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Proceed to Safety

Large Numbers  

The earlier parts of this article are very well summarised

by Adam Townsend in the wonderful article in UCL's

chalkdust magazine

This page begins with million, billion, etc., proceeds through Googolplex and Skewes' numbers (organised into "classes" based on the height of the power-tower involved), then moves on through "tetration", the Moser and the "Graham-Rothschild number", on to lesser-known hierarchies of recursive functions, the theory of computation, transfinite numbers and infinities. If it's a number and it's large, it's probably here.

Contents

Author's Introduction

Class 0 Numbers (like 3)

Class 1 Numbers (like 100)

Class 2 Numbers (like googol)

The -illion Names

Conway-Wechsler Extension

Knuth -yllion System

Class 3 Numbers (like googolplex)

Class 4 Numbers

Skewes' Number

Higher Classes

The Quality of Uncomputably Larger

Power Towers

Inventing New Operators and Functions

Function Hierarchies

Why Function Hierarchies Require a Transfinite Ordinal Index

Why There are Competing Function Hierarchies

Beyond Exponents: hyper4 (Tetration)

Extension to reals

A "logarithm" for hyper4

Hyperfactorial and Superfactorial

Higher hyper Operators

The "Generalised Hyper" Function

Bowers' Array Notation (3-element Subset)

Knuth's Up-arrow Notation

A Partial Ordering for Knuth Up-Arrows

A Partial Ordering for the Hyper Function

Composed Up-Arrow Notation

Steinhaus-Moser-Ackermann Notation/Functions

Ackermann's Function

The Mega and the Moser

The Fast Growing Hierarchy

Goodstein Sequences

The various "Graham's number"s :

The "Graham-Rothschild Number"

The "Graham-Gardner Number"

The "Graham-Conway Number"

Goodstein Sequences (strong)

Friedman Block Subsequence Length

Superclasses

Conway's Chained Arrow Notation

A Partial Ordering for short Conway chains

More Bowers Constructions :

Bowers' Extended Operators

Bowers' Array Notation (4-element Subset)

Bowers Arrays with 5 or More Elements

Generalised Invention of Recursive Functions

Formal Grammars

TREE{3}

Friedman's SSCG()

The Lin-Rado/Goucher/Rayo/Wojowu Method

Lin-Rado Busy Beaver Function

Beyond BB Function

Oracle Turing Machines

Declarative Computation and Combinatory Logic

Adam Goucher's Ξ(n)

Computation by Formal Logic and Set Theory

Peano Arithmetic

The von Neumann Construction

Forming Predicates

Not So Fast!

Rayo's Calculus

Formulas

Direct Declaration of the Existence of a Number

Doing Maths in First-Order-Logic and Set Theory

Truth and Uniqueness

Rayo's Number

Variable Assignments

Gödel-Coding

Rayo-nameability

Rayo's Number

BIG FOOT

The Frontier

Transfinite and Infinite Numbers

Ordinal Infinities

The First Cardinal Infinity: Aleph-Null

The Ordinal "Countable" Infinities

Epsilon-Null

All Ordinals Countable by Reordering

Aleph-One

The Continuum

The Continuum Hypothesis

The Power Sets of the Continuum

Inaccessible Infinities

Footnotes

Bibliography and other References

Other Links


Author's Introduction

Large numbers have interested me almost all my life.

This page covers all the huge numbers I have seen discussed in books and web pages, and it actually does so in numerical order, as near as I can tell (see the uncomparable and superclass 5 discussions).

One important thing to notice is that all discussions like this ultimately lead to difficult and unsolved problems in the theory of algorithms and computation. This page ends with Turing machines just before crossing over to the transfinite numbers. If you want to learn something about the theory of algorithms and computation, get two or more fairly knowledgeable people to compete at describing the highest number they can, and then stand back!. One such competition (detailed in a footnote) took only a few days to move beyond the range of everything discussed in the first two-thirds of this webpage, and then spent another few years discussing formal proofs.

This page is meant to counteract the forces of Munafo's Laws of Mathematics. If you see room for improvement, let me know!

Classes

First of all, I'm going to define what I call "classes" of numbers. This is a version of the "levels of perceptual realities" in an article On Number Numbness by Douglas Hofstadter [40], [42]. I have set arbitrary breakpoints (6, 106, etc.) that are based on well-researched results regarding how perception and cognition handle numbers and quantities.

[...]If [the] numbers [have] millions or billions of digits, the numerals themselves (the colossal strings of digits) would cease to be visualizable, and your perceptual reality would be forced to take another leap upward in abstraction — to the number that counts the digits in the number that counts the digits in the number that counts the objects concerned.[...]

It is a powerful and basic concept but usually goes unsaid. I think you'll agree that something like this makes sense (though perhaps you might choose your own arbitrary points of separation between the subitisable and the quotidian). In addition to these distinctions that relate to perception and cognition, the categories correspond to computational abilities, such as whether it is possible for your computer to store enough digits to resolve the effect of adding 1, whether it is possible to resolve the effect of multiplying by 2, and so on. Almost all numbers that are easy to make simple statements about (such as which of two numbers is larger) can be put into the class system.

All numbers that anyone ever has to deal with in any practical application (unless you count abstract mathematics and nerdy one-upmanship contests as practical :-) are members of one of the first four classes. Two, a hundred, googol, and googolplex are examples from classes 0 through 3, respectively.

Class-0 Numbers

(the concept of subitising)

Class-0 numbers are those that are small enough to have an immediate intuitive or perceptual impact. Perceiving such a number is called subitising, and for most purposes the limit has been shown to be somewhere from 5 to 9 (see Kaufman [31] and Miller [32]). I'll be a bit conservative here and place the limit at six. So, the numbers 1 through 6 are class 0.

Experiments with animals, when properly set up and conducted, demonstrate ability to identify numbers of objects and exhibit different behavior based on whether the number of objects is equal to some specific value — for example, pressing a lever only when five objects are present. Such experiments also show that the animal's ability to perform the feat falls off sharply between 4 and 8: the task can almost always be performed reliably when the number is 4, and can seldom be performed reliably when the number is 8 (with intermediate results in-between).

In the days of anthropological research by scientists from Europe who were encountering other cultures for the first time, it was discovered in some cases that a group of people had words for a few of the smaller numbers (say, up to three) but not beyond that. Due to a misunderstanding of the difference between having a word for a number and being able to perceive and understand precise quantities, this led to some myths that the people in such cultures couldn't count any higher than three or some other small number. Such a belief or myth reflects the basic truth that there is some additional abstraction or understanding involved when an amount is greater than what is subitizable.

One way to see this phenomenon for yourself is to use flash cards (or a computer program set up to simulate flash cards) that present pictures of objects that can be counted and placed in random arrangements — but look at the picture only long enough to see it, and not long enough to do any type of counting. It is not allowed to use regular arrangements like grids, or any other distinguishing attributes such as multiple colours or shapes. After the picture is hidden, try to answer how many objects there were. You then try to count the number of objects in your mental image of the picture you've just seen. If the number of objects is a class 0 number, you'll usually be able to give the right answer. As you increase the numbers of objects, your counts will be less and less likely to be correct. Obviously, this gives a rather fuzzy definition of "class 0", but the value you get will almost always be consistent with Miller's result of "seven plus or minus two". [32]

Class-1 Numbers

Class-1 numbers are those that are small enough to be perceived as a bunch of objects seen directly by the human eye. What I mean by "seen directly" is that it is possible to see the number as a set of separate, distinct objects in a single scene (no time limit, but the observer and the objects cannot move). 100 is a class-1 number because it is possible to see 100 objects (goats for example) in a single scene. The limit for class-1 numbers will vary depending on the use of colours, etc. and the quality of one's vision, but for black dots on white background most people would probably be able to see around a million, 1,000,000 or 106. You can just barely put 1,000,000 dots on a large piece of paper and stand at a distance such that you can perceive each individual dot as a distinct dot, and at the same time be within viewing distance of the other 999,999 dots. (I have actually done this, just for fun!) As with Class-0 the definition is fuzzy, some people have better vision and could manage 10,000,000 dots or even more.

The earliest conscious communication of numbers between humans was probably limited to class-0 and very low class-1 numbers, because of simple physical methods of counting (like fingers and toes). The first written number systems consisted of tally marks and extended into the class-1 range. (Methods involving objects or symbols that each count for 5, 10 or larger values, came later, see below.)

Class-1 numbers include all of the quotidian (everyday) quantities bigger than the subitizable, and because they occur so often, people can comfortably handle or perceive them due to experience and familiarity. For values in class 1, it is easy to distinguish the magnitude of the value just by looking at it. Most people have realised that, if they walk into a room with 85 people, although they can't tell it's exactly 85, they know right away it's somewhere around 75 to 100. No thought or calculation is necessary. This is an immediate perception of magnitude, and the ability extends to numbers up into the thousands and tens of thousands, with less percentage accuracy as the amounts increase. A person in a stadium with 10,000 people will have a fuzzier magnitude perception (they might guess anywhere from 3,000 to 30,000). By the time we get to numbers like 108 (the number of blades of grass in an acre) a person is probably about as likely to believe "10 million" (107) as "a trillion" (1012) unless they take the time to do some calculations (Fermi estimation would be adequate).

Class-1 numbers also include most types of things that people aggregate or count with the passage of time. If you have kept count of how many times you have done something (e.g. jogging) or the number of things in a collection (e.g. stamps) it probably numbers in the class 1 range. The actual act of counting usually wears out before exceeding class 1, partly because of the difficulty of accurately remembering the digits. (Supposing you need to remember the number from one day to the next — no written or other aids, keeping count of the number of days you have jogged this year is much easier than keeping count of how many steps you have taken this year — once that number gets into 6 or 7 digits mistakes are very likely for most people).

Symbolic representations of numbers soon became common. The earlier systems were just tally-marks with lots of different symbols, like one symbol to represent 1's and another to represent 10's, etc. Roman numerals are the most-used example of this. Often, different types of physical objects (like round and flat stones) were used for counting. Many examples are described in [46]. With symbolic systems it became easy for people to express, write, and do arithmetic with numbers throughout the class-1 range. Such representation systems usually reached their limit right around 1,000,000 for the same reasons that class-0 perceptive abilities are limited to 6: it is difficult to keep track of lots of different types of symbols/objects at once, and 5 or 6 types of symbols/objects is a practical limit — but the limit also existed because there was little need to deal with larger numbers. As in modern times, larger units were used when smaller ones were inconvenient — one does need to worry about the last few centimetres when considering a distance of 123.4 km.

Class-2 Numbers

Class-2 numbers are those that can be represented in exact form using decimal place-value notation (or another small integer base, like base 2, 16 or 60). Typically this depends on how the digits are recorded and what you need to do with them. Since I used 6 as the upper limit of class 0, and 106 = 1000000 for the upper limit of class 1, I'll just continue the pattern and say that the class-2 numbers go from 106 to about 101000000.

Place-value notation was popularised in the Arabic culture (but came from India, and perhaps from China before that, again see [46]). It opened up the range of class-2 numbers to anyone who wanted to use them. It was no longer necessary to come up with new symbols for each successive power of 10. Generalizations in arithmetic rules were obvious: adding 2000+7000 was not only analogous to adding 2+7, it was essentially the same thing. Handling huge numbers became easy. To make an exact calculation about thousands of objects, only a handful of objects (the digits) need to be manipulated.

Googol is a class-2 number, as are the various large prime numbers used in cryptography, all of the known Perfect numbers (until 1997!), the Fermat numbers with known factorization, etc. All of the large physical constants like 6.02×1023 (Avogadro's number) and 1080 (the number of protons in the universe) are class-2. So are most of the numbers with names ending in -illion, like vigintillion (1063), centillion (10303), and on up to the somewhat contrived milli-millillion (103000003) (which, by my admittedly arbitrary decision, is a bit beyond the class-2 range).

The Big Number Names of Nicolas Chuquet

(number names based on Latin)

The word million comes from around 12702, and entered the English language around 13706. The names billion, trillion, and so on up to nonillion, plus the general idea of continuing with Latin-derived prefixes all first appear in the late 15th century, in writing by Nicolas Chuquet, a French mathematician living in Lyon from 1480 until his death in 1488. (There were also the longer forms bymillion and trimillion used as early as 1475 by Jehan Adam, but these never caught on). Follow this link for more details: Origins of the Chuquet number names.

Peletier's Proposal and the Short Scale

In 1549 Jacques Peletier repeated the suggestion that billion should be one million million = 1012, and trillion for 1018 and so on. He also introduced1,2 the use of milliart, billiart and so on to represent the skipped-over powers of 1000, like 109 and 1015.

The long scale is Chuquet's original system, and has digits grouped 6 at a time, thus trillion is a million times larger than billion. This is the "billion=1012 system". Peletier's names for 10(6N+3) (in the English spelling, milliard=109, billiard=1015, etc.) are compatible with this system.

The use of number-names during the following few centuries eventually led to widespread usage of billion to mean 109, trillion for 1012, and similar redefinitions of the higher names. These definitions are the short scale or "billion=109 system". Follow this link for more on the history of short vs. long scale. Here is a related video by Numberphile: How big is a billion?.

Zillions: Big-Number Words as a Hyperbolic Adjective

While the confusion between short and long scale was becoming well-established, the big-number words ending in -illion were also becoming popular for the purpose of espressing an excessively or unimaginably large, or even infinite, quantity. This is a type of usage that was already common for hundreds, thousands, myriads and millions. For example, OED's [43] HUNDRED heading 2 a. begins: "Often used indefinitely or hyperbolically for a large number: cf. thousand. (With various constructions, as in [heading] I.)", and then gives nine quotations dating from 1300 AD to 1885. In the following table I show the first documented use of each number-name in both the literal sense and in this "superlative" sense.

(It should be noted that zillion more generally can refer to far larger things. For example, Howard DeLong[35] used the term "zillion" to refer to an iterated Ackermann function of some other really large number c1.[50]

Standard Accepted Names and SI Prefixes

This table shows all positive powers of ten that have authoritatively accepted names in English (by [43]) up to Chuquet's highest name nonillion. The numeric values here follow the billion=109 system ("short scale"). I am also including a few other non-powers of 10 that have names in English, but leaving out many base-20 constructions and other names less than 100, about which you can read plenty in [46]. I include all former and current official SI prefixes because they are quasi-"words" that have a purely numerical meaning. The dates of first literal and superlative usage are largely from OED [43] but are augmented as indicated in the footnotes.

The Standard Names and SI Prefixes

N N in Latin 3,18 103N+3name for 103N+3 first
literal
usage [43]
first
superlative
usage [43]
SI prefix(es)20
101 ten deca- or deka- (da,dk)
102 hundred 950 AD 1300 hecto- (h)
10×12great hundred 1533
122 gross 1411
0 103 thousand971 AD 1000 kilo- (k)
210 1024 kibi- (ki)
123 great gross1640
104 myriad 1555 1555 myria- (my)
1unus 106 million 1370 1362 Mega- (M)
220 1048576 Mebi- (Mi)
2duo 109 great million,
milliard,
billion
1625,
1793,
169021
?,
182322,
?
kilomega-, Giga- (G)
230 1073741824 Gibi- (Gi)
3tres 1012trillion169021184723 megamega-, Tera- (T)
240 1099511627776 Tebi- (Ti)
4quatuor1015quadrillion 167421185523Peta- (P)
250 1125899906842624 Pebi- (Pi)
5quinque1018quintillion 167421185523 Exa- (E)
260 1152921504606846976 Exbi- (Ei)
6sex 1021sextillion 169021185523 Zetta- (Z)
270 1180591620717411303424 Zebi- (Zi)
7septem 1024septillion 169021 ? Yotta- or Yotto- (Y)
280 1208925819614629174706176 Yobi- (Yi)
8octo 1027octillion 169021185523
9novem 1030nonillion 169021 ?

Chuquet left it to others to work out the details of extending the names beyond nonillion. Although there is much discrepancy between the actual number-names in Latin and the -illion names Chuquet listed, it was nevertheless understood that Latin number-names were to be used to extend the names as needed. Using Latin for prefixes goes smoothly as far as vigintillion. The following names are found in many dictionaries19; vigintillion and centillion are a little more common than the others. Some popular non-dictionary sources have made reference to millillion and milli-millillion (mostly due to Henkle/Brooks, and Borgmann [34]).

Larger Standard Names Beyond Chuquet's Nonillion

N N in Latin 3,18103N+3name for 103N+3
10 decem 1033 decillion
11 undecim 1036 undecillion
12 duodecim 1039 duodecillion
13 tredecim 1042 tredecillion
14 quattuordecim 1045 quattuordecillion
15 quindecim 1048 quindecillion, quinquadecillion
16 se(x)decim 1051 sexdecillion, sedecillion
17 septemdecim 1054 septendecillion
18 duodeviginti24 1057 octodecillion
19 undeviginti24 1060 novemdecillion, novendecillion
20 viginti 1063 vigintillion
10100 "googol" = ten duotrigintillion
100 centum 10303 centillion
1000 mille 103003 millillion
1000000 decies centena milia 103000003 milli-millillion
1010100 "googolplex"

The Conway-Wechsler System

Chuquet's names are notable for:

The Henkle/Brooks names of the late 19th century fall short of that mark on one or two counts.

Today it is useful to consider systems proposed by those other than "a respected well-known mathematician". The vast majority of huge numbers in use by the general population are in incremental games and similar recreations. Most of the work in extending the frontiers of practical computation with large numbers has been done in software libraries that facilitate such games. Therefore, we should not aspire to the four advantages of Chuquet I just listed, but instead something more like this:

In order for a system of words (names) to be useful by a set of people (perhaps sharing a specific field of application, or having cultural links such as a shared language), a system of words/names should do well by the following measures:

The only modern-day system with equivalent qualifications is the one worked out by Conway and Wechsler, and described in [44], then refined slightly by Miakinen. It extends the Chuquet names arbitrarily far, and surpasses Henkle/Brooks and the other ad-hoc systems by having better spelling, greater consistency, and avoiding hyphens. It was developed by John Horton Conway and Allan Wechsler after significant research into Latin5 and careful consideration of all the rules for combining syllables (called "assimilation" or "liaison"). Olivier Miakinen4,9 refined it to fix a few minor problems, as described below.

The system is based on the short scale (billion=109) but the names could easily be used in a long scale system. A number name is built out of pieces representing powers of 103, 1030 and 10300 as shown by this table:

1's 10's 100's
0 - - -
1 un n deci nx centi
2 duo ms viginti n ducenti
3 tre * ns triginta ns trecenti
4 quattuor ns quadraginta ns quadringenti
5 quin ns quinquaginta ns quingenti
6 se sx n sexaginta n sescenti
7 septe mn n septuaginta n septingenti
8 octo mx octoginta mx octingenti
9 nove mn nonaginta nongenti

The rules are:

The 1's column in combination with deci in the 10's column seem designed to replicate the names in established usage (shown above, here and here), with three differences. Miakinen comments 4 on these, which concern 15, 16, and 19:

N standard Conway-Wechsler Miakinen's opinion
15 quindecillion quinquadecillion Recommends against C-W as there are far fewer quinquadecim in Latin than quindecim
16 sexdecillion sedecillion Agrees with C-W since sedecim is seen in Latin more than sexdecim
19 novemdecillion novendecillion Accepts C-W since both novendecim and novemdecim appear in Latin about equally often, and using an n instead of m brings it in line with septendecillion

I agree with Miakinen and thus I have put quin in row 5 of the 1's column above, instead of quinqua.

Going beyond vigintillion into territory newly covered by Conway-Wechsler, some near-ambiguities arise. For example 10261 is sexoctogintillion and 102421 is sexoctingentillion. Then there's 10309 = duocentillion while 10603 = ducentillion; and similarly 10312 = trescentillion while 10903 = trecentillion.

This system seems to have been widely adopted, based on the diversity of results I find with online searches in 2022 (including a lot of videos related to incremental games and who-knows-what else). It seems that the subtleties of spelling (and probably pronunciation) haven't been too much of a concern, as I find more spelling errors that are due to other causes.

The Conway-Wechsler system extends to arbitrarily high values. After setting out the rules above, the authors continue7:

With Allan Wechsler we propose to extend this system indefinitely by combining these according to the convention that "XilliYilliZillion" (say) denotes the (1000000X + 1000Y + Z)th zillion, using "nillion" for the zeroth "zillion" when this is needed as a placeholder. So for example the million-and-third zillion is a "millinillitrillion."

As their example shows, the beginning parts of the standard names such as million and trillion are used for the "1" and "003" parts (respectively) of the number 1,000,003, with the placeholder "nilli" for the central "000" portion. This is the "1,000,003rd zillion", which is 103×1000003+3=103000012. In general, when naming 103N+3, the rules above are to be used for each group of 3 digits in the number N.

For another example, consider 1019683: this is 103×6560+3, so N=6560. That breaks up into a "6" part (the standard sextillion) and a "560" part (sexagintaquingentillion by the above table and rules); these are combined to form sextillisexagintaquingentillion which is the full Conway-Wechsler name for 1019683.

Their name for googolplex is ten trillitrestrigintatrecentilli....trestrigintatrecentilliduotrigintatrecentillion; with the "...." replaced by 30 additional repetitions of "trestrigintatrecentilli". This name is two words and 3+766 letters long.

See more examples of Conway-Wechsler number names here.

There have also been numerous personal or ad-hoc Chuquet extensions, follow that link for more.

A Practical Alternative

If the above tables seem a bit much to deal with, here is my modest proposal for a simpler naming system:


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Japanese readers should see: 巨大数論 (from @kyodaisuu on Twitter)

If you like this you might also enjoy my numbers page.


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