# Counting Really,Really,REALLY High

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Like many I have long been fascinated with extremely large numbers.

Archimedes is perhaps the father of this fascination,perhaps only the earliest well-remembered person to have it.Despite living in a civilization that named no number larger than the myriad (10000) he managed in his Sand Reckoner to devise a numbering system that reached

100002x(100004)

compared with the Indian numbering system which never unequivocally extended beyond the Adant Sanghair at
1039

with the Asankhyeya beyond that ambiguous:the value is given at
10140

by past Guinness editions but sources differ[WARNING...PDF LINK]...some well below that, some far beyond Archimedes.

While the Hindu-Arabic numerals have prevailed in modern civilization, the Roman name millia for a thousand has proven the basis for number names in many languages.By the 13th century the "million",the "big thousand",had emerged as a name for a thousand thousands (1,000,000 or 106).

Nicolas Chuquet,in the 15th century,is the first person recorded to have written of and explained the further generation of large-number names,delineating the billion,trillion,and quadrillion as the second, third,and fourth powers of a million.With the intervening thousand-multiples dubbed a milliard,billiard,etc. this system was established,only to be overtaken in many countries by human laziness, demanding a new "illion" name every three instead of six orders of magnitude,all in the same instead of alternating formats,destroying the coherence of the implicit number being the power of a million it represented and harshly cutting the capacity of the system to deal with truly immense numbers.

On this website I detail my own system of large number naming, which builds on the traditional/original "illions" and "illiards" even though they have almost been driven from the English language, and I take delight in conscripting the scientific prefixes mainly devised for that other monument to human laziness,the metric system (a simplistic farce conceived by eccentrics at the behest of madmen, and imposed by dictators for the convenience of idiots).

To lay out the matter:

### Original Development

The number names up to the nonillion definitely existed by the 17th century.Further development along the lines prescribed by Chuquet exploded in the 19th century,with the "illions" reaching the 10th (decillion),20th(vigintillion),and 100th(centillion) powers of a million; these names being applied to half that power,times a thousand,by devotees of the short scale.

These are the numbers one is likely to find in dictionaries, the ones between vigintillion and centillion usually absent; by 1860 a Professor William Downs Henkle(1828-1881) had come up with a naming scheme that proceeded with examples all the way to the millionth "illion",which he dubbed the milli-millillion.Munafo has found Henkle's recitation of previous work and additions thereto in an 1860 Ohio Educational Monthly here...he regarded the duodecillion (cited in Pike & Leavitt by 1826),as the then accepted limit of common usage before extensions,but cited Alfred Holbrook (1816-1909) as the coiner of "millillion" for the thousandth illion in his 1859 self-published book "The Normal" (page 306).He also credited "centillion" as a coinage of Noble Heath in his A Treatise on Arithmetic, copyright 1855,though the word has been dated by a dictionary to 1852. Another of Henkle's sources is Benjamin Greenleaf's National Arithmetic,which includes "Vigintillion" in a short-scale list in a discussion (page 15) of short and long scales...the preface to the 1847 edition indicates the first was twelve years previously.

Henkle's list was republished by Edward Brooks in The Philosophy of Arithmetic(1876),retained in the (1904 edition). (It is on page 571 as an appendix after the index).
Borgmann (see below) cited the latter edition in 1968, using only the short-scale meaning though Henkle originally cited the long.

The names beyond "centillion",however,remained unaccepted curiosities.

#### The Googol Interruption

At a date variously given between 1920 and 1938,mathematician Edward Kasner asked his young nephew to think up a name for 10100,and the boy came up with "Googol".When Kasner published a book introducing the term in 1940 he did not give a reason for the choice,but as a former number-fascinated child I might guess that the resemblance between the numeral 0 and the letter O and the presence of a hundred zeroes in the number, along with its impressive size,could have suggested a name with "ooooooh".(An alternative hypothesis might be that the breathtaking size of the number would evoke an expression like the "goo-goo- googly eyes" of the cartoon character Barney Google). Following up on this came the name "googolplex" for ten raised to this number.

10100 is well within the range of the already-created number names,though as 100 is not a multiple of 6 or even 3 it was not an even "illion" or "illiard".On the short scale it is ten duotrigintillion and on the long scale ten sexdecilliards.But as the canonical names for the number were not and are not in typical discourse,as the number exceeds the subatomic particles in the Universe,and even further exceeded what was known circa 1940, the irregular name had a chance to catch on and distract people. And the unfortunate disagreement as to whether a centillion is a thousand times the cube of a "googol" or the sixth power of a "googol" inhibited ready citation of the centillion as filling the "biggest accepted number" position.The Guinness book has mentioned the "googol" out of its sheer notoriety despite its never being a recordholder,switching from the long-scale proper name "10,000 sexdecillion" to short-scale "10 duotrigintillion" with the 1976 edition.

The high profile of this disconnected and arbitrary amount and name has disrupted public understanding of number naming systems.

### Alternative Extensions

Over the years people have addressed the Henkle name-set with eyes toward both refinement and extension.A significant contribution has been made in the pages of Word Ways,a magazine whose focus is recreational linguistics rather than mathematics.Founding editor Dmitri Borgmann highlighted the Henkle set asking about potential correction of Henkle's Latin in 1968,and the next year Rudolf Ondrejka contributed a list with tweaked Latin and an extension to the milli-millimillillion. During his decades-long editorship of Word Ways,A. Ross Eckler Jr. (also a correspondent of mine on supercentenarian research) published three articles in extending and refining number names by John Candelaria of California...the first two in 1975 and 1976 comprising a significant but comparatively modest extension with different nomenclatural conventions,and the third in 1983 redefining the lower reaches and going dramatically further.Munafo has reprints at his site of the Candelaria articles that can be found at Butler University Digital Commons in PDF format here, here,and (after Ondrejka had responded by extending his system in this 1976 article) here. Candelaria submitted a further manuscript that exponentially expanded his system in 1986 that was not published but has been provided to me by Mr. Eckler,and particulars will be addressed below.

Others have also proposed naming systems that build on the "illions" though most accept the short scale that leaves out the "illiards" and thereby cut the efficiency of their nomenclature in half.Donald Knuth of up-arrow-notation fame revived the "yllion" spelling Chuquet used and the myriad base of Archimedes to create a system where each "yllion" was the square of the previous,Russ Rowlett of UNC proposed Greek roots be substituted for Latin,and so forth.

However,the "installed base" of powers of 1,000,000 from 10 to 100 relies on the following unit prefixes:

• un- for one,
• duo- for two,
• tre- for three,
• quattuor- for four,
• quin- for five,
• sex- for six,
• septen- for seven,
• octo- for eight,
• novem- for nine,
preceding the following tens prefixes:

• dec- for ten,
• vigint- for twenty,
• trigint- for thirty,
• quatrigint- for forty,
• quintigint- for fifty,
• sexagint- for sixty,
• septuagint- for seventy,
• octogint- for eighty,
• nonagint- for ninety.

Thus,the 82nd power of a million (1,000,00082) is a duooctogintillion.John H. Conway(1937-2020) and Wechsler proposed a hundreds-place system in which tens prefixes would be followed by an "a" after the "t" at times,while Henkle omitted the "int" entirely (using "vigillion","trigillion",etc.)...a middle course of streamlining seems wisest to me,eliding the "t" where inconvenient.

### My Personal Interest

I was born in 1961,and literate and numerate from an early age. My father taught me the numbers to quadrillion,maybe quintillion; mathematics professor John Danskin,a friend of the family,told me the names up to decillion.At the time I knew only the short scale. By 1972 I had discovered the vigintillion in a dictionary,and had my first Guinness Book,which mentioned the centillion and Ondrejka's milli-millimillillion...each in those editions given only their long-scale meaning.(Ondrejka's 1968 article provided for both while Candelaria assumes only the short scale;his 1976 article,never mentioned in Guinness,responded to Candelaria in short scale terms).

I was naturally interested in the expansion of numbering, though like Robert Munafo and "Sbiis Saibian" I went through a phase of naive reinvention, renaming the centillion the "milyotillion" for no particular reason (see "Not Arbitrary" below) and banging out about five pages of names (still using the short scale) on my typewriter in the 1970s...I just didn't want the number names to stop.

I kept pace with the Guinness Book (there are only two years since then I don't have yet) and noted various inclusions...for years a reference appeared "the number Megiston...is considered too great to have any physical meaning",the "..." sometimes merely the word and digits "written 10" and sometimes more correctly with a circle around the "10" as proposed when it was named (by 1950) by Hugo Steinhaus (1887-1972) but never with a pentagon as provided when Leo Moser (1921-1970) refined the notation.I assumed this meant that "Megiston" was simply a name and the circled 10 a symbol for "a number too great to have any physical meaning",and it was not until I discovered Susan Stepney's large-number webpage in the late 1990s that I learned it referred to a specific value.

I may digress into explication of the Megiston in the future, as I am interested in how it ranks compared to the numbers I name below,but this is out of chronology compared to the explanation of the genesis of my system.

With the 1979 edition the Guinness Book updated to replace the second Skewes Number

101010103

(as given,though this is actually an approximation) with Graham's Number as "the largest number in a mathematical proof" (Munafo has discovered that Skewes was actually "outnumbered" in 1962) and referred to Knuth's arrow notation as required to express it. I later saw the notation explained in Scientific American, Stepney's page linked above explains it as well.

For years in the 1980s Guinness referred to John Candelaria, "The Man Infinity Fears",and his system culminating in the "milli-decilli-fiveillionillion".I have entered into correspondence with him and he has offered to explain his system,but his 1986 manuscript that I got hold of in 2010 does not include this number, and from what I can tell of his system I'm not sure how it would be generated,but if there was such a number I expect a milli-decilli-sixillionillion would be much larger.

#### The Candelaria Manuscript

Candelaria is obsessed with the exclusive and consistent use of Latin roots.As such,he uses as multiplicative prefixes what are generally familiar as divisive ones...milli-,centi-,and deci-.(In discussion in his article of Graham's number and hyper operators,he uses "quadriation", "quintiation","sextiation","septiation",etc. where most use "tetration", "pentiation",etc.).Nonetheless English cardinal-number names make their appearance in names like "millifourillion","millifiveillion",etc.

The "millillion",in his system,is the millionth "illion",and with no "illiards" separating them this is a mere 103000003.The "millioneillion" is the million millionth "illion",the "millitwoillion" is the million million millionth "illion",and so to the "millimillionillion",which is the million-to-the-millionth-power illion. At this point the "levels" of exponentiation in giving the number of zeroes give way to "layers".

The millimillillionillion follows in due course,but what would be the "millimillioneillionillion" is streamlined to "millillioneillion". The "layers" of exponentiation grow to a million with the millillimillionillion,at which point this is deemed a "height".

The number of illions taken to reach the millillimillillionillion raised to the millionth power reach the millioneillioneillion,which raised to the millionth power reaches the millioneillitwoillion,which sets off asymmetrical growth, the millioneillimillionillion ordinal of illions rising to its millionth power reaching the millioneillimillillionillion at which point the million "heights" become an "elevation".

The millioneillimillioneillionillion occurs without streamlining, and the illion the millionth power as many times down is a millioneillimillillioneillionillion. The "elevations" reach a million with the millioneillimillillimillionillionillion,which kicks off the "altitude" phase of exponentiation and has as its millionth-power-number illion the millioneillimillillimillillionillionillion,but when you get to the millioneillimillillimillioneillionillionillion,a millionth power further lies the millitwoillioneillion.

When the "one" to the right reaches a million and then a millillion, the "altitudes" stop,the "perigees" start,and the tables end.

In the text he explains that when the two enclosed incrementing numbers both reach a millioneillioneillion,the millillioneillioneillion is reached,and only when the two included ones each reach that number in turn is there a millioneillioneillioneillion.He says that in Graham's number terms this is about G-3,and in order to express Graham's number itself (G-64) he would need a name with about G-1 "ones" in it.

Caveat:I may have gotten confused along the line,and welcome his corrections on this.

In July 2010 he wrote to me explaining further his "T" and cross notations,an alternative means of expressing Knuth-arrow functions, and indicated that the Guinness references were obsolete,though he did not explain how the number referenced there fit into his other nomenclatures.

### Objectives For Extension

I did not see the Candelaria manuscript until 2010,and began developing the system I use in the 1980s.Certain standards,some applied in retrospect,have guided me.

Systematic--The names used must be readily comprehensible, related to the numbers they express in a viable sequence.Slavish adoption of a particular root system regardless of implications in English is erring on one extreme,so no "divisive" Latin;throwing in terms from a plethora of cultures until you can't find any more is another extreme to avoid,the implication of one language's words being more augmentive than another's is absurd.A computation method should be implicit;no "bubebillion" for the billionth Busy Beaver number.

Backward compatible--I am (besides insisting on long-scale meanings,which are the originals anyway) not disturbing the "installed base" of the first 100 "illions".I see nothing wrong with the "backward counting" of intercalated names (see below).I am not "starting from scratch",else a systematic approach would be to rename a million an "aion" and then go "bion" to "zion","aaion" to "azion" to "baion" to "zzion" to "aaaion" and so forth.

Not Arbitrary--No throwing of random syllables at numbers hoping they will stick disregarding the systematic approach.This is what happens with the "Googol" and derivatives,and many other names assigned by Jonathan Bowers to his "Infinity Scrapers" (so what does each "mea" before the "lokka" mean anyway?)...else I could just say "the sum of all numbers named by Bowers on his Infinity Scrapers and "Illion" pages is an Oompaloompakoopatrooparamalamadingdong,its factorial is an Oompaloompakoopatrooparamalamadingdongbang, and when you load that as every term into one of his "Exploding Arrays" measuring that many in each of that many dimensions,you get a Wopbopaloopbobadabingdonghey!"...totally arbitrary. The apogee of arbitrary naming of course is Frank Pilhofer's "I define a Frank as one more than the largest number you can come up with",which opens the door to a megafrank of paradoxes.

Candelaria's layer/height/altitude/etc. seem arbitrary choices to me for all his dedication to systematic roots.

Words Work--The underlying philosophy of having a number naming system is that using names can indeed convey value more efficiently than words or symbols...scientific notation is (in the lower reaches) for those who can't find the name of a number, or where the individual name happens to be impractical.(Subtracting one from a giant number results in a giant name pretty quickly!)

Consider,for example,the issues involved in surrounding a number N with an upward arrow,a rightward arrow,a circle (or concentric circles),a picture of an explosion,and an arrow returning to the base of the upward arrow,and explaining that this means the following:

1. Like a number in a Moser polygon,N is raised to itself.
2. This number is then placed before and after as many Knuth upward arrows.
3. The resulting number is then repeated as many times separated by (John H.) Conway right arrows.
4. The then resulting number is placed within that many Moser polygons each of that many sides.
5. This resulting number is then used as every term of a Bowers Exploding Array measuring as many in each of as many dimensions.
6. The entire cycle of operations repeats N times.

You can figure out how these operations work from Stepney's and Bowers's web pages,but the representation with symbols is much more involved than simply defining a word to do all this...I'll use "popble",contracted from elements of "point", "polygon",and "blast".

It should be readily apparent that any integer greater than 1 popbled is incalculably greater than Graham's Number...if you popble 2,you have 4^^^^4 repetitions of 4^^^^4 separated by Conway arrows,and even 4 repetitions of 3 separated by Conway arrows are bigger than Graham's Number,and you haven't even gotten to the Moser or Bowers stages of the first cycle yet.(Bowers volunteers that 2 popbled easily exceeds a gongulus (defined on his "Infinity Scrapers" page)).

I mention this not to just call a million popbled a "Popblillion" and go home,but to demonstrate that a word can say a lot once understood.And we have much bigger numbers to popble.

#### 2016 DEFINITION OF POPBLE

After due consideration the double use of the Moser function did not sit well with me,and I amplified the power of the popble by substituting use of the Ultrex Function for the first step*.So

1. N is ultrexed N times to bound N.
2. This number is then placed before and after as many Knuth upward arrows.
3. The resulting number is then repeated as many times separated by (John H.) Conway right arrows.
4. The then resulting number is placed within that many Moser polygons each of that many sides.
5. This resulting number is then used as every term of a Bowers Exploding Array measuring as many in each of as many dimensions.
6. The entire cycle of operations repeats N times.

*As of 2020 I am considering substituting the more powerful SHOT Function or LHOT Function.

### NEW 2020 DEFINITION OF POPBLE

After due consideration the implied use of the Conway-Guy function seemed obviously improvable under Ultrex principles.

1. N is ultrexed N times to bound N.
2. This number is then placed before and after as many Knuth upward arrows.
3. The resulting number is then used as the base,bound, and number of iterations in the Ultra Conway-Guy Function,using Conway right-arrows..
4. The then resulting number is placed within that many Moser polygons each of that many sides.
5. This resulting number is then used as every term of a Bowers Exploding Array measuring as many in each of as many dimensions.
6. The entire cycle of operations repeats N times.

*I may create a function that substitutes the more powerful SHOT Function or LHOT Function.

Constraints--If we are going to have words as names, they can't veer into appearing as mathematical expressions despite their mathematical definitions.The only punctuation marks within a word that don't look out of place are the hyphen and apostrophe...and they must be used sparingly and to effect.(Henkle used hyphens very early,and others have copied this).

The Creation--I've probably got the legal-pad tearsheets around somewhere,but I devised my system at some time in the late 1980s,when Lunacon was at the Tarrytown Marriott and Isaac Asimov was still attending.I had read his 1963 column "T-formation" which explained numbers in powers of 1012 from my father's collection,and in college seen his "sequel" called "Skewered!" which wrote of Skewes' Number,and wanted to show him my innovation. At that time,though Guinness had credited the apocryphal "dea" prefix, the World Almanac showed up to "exa",so my system went up to "exillion" before the first hyphenations,the zettillion and yottillion were added after 1991.Not until 2010 did I discover that Bowers had (some time later,I believe) used these names for different numbers.

But,let us proceed.

### The System

Now at last my own system for the naming of giant numbers.

I despair of getting it accepted by the mathematical community, in part because it is most effective in combination with the moribund European system,or long scale,as noted above...each name's implicit number is the power to which a million is raised,a billion is a million million and so on...but I am not about to "short-scale" it.

I have had contributions detailing it as a proposal thrown back in my face by wikiculture types obsessed with guaranteeing all information comes to them secondhand or worse.

My innovation,then,begins at the current accepted terminus,the centillion,which is defined as 10^600,the hundredth power of a million,rather than 10^303 in the far more accepted American system.

Each intervening thousandfold is an "-illiard" of the preceding "-illion".

To commence with one can simply continue with the scientific prefixes:
10000001000= 1 kilillion
10000001000000= 1 megillion
10000001000000000= 1 gigillion
10000001012= 1 terillion
10000001015= 1 petillion
10000001018= 1 exillion
10000001021= 1 zettillion
10000001024= 1 yottillion
and with the addition of new prefixes by the 2022 CGPM
10000001027= 1 ronnillion
10000001030= 1 quettillion

#### Intercalated names

Within the major names must of course come many in-between names. The "unit" and "tens" prefixes mentioned above work for the last two digits of the name's implicit number (the power to which it raises a million).For clarity's sake,Greek prefixes multiply larger quantities that analogous Latin prefixes add to.

1000000202= 1 duobicentillion
1000000303= 1 tretricentillion
1000000404= 1 quattuorotetracentillion
1000000505= 1 quinpentacentillion
1000000606= 1 sexhexcentillion
1000000707= 1 septhepcentillion
1000000888= 1 octooctaginoctacentillion
1000000109= 1 noncentillion
1000000900= 1 nonacentillion
1000000909= 1 nonnonacentillion
1000000999= 1 novemnonaginnonacentillion

Beyond a kilillion,the prefix "hec" is used for hundreds of thousands,etc. of powers of a million,to avoid the "divisive" connotations of "centi" where it is not part of the "installed base".The full "hecto" is reserved for use in hyphenated forms, see below.

10000001001= 1 unkilillion
10000001002= 1 duokilillion
10000002000= 1 bikilillion
10000001010= 1 decikilillion
10000001100= 1 cenkilillion
10000001234= 1 quattuorotriginbicenkilillion
100000010000= 1 deckilillion
1000000100000= 1 heckilillion
100000020200= 1 bicenbideckilillion
1000000122222= 1 duoviginbicenviginheckilillion
1000000222222= 1 duoviginbicenviginbiheckilillion
1000000300033= 1 tretrigintitriheckilillion
The "ti" erodes before adjacent operators only.
1000000555555= 1 quinquinquaginpentacenquinquaginpentaheckilillion
10000001234567= 1 septsexaginpentacenquattuorotriginbiheckilmegillion
100000010000000= 1 decmegillion

The prefix deca is like hecto reserved for hyphenated forms.

A few details need to be ironed out but I hope you get the picture.

#### Enter the hyphen

With the conclusion of the accepted prefixes at the quettillion, the 10^30th power of a million,itself of course but a tenth of a percent of a quettilliard,a new strategy soon becomes necessary... as in this system 10^30 is a quintillion,not a nonillion, a million raised to 10^31 is a decaquintillio-illion. A thousand of these are a decaquintillio-illiard,a thousand of those a decaquintillio-million,and thereafter new elements go after the hyphen until the pre-hyphen part must be incremented.

10000001032= 1 hectoquintillio-illion
The "hecto" here represents a hundredfold increase of the raw number before the hyphen,rather than being an element in the name itself; not needed for a quadrillion,but when you get to a kilillio-illion...
1000000100 x(10000001000)= 1 hectokilillio-illion
while
10000001000000100000= 1 heckilillio-illion

A big difference.

100000010000001000000= 1 megillio-illion
100000010000001000000000= 1 gigillio-illion
100000010000001012= 1 terillio-illion
100000010000001015= 1 petillio-illion
This generation of course persists through the quettillia-illion, which is a million raised to a thousand times a million raised to 10^30.

At this point a second generation of hyphens becomes necessary; the amount before the last hyphen being (consonants after the final vowel deleted) the amount to which a million is raised, independent of any multipliers then applied after the hyphen.

The decaquintillio-illio-illion is of course

1,000,0001,000,0001031

Still short of the Skewes numbers,but incredibly much larger than the googolplex.The yottillio-illio-illion is
1,000,0001,000,0001,000,0001024
beside which both Skewes numbers are inexpressibly small trifles.

This then was the system as I had it after the official adoption of the zetta- and yotta-prefixes in 1991 reached me,with the principles having been drawn up years earlier.

With the November 18,2022 approval (PDF LINK---SEE RESOLUTION 3) of the proposal to introduce ronna- and quetta- prefixes the quadrillia-illion became the ronnillion and the quintillio-illion the quettillion.

A yottillia-illia-illia-illia-illion may be an extremely large number for most uses,albeit only one part in the millionth power of a million of a yottillia-illia-illia-illia-megillion,but I realized that further extension of the system would be needed to overcome the lengthening of the names at this point (obviously intercalated names have become utterly unwieldy already) and the apostrophe was the available tool.

### The System Extensions of 2010

In 2010 I finally got busy codifying the notions I'd had in this direction.

#### Enter the apostrophe

The easy application of the apostrophe is after a prefix that denotes the number of layers of exponentiation.Where the "googol" is ten sexdecilliards and a million raised to this is a

decasexdecillia-illion = 100000010100

Inserting an apostrophe (besides transferring the accent to "ca") yields a

deca'sexdecillia-illion = 1000000(1099)(1099)(1099)(1099) (1099)(1099)(1099)(1099)(1099)(1099)

Unless your font size is optimized for the legally blind, or your browser does not change font size for superscripts, I guess I have to clarify that that's ten layers of (1099) exponents.

Of course the prefix to the apostrophe (always before a name's final hyphen) can itself be a hyphenated-number exponent...a sexdecillia-illio'sexdecillia-illion is 1000000 raised to 1099 in a tower (1000000 raised to 1099) levels high and a yottillio-illio'yottillio-illion is 1000000 raised to (10000001024) in a tower (100000010000001024) levels high.

Can there be further apostrophe-and-hyphen chaining?
Of course.
Is this the most efficient way of keeping the numbers growing?
Of course not!

#### The verbal particles

We now reach the verbal particle operators,and some arbitrariness in choosing them is unavoidable.However,I take solace in every vowel following the letter D resulting in a particle meaning "of" in some European language or other.

Each particle,invoked by a prefix designating a number larger than 1, modifies the term immediately to its right.Apostrophes and hyphens in such prefixes relate only to the next operator/term..

##### The da- operator

The da- operator is an exponent to which the element before the following apostrophe is raised.Thus,compared to the aforementioned yottillio-illio'yottillio-illion,a yottilliodayottillio'yottillio-illion raises a million to a yottillion not the yottillionth power of a million times,but the yottillionth power of a yottillion times.

##### The de- operator
The de- operator is a number of times the da- operator is raised to itself.Compared to the aforementioned yottilliodayottillio'yottillio-illion, a yottadeyottilliodayottillio'yottillio-illion raises a million to a yottillion not the yottillionth power of a yottillion times,but the (yottillion raised to itself 1024 times)th power of a yottillion times.
It may be noted that these operators are to some extent taking on the role of Knuth arrows.
##### The di- operator

The di- operator is a number of times the raising of the de- operator to itself that many times is repeated.

##### The do- operator

The application of the do- operator to the di- operator involves the di- operator being raised to itself dido times.

##### The du- operator

The du- operator is the well-known "hyper" operator. Thus,a number beginning triyottilliadubiyottilliado... would see the (biyottilliard) "do" operator "triyottilliardated" to a triyottilliard before application to the "di" operator.(The operator should be seen as specifying the number of Knuth arrows).

##### The sha- operator

The sha- operator is based on my SHOT Function,which juices up hyper-operations. Thus,a number beginning triyottilliashabiyottilliadu... would see the (biyottilliard) "du" operator SHOT to the bound of a triyottilliard before application to the "do" operator.

##### The pa- operator

Now we come to the patriarch of operators, the "pa" operator...in the tradition of my favorite Norwegian folktale, Den Syvende Far i Huset (The Seventh Father in the House)...where a traveller,seeing a man outside a farmhouse,asks if he can stay the night and the man refers him to his father inside the house,and only after the issue is referred six times in turn to a previous generation does the incredibly wizened ultimate ancestor give permission.

"Pa" is short for popble.

Remember popble?

This operator is the number of times the "du" operator is popbled before application to the "do" operator...or the "sha" operator, if present,popbled before application to the "du" operator.

But it's a very large number.

#### Nesting operators

Of course,for some numbers can never be large enough,and if you've read this far you're likely to be among them.It is of course possible to nest operators subject to nothing being invoked at a level unless all subsequent operators have been invoked.Within a name as a whole,a final apostrophe must precede a final hyphen,a final da-operator precede a final apostrophe,etc,and this holds within nested layers.But what precedes an operator can be a prefix:
exapa
a number from the final vowel:
exilliapa
a hyphenated number:
exillia-illiopa
an apostrophized/hyphenated number:
nonayottillia-illia'octoyottillio-illiapa
a number with a da- operator before the apostrophe:

And so forth...these all shown as prefixing a pa- operator but they can be applied to whatever particle operators a number has in its name,in the proper sequence.

#### Meta-operators

Of course,there can be further operators that work on other operators:

##### The re- operator

The "re" operator relates to repetitions...
Re- operators do not work alone.
Re- operators work in relation to the next final non-re operator.
The basic format is re(repetitions)re(repeated string)[operator], with the function of the closing operator being performed on the repeated string a number of times dictated by the repetitions string to reach a final value at which the closing operator actually performs its work on the next operator.

Thus,

reyotta'yottillia-illiorezetta'biyottillia-illiapa

involves a zetta'biyottillia-illiard being popbled a yotta'yottillia-illion times to reach the number of times the du-operator of the number is popbled before application to the do-operator.

An arithmetical prefix before the first of a pair of re- operators means that the number reached by applying the repetition string to the repeat-string has the repetition string applied to it again,and this is repeated until this is done the number of times derived from the prefix before the product is applied to the next operator.However, if a hyphenated re- operator precedes it,it applies the repetitions number to the repetition string. Thus,

trireyotta'yottillia-illiorezetta'biyottillia-illiapa

involves a zetta'biyottillia-illiard being popbled a yotta'yottillia-illion times,that number being popbled a yotta'yottillia-illion times,and that number being popbled a yotta'yottillia-illion times to reach the number of times the du-operator of the number is popbled before application to the do-operator...or sha-operator popbled before application to the du-operator.

It might be noted in passing that combination of "re-" and "du-" operators is all it takes to pass Graham's Number without any need to invoke the more powerful "pa-" operator,or even the "sha-" operator.

A re- operator prefixed to the first of a pair of re- operators means the "next final" operation is first applied to the repetitions string the number of times in the repetitions string to reach the actual repetitions number used on the repeat-string. Thus,

rereyotta'yottillia-illiorezetta'biyottillia-illiapa

involves a yotta'yottillia-illion being popbled a yotta'yottillia-illion times,and this product-popbling being repeated a yotta'yottillia-illion times,to reach the number of times a zetta'biyottillia-illiard is popbled to reach the number of times the sha-operator of the number is popbled before application to the du-operator.

A hyphenated re- operator prefixed to the first of a pair of re- operators means the "next final" operation is applied to the repetitions string the number of times in the repetitions string, and this cycle repeated on the product of each set of so many repetitions the number of times in the repetitions string,to reach the actual repetitions number used on the repeat-string.A prefix to this hyphenated re-operator multiplies this.

An apostrophized re- operator can be prefixed to a hyphenated re- operator that is followed by a pair of re- operators,and signifies the "re-re" operation being performed on the product of the "re-re" operation performed the product of the "re-re" operation number of times, the product of the "re-re" operation number of times.

At this point we had best demonstrate on a sample number,and by this time the number names have grown unavoidably large,though the factor by which they are shorter than the numbers they name has grown transastronomically.The name of the sample number is within the next pair of horizontal lines...

This combines nesting operators and simple and complex meta-operators. Let us begin.

"TNRSF" means "The Number Reached So Far".

Since there is a re're-re situation,we first locate the repetition string, nonayottillia-illia'nonayottillia-illiadooctoyottillia-illia'yottillia-illiadiyottillia'yottillia-illiodetriexillia-yottilliodayottillia-illia'yottillia-millia and confirm that the next final operator is a "pa".

Calculating the value of the repetition string,we take a million and raise it to a power tower of a nonayottillia-illiard nonayottilliards, and multiply by 1000 to get the nested "do" operator.
The nested "di" operator is a million raised to a power tower of an octoyottillia-illiard yottilliards,all multiplied by 1000.
The nested "di" operator is raised to the nested "do" operator, to get the number of times the process of raising the nested "di" operator to itself in a power tower,raising the resulting number to itself as many times as itself,and so on is repeated.
The resulting number is the number of layers in the power tower by which the nested "de" operator (a million raised to a power tower of a yottilliard yottilliards) is raised to itself.
The nested "da" operator (a triexillia-yottillion,or a million raised to the 3,000,000,000,000,000,000th power of a million, all multiplied by 1,000,000,000,000,000,000,000,000) is raised to the resulting number to produce the number to which a yottillia-illiard (a million raised to the 1,000,000,000,000,000,000,000,000th power of a million,times a thousand) is raised to produce the number of layers in the power tower of yottilliards to which a million is raised.
The resulting number is multiplied by 1,000,000,000.

We have now calculated the repetition string of the "pa" before meta-operation..

Since the next operator is "pa",we must popble this number.

As noted before,popbling a number N involves N cycles of operation, and where multiple popbling occurs the second popbling (of N-popbled) involves N-popbled cycles,the third N-popbled-popbled cycles,and so on until the required number of popblings is completed.

In each cycle:
The input number is ultrexed itself times to a bound of itself.
This number is placed before and after as many Knuth arrows.
The resulting number is the first term,and the number of terms in,a Conway chain separated by horizontal arrows,as well as the number of times the creation of such a chain is repeated by the Ultra Conway-Guy function. (Recall that four threes, separated by three arrows,pound Graham's Number into subatomic dust, and each term added expands more than all previous terms).
The then resulting number is placed in as many Moser polygons each of as many sides.In resolving these from the triangle level,of course, each square has enormously more triangles,each pentagon enormously more squares,each hexagon enormously more pentagons,etc.,than the last.And the final square of the final pentagon has far more triangles than all previous squares combined.
The number resulting from the full resolution of the Moser polygons is each term of,and the number of terms in each dimensions of,and the number of dimensions in,a Bowers Exploding Array...and even the linear Bowers arrays outstrip Conway chains by exponentially increasing amounts from the fifth term onwards,and each row,each plane,each dimension adds more than all previous together.

This cycle of operations repeats,in the first case in this instance, 1,000,000,000,000,000,000 times the number of the repetition string (given the "exa" before the first "re" of the pair).

The hyphenated "re-" is now invoked.

TNRSF is popbled TNRSF times,to yield a number popbled that many times, this cycle repeated TNRSF times.

This yields "the product of the re-re operation",referenced above..

The apostrophized "re-" is now invoked.

TNRSF is treated as the repetition string,and all foregoing operations are repeated on it TNRSF times.This yields a number on which those operations are again performed that many times.

The "yotta" before the apostrophized "re" is now invoked, multiplying the repetitions 1,000,000,000,000,000,000,000,000-fold.

We have now completed the meta-operation on the repetitions string of the "pa" operator.

We now have the number of times we popble the repeat-string of the "pa" operator,which is itself nonayottilliadanonayottillia-illia'nonayottillia-illia (calculated by raising a million to a power tower of a nonayottillia-illiard-to-the-nonayottilliardth-power nonayottilliards,and multiplying the result by 1000).

Popbling that number the meta-operated repetitions string number of times,we have the calculated "pa" operator.

We use the "pa" operator as the number of times to popble the "du" operator,nonayottilliadayottillia'yottillia-exillia (raise a million to a power tower of a yottilliard to the nonayottilliardth power yottilliards,multiply by the 1,000,000,000,000,000,000th power of a million,and multiply by 1000).

At this point we reach more (simple) meta-operators.These are resolved before application of the "du" operator.

Because there are re-operators,the "do" operator works on itself before being applied to the "di" operator.The repetitions string is yottillia-billia'yottillia-illiadayottillia-illia'yottillia-illio and the repeat-string is triyottillia'yottillia-illia so a triyottillia'yottillia-illiard (a million raised to a power tower of the 3,000,000,000,000,000,000,000,000th power of a million, times a thousand,yottilliards,all times a thousand) is raised to itself itself-raised-to-itself times (a power tower), the resulting number raised to itself itself-raised-to-itself times, this repeated a yottillia-billia'yottillia-illiadayottillia-illia'yottillia-illion times.

The "du" operator calculated above is now applied to this "do" operator,which is "calculated-du-operator-ated" to the calculated-du-operator.

This finally calculated "do" operator is the one applied to the "di" operator,which is bizettillia-illia'exillia-trillio (calculate by raising a million to a power tower of a bizettillia-illiard exilliards,and multiply by 1,000,000,000,000,000,000). The "di" operator is raised to itself in a power tower of itself layers, and this operation repeated on the resulting number, a number of times equal to the "di" operator raised to the calculated "do" operator.

This calculated "di" operator is applied to the "de" operator,
yotta'yottillia-illio (a million raised to a power tower of 1,000,000,000,000,000,000,000,000 yottilliards) which is raised to itself in a power tower of a number of layers equal to the calculated "di" operator,and this operation repeated on the resulting number, a number of times equal to the "de" operator.

The "da" operator,octoyottillia-illia (a thousand times the octoyottilliardth power of a million) is now raised to itself a number of times equal to the "de" operator.

TNRSF (the calculated "da" operator) is the exponent to which one raises a nonayottillia-illiard (a thousand times the 9,000,000,000,000,000,000,000,000th power of a million) to get the number of layers in a power tower of yottillions to which one then raises a million.

Multiply by the 1,000,000,000,000,000,000th power of a million.

Multiply by 1,000.

### System Extensions of 2011

Beyond a multiplicative prefix to an apostrophized re- operator, a hyphenated prefix goes yet further...after the "re-re" operation being performed on the product of the "re-re" operation performed the product of the "re-re" operation number of times, the product of the "re-re" operation number of times,the product of this is then treated as the product on which the operation is performed that many times,and this continues on each product for a number of iterations indicated by that hyphenated prefix.

Thus,the above sample number would be much larger if it had started "Yotta-re'" rather than "Yottare'" (and of course there can be "Yottillia-re'" or "yottillia-illio-re'" et cetera).

##### The pre- operator

The pre- operator is the meta-operator's meta-operator; it is employed only on re- operators that have been through the above extensions,in order to pump up the number of re-iterations performed with the prefixed re' operator.

As ever,one looks ahead to determine what function the meta-operator is being applied to,and this function is applied to the second of a pair the number specified in the first of a pair of pre- operators.

preyottilliapreyotta-re'

would,if a pa- operator were involved,popble 1,000,000,000,000,000,000,000,000 a thousand times the 1,000,000,000,000,000,000,000,000th power of a million times to get the number of iterations the re' operator would use starting with the "re-re" operation being performed on the product of the "re-re" operation performed the product of the "re-re" operation number of times,the product of the "re-re" operation number of times.

A direct prefix to the first of a pair of pre- operators functions as it does for the re- operators,while a "re" prefixed to a pair of pre-operators functions as does the hyphenated re-operator;this in turn can take an arithmetical prefix.

Thus,the sample number above could have begun

Yottilliarepreyottillia-illiapreyotta-re'

Keep counting...

Louis Epstein

I note that I originally drafted my page on coming upon the pages of Stepney, Robert Munafo,and Matt Hudelson....this expansion was revised January 22,2011, with clarifications on Graham's number and the addition of pre- operators to the 2011 extensions.
Two typos were fixed at the end of 2015,link to Titled Numbers added January 1st 2016.On January 8th 2016 links to the book by Brooks and the Henkle list were added,on January 15 link to Holbrook's book and on January 18th links to Heath's,Pike's,and Greenleaf's.
Enhanced definition of popble published February 4,2016.
On the death of John Horton Conway,his dates were added and misattribution of the number naming system to the OTHER mathematician John Conway corrected,April 12th 2020.

Mention of SHOT Function linked April 25th 2020,and "sha" operator employing it in amplifying numbers added April 26th 2020.Copyedit May 14th.
Active proposal of new prefixes and its effects noted May 16th,corrected May 17th.

Popble again redefined by the Ultra Conway-Guy Function June 6,2020.