the next is entirely a subjective choice when taking an agreed-upon succession of ascending scale to mean a very precise thing, especially when the consequences of making a different choice than someone else are virtually nonexistant, and any choice that at least orders the symbols in the same ascending pattern in an internally consistent and logically sound fashion is basically as 'correct' as any other. So while there are multiple tables even in the one blog-post I am rehashing, with multiple columns denoting sources and the names used for the varying scales, I will try my best to show what is the 'modern accepted' version if multiple interpretations exist, and leave any further digging to whomever reads this and finds it inadequately detailed.

For those tens-power-units that are noted in 'common' usage (though their frequency is almost certainly diminishing the higher one goes up the scale, for understandable reasons), virtually all beyond 万 are effectively raised-powers of 万=10⁴ itself.

The blog post begins with the line

子供の頃、算数の教科書で大きい数の単位を知って感動した。日本語は素晴らしいと思ったが、実はこれらは全て中国から来たものだと後に知った。以下は日本語の数の表である。

Which I am, as a rank amateur, choosing to translate as:

As a child, I was moved [tr. emotionally] upon learning the large number-units in an arithmetic textbook. [At the time] I had thought 'Japanese is wonderful', but I later learned that, in fact, these were all Chinese in origin. Below is a table of Japanese numbers.

I am adapting the actual table that follows that opening line into a more compact form, leaving out the additional columns that specify the prefix-scale applied to yield intermediate ten-powers between one myriad-power and the next, and rather focusing on the myriadic powers themselves. I am assuming (perhaps incorrectly) that anyone reading this far is either familiar enough with 一十百千 to not need the additional explainer for these sub-myriad powers, or is in any case only concerned with the big numbers, not those 雑魚 (ざこ/zako). Also note that because all these powers are above 千, when they are actually being used productively to form a proper numeral quantity, a prefix of 一 would be used to denote 1*10^x.

(By 'adapting', I really mean translating/editing the original table, as an HTML entity. It's big enough that I would probably want to write a script to generate the HTML from scratch if I were going down that road, and the HTML structure itself is relatively hard to interpret as the IP of the author, since it is very formulaic and merely encapsulates existing facts rather than putting an authorial spin or creative interpretation on them. Not that I am saying 'steal from this person', I just believe that the table itself can't reasonably be designated as a creative opus and consitute what I am doing below as plagiarism. Even the adaptation of the HTML is being done by hand rather than script or, worse-yet, via AI, and is rather painstaking as-is, so I would argue that even if the original is IP, my version is a transformative product.)

Kanji	Value
万（まん）	104
億（おく）	108
兆（ちょう）	1012
京（けい）	1016
垓（がい）	1020
𥝱（じょ）*	1024
穣（じょう）	1028
溝（こう）	1032
澗（かん）	1036
正（せい）	1040
載（さい）	1044
極（ごく）	1048
恒河沙（ごうがしゃ）	1052
阿僧祇（あそうぎ）	1056
那由他（なゆた）	1060
不可思議（ふかしぎ）	1064
無量大数（むりょうたいすう）	1068

^*: There are multiple notes on this particular entry. The first is that, perhaps due to font or unicode-related issues, the orignal blog-post by Takasugi-shi used a now-dead image link for this character, which I have done my best to find the corresponding Unicode glyph to replicate. The other is that there is an extended aside in the body of the blog itself, that explains that the value of 10²⁴ was originally (and properly, according to Takasugi-shi) designated by the character 「秭（し）」, which [the author alleges] was, in the mid-17th century, mistakenly rendered as the slightly-similar-looking 「𥝱（じょ）」, which has by the modern day become the de-facto tens-power-unit kanji for 10²⁴. While I can't comment on whether this explanation is historically accurate, I have no particular reason to fact-check the source that I am only attempting to preserve, not audit, and so will leave the Internet Archive link and its accompanying list of reference-links as the place to go to figure out what is really going on here. Another source linked to by the blog post in question, which can only be found in an even earlier wayback machine snapshot, corroborates this story, though it is unclear if this is the same source that Takasugi-shi is drawing on or whether this is one of multiple sources for this claim.

Now, onto the main point of interest I had in first finding this page, which, like young Takasugi-shi, I was moved upon finding.

無量大数 (muryoutaisuu, lit 'immeasurable large number), at 10⁶⁸ (or 10⁸⁸ depending on which system you are employing or what source you are treating as authoritative), is already pretty darn big, I would say. There aren't a lot of things you can reasonably count that high within practical reason, though unlike numbers such as one googol (10¹⁰⁰) which are effectively 'monolithic' large numbers without any nearby ten-powers you can ease yourself down to without referencing googol itself, it is just the last turtle in the chain of myriads-of-myriads. But it is by no means the end of the story, and we can go even deeper.

The end of the blog post has a huge ringer in the form of a list of numbers from a Buddhist source that grow unbelievably fast and which quickly outstrip the number of atoms in the observable universe and any other quantity that forms a basic upper-bound on how large a number you would ever need to worry about if you were purely a pragmatist. With the exception of the first number listed in the table that follows, 洛叉（らくしゃ） as 10⁵, the progression is based on 10^7(2x). In other words, instead of scaling by our basic myriad (which in this case, the base-case is 10⁷ rather than 10⁴ but it hardly matters by the end due to the sheer magnitude of the numbers being presented), we square each one to get the next. In comparison, if we were doing this the whole time in our basic set of numbers, 無量大数 being our base myriad multiplied by itself an additional 16 times, we would instead have our base myriad squared 16 times, or 10^4x216 = 10²¹⁸ = 10²⁶²¹⁴⁴. If you could write one zero per second without any breaks or interruptions, that would take slightly over 3 days to write down in Arabic numerals (and though it's begging the question, would take only four characters and 40 total strokes to write in Kanji if that were the actual value of 無量大数).

I might later do a deeper dive into the significance or origins of the names of the numbers given below, which for time constraints, I will merely leave as kanji with hiragana ruby annotations and not transliterate the whole list quite yet (though I might go back and edit this post later for the sake of finishing it more quickly but not leaving it an incomplete product). The majority I believe to be derived from Sanskrit terms with significance rooted in Buddhism, rendered into either Japanese Kanji or Chinese Hanzi (which I may, if I do compile a follow-up post to this one, explore further).

Without any further ado, here is the insane table, this time ripped mostly one-for-one from the revivified blog-page of Takasugi-shi (with the column-headers translated but otherwise unaltered).

Numerical Value	Lexical Form
105	洛叉（らくしゃ）
107	倶胝（くてい）
107×2 = 1014	阿庾多（あゆた）
107×22 = 1028	那由他（なゆた）
107×23 = 1056	頻波羅（びんばら）
107×24 = 10112	矜羯羅（こんがら）
107×25 = 10224	阿伽羅（あから）
107×26 = 10448	最勝（さいしょう）
107×27 = 10896	摩婆羅（まばら）
107×28 = 101792	阿婆羅（あばら）
107×29 = 103584	多婆羅（たばら）
107×210 = 107168	界分（かいぶん）
107×211 = 1014336	普摩（ふま）
107×212 = 1028672	禰摩（ねま）
107×213 = 1057344	阿婆鈐（あばけん）
107×214 = 10114688	弥伽婆（みかば）
107×215 = 10229376	毘攞伽（びらか）
107×216 = 10458752	毘伽婆（びかば）
107×217 = 10917504	僧羯邏摩（そうがらま）
107×218 = 101835008	毘薩羅（びさら）
107×219 = 103670016	毘贍婆（びせんば）
107×220 = 107340032	毘盛伽（びじょうが）
107×221 = 1014680064	毘素陀（びすだ）
107×222 = 1029360128	毘婆訶（びばか）
107×223 = 1058720256	毘薄底（びばてい）
107×224 = 10117440512	毘佉擔（びきゃたん）
107×225 = 10234881024	称量（しょうりょう）
107×226 = 10469762048	一持（いちじ）
107×227 = 10939524096	異路（いろ）
107×228 = 101879048192	顛倒（てんどう）
107×229 = 103758096384	三末耶（さんまや）
107×230 = 107516192768	毘睹羅（びとら）
107×231 = 1015032385536	奚婆羅（けいばら）
107×232 = 1030064771072	伺察（しさつ）
107×233 = 1060129542144	周広（しゅうこう）
107×234 = 10120259084288	高出（こうしゅつ）
107×235 = 10240518168576	最妙（さいみょう）
107×236 = 10481036337152	泥羅婆（ないらば）
107×237 = 10962072674304	訶理婆（かりば）
107×238 = 101924145348608	一動（いちどう）
107×239 = 103848290697216	訶理蒲（かりぼ）
107×240 = 107696581394432	訶理三（かりさん）
107×241 = 1015393162788864	奚魯伽（けいろか）
107×242 = 1030786325577728	達攞歩陀（たつらほだ）
107×243 = 1061572651155456	訶魯那（かろな）
107×244 = 10123145302310912	摩魯陀（まろだ）
107×245 = 10246290604621824	懺慕陀（ざんぼだ）
107×246 = 10492581209243648	瑿攞陀（えいらだ）
107×247 = 10985162418487296	摩魯摩（まろま）
107×248 = 101970324836974592	調伏（ちょうぶく）
107×249 = 103940649673949184	離憍慢（りきょうまん）
107×250 = 107881299347898368	不動（ふどう）
107×251 = 1015762598695796736	極量（ごくりょう）
107×252 = 1031525197391593472	阿麼怛羅（あまたら）
107×253 = 1063050394783186944	勃麼怛羅（ぼまたら）
107×254 = 10126100789566373888	伽麼怛羅（がまたら）
107×255 = 10252201579132747776	那麼怛羅（なまたら）
107×256 = 10504403158265495552	奚麼怛羅（けいまたら）
107×257 = 101008806316530991104	鞞麼怛羅（べいまたら）
107×258 = 102017612633061982208	鉢羅麼怛羅（はらまたら）
107×259 = 104035225266123964416	尸婆麼怛羅（しばまたら）
107×260 = 108070450532247928832	翳羅（えいら）
107×261 = 1016140901064495857664	薜羅（べいら）
107×262 = 1032281802128991715328	諦羅（たいら）
107×263 = 1064563604257983430656	偈羅（げら）
107×264 = 10129127208515966861312	窣歩羅（そほら）
107×265 = 10258254417031933722624	泥羅（ないら）
107×266 = 10516508834063867445248	計羅（けいら）
107×267 = 101033017668127734890496	細羅（さいら）
107×268 = 102066035336255469780992	睥羅（へいら）
107×269 = 104132070672510939561984	謎羅（めいら）
107×270 = 108264141345021879123968	娑攞荼（しゃらだ）
107×271 = 1016528282690043758247936	謎魯陀（めいろだ）
107×272 = 1033056565380087516495872	契魯陀（けいろだ）
107×273 = 1066113130760175032991744	摩睹羅（まとら）
107×274 = 10132226261520350065983488	娑母羅（しゃもら）
107×275 = 10264452523040700131966976	阿野娑（あやしゃ）
107×276 = 10528905046081400263933952	迦麼羅（かまら）
107×277 = 101057810092162800527867904	摩伽婆（まかば）
107×278 = 102115620184325601055735808	阿怛羅（あたら）
107×279 = 104231240368651202111471616	醯魯耶（けいろや）
107×280 = 108462480737302404222943232	薜魯婆（べいろば）
107×281 = 1016924961474604808445886464	羯羅波（からは）
107×282 = 1033849922949209616891772928	訶婆婆（かばば）
107×283 = 1067699845898419233783545856	毘婆羅（びばら）
107×284 = 10135399691796838467567091712	那婆羅（なばら）
107×285 = 10270799383593676935134183424	摩攞羅（まらら）
107×286 = 10541598767187353870268366848	娑婆羅（しゃばら）
107×287 = 101083197534374707740536733696	迷攞普（めいらふ）
107×288 = 102166395068749415481073467392	者麼羅（しゃまら）
107×289 = 104332790137498830962146934784	駄麼羅（だまら）
107×290 = 108665580274997661924293869568	鉢攞麼陀（はらまだ）
107×291 = 1017331160549995323848587739136	毘迦摩（びかま）
107×292 = 1034662321099990647697175478272	烏波跋多（うはばた）
107×293 = 1069324642199981295394350956544	演説（えんぜつ）
107×294 = 10138649284399962590788701913088	無尽（むじん）
107×295 = 10277298568799925181577403826176	出生（しゅっしょう）
107×296 = 10554597137599850363154807652352	無我（むが）
107×297 = 101109194275199700726309615304704	阿畔多（あばんた）
107×298 = 102218388550399401452619230609408	青蓮華（しょうれんげ）
107×299 = 104436777100798802905238461218816	鉢頭摩（はどま）
107×2100 = 108873554201597605810476922437632	僧祇（そうぎ）
107×2101 = 1017747108403195211620953844875264	趣（しゅ）
107×2102 = 1035494216806390423241907689750528	至（し）
107×2103 = 1070988433612780846483815379501056	阿僧祇（あそうぎ）
107×2104 = 10141976867225561692967630759002112	阿僧祇転（あそうぎてん）
107×2105 = 10283953734451123385935261518004224	無量（むりょう）
107×2106 = 10567907468902246771870523036008448	無量転（むりょうてん）
107×2107 = 101135814937804493543741046072016896	無辺（むへん）
107×2108 = 102271629875608987087482092144033792	無辺転（むへんてん）
107×2109 = 104543259751217974174964184288067584	無等（むとう）
107×2110 = 109086519502435948349928368576135168	無等転（むとうてん）
107×2111 = 1018173039004871896699856737152270336	不可数（ふかすう）
107×2112 = 1036346078009743793399713474304540672	不可数転（ふかすうてん）
107×2113 = 1072692156019487586799426948609081344	不可称（ふかしょう）
107×2114 = 10145384312038975173598853897218162688	不可称転（ふかしょうてん）
107×2115 = 10290768624077950347197707794436325376	不可思（ふかし）
107×2116 = 10581537248155900694395415588872650752	不可思転（ふかしてん）
107×2117 = 101163074496311801388790831177745301504	不可量（ふかりょう）
107×2118 = 102326148992623602777581662355490603008	不可量転（ふかりょうてん）
107×2119 = 104652297985247205555163324710981206016	不可説（ふかせつ）
107×2120 = 109304595970494411110326649421962412032	不可説転（ふかせつてん）
107×2121 = 1018609191940988822220653298843924824064	不可説不可説（ふかせつふかせつ）
107×2122 = 1037218383881977644441306597687849648128	不可説不可説転（ふかせつふかせつてん）

I probably don't need to tell you how insane it is that, even if these numero-lexical entities themselves are even more obscure than the basically-unused 無量大数 (at least, I imagine that any usage it entertains is purely as 'the biggest number that exists' rather than the precise value of '1e68'), the fact they even existed at all, however historically, is definitely impressive.

If any of this stuff is interesting to even one person [other than myself], I'd say it was worth it to keep this list of numbers alive into a new era.

Stay tuned for the announcement of my upcoming incremental idle game where you win if your score reaches 不可説不可説転. /hj.