Large Numbers

First page . . . Forward to page 3 . . . Last page (page 7)

The Conway-Wechsler System

Rules for one system extending up to 10³⁰⁰⁰ are given in The Book of Numbers by Conway and Guy. This system was developed by John Conway and Allan Wechsler after significant research into Latin⁵ but Olivier Miakinen⁴ has refined it, as described below.

The name is built out of pieces representing powers of 10³, 10³⁰ and 10³⁰⁰ as shown by this table:

The rules are:

- Take the power of 10 you're naming and subtract 3.

- Divide by 3. If the remainder is 0, 1 or 2, put one, ten or one hundred at the beginning of your name (respectively).

- Break the quotient up into 1's, 10's and 100's. Find the appropriate name segments for each piece in the table. (NOTE: The original Conway-Wechsler system specifies quinqua for 5, not quin.)

- String the segments together, inserting an extra letter if the letter shown in parentheses at the end of one segment match a letter in parentheses at the beginning of the next. For example: septe(mn) + (ms)viginti = septemviginti; se(sx) + (mx)octoginta = sexoctoginta. For the special case of tre, the letter s should be inserted if the following part is marked with either an s or an x.

- If the result ends in a, change the a to i.

- Add llion at the end. You're done.

Many of the resulting names are only slightly different. For example 10²⁶¹ is sexoctongintillion and 10²⁴²¹ is sexoctingentillion. Then there's 10³⁰⁹ = duocentillion and 10⁶⁰³ = ducentillion.

There are a few minor problems⁴,⁵ with the names quinquadecillion, sedecillion, and novendecillion (which are better known as quindecillion, sexdecillion and novemdecillion respectively). Miakinen explains that sedecillion and novendecillion are more true to the "rules of assimilation" in Latin, and thus the Conway-Wechsler version is better. But he also explains that quinquadecillion should be quindecillion because the Latin for 15 is "quindecim", not "quinquadecim", and proposes a similar change to all the Conway-Wechsler names involving the quin prefix; I have adopted his suggestion here.

Also of note is Wechsler's comment about an important omission in Conway's book:

The presentation in The Book of Numbers was designed to leave the impression that the system up to 999 was pre-existing, although in fact we invented a lot of it; you will note that Conway carefully never says the system is ancient. -- Allan Wechsler⁵

Undoubtedly The Book of Numbers will trip up one or two future etymology researchers, but the inconsistency with quindecillion, sexdecillion and novemdecillion should make the truth obvious.

Other Chuquet-Like Systems

There seems to be a cult-like appeal to the Chuquet number names, evident in the fact that many people have been inspired to create lots of number-names to extend the above table further.

I myself did this at age 10, inventing names like bigintillion=10⁹³, cigintillion=10¹²³, etc. (starting with different letters of the alphabet) to permit successive sets of ten names like unbigintillion=10⁹⁶, duobigintillion=10⁹⁹, and so on. This system had plenty of problems — for example, in a hypothetical standard American English pronunciation, at least two of cigintillion, kigintillion and sigintillion must sound the same — but that didn't stop me. It is also redundant to the existing standard — my bigintillion is the same as the standard trigintillion, etc. Within a month or two I decided scientific notation was better.

The earliest concerted effort I know of started with a "Professor Henkle", whose ideas for number-names up to the above-listed milli-millillion=10^3000003 were published in 1904 by Brooks [27]. These were listed in 1968 by Borgmann [28] in the premiere issue of his Word Ways: the Journal of Recreational Linguistics. This sparked several Chuquet-like extension proposals including those by Ondrejka in 1968 [29] and Candelaria in 1975 and 1976 [30],[31].

John Knoderer created a set of number names between vigintillion and centillion, differing somewhat from the table above. The names are listed on his Numbering Systems & Place Values page. A similar list is provided by Sally Berriman.

An alternative based on Greek prefixes has been proposed by Russ Rowlett, who suggests using the names tetrillion=10¹², pentillion=10¹⁵, hexillion=10¹⁸, heptillion=10²¹, oktillion=10²⁴, ennillion=10²⁷, dekillion=10³⁰, and so on.

These systems have plenty of differences, and some ambiguities. Going beyond 10³⁰⁰⁰ (or 10⁶⁰⁰⁰ if you choose the billion=10¹² system a.k.a. "long scale") produces even more confusion, mainly from the use of a large number of prefixes strung together and having to remember what order they go in. For example, in the system deveoped by Landon Curt Noll. He also provides a page to convert numbers to names with downloadable source code for UNIX geeks like me. To give two examples of his system, 10¹⁹⁶⁸³ is "one sexmilliaquingensexagintillion" and 10¹⁰¹⁰ = 10^10000000000 is "ten tremilliamilliamilliatrecentretriginmilliamilliatrecentretriginmilliatrecendotrigintillion". His system has a name for googolplex that is two words long, "ten tremilliamilliamillia...milliatrecentretriginmilliatrecendotrigintillion"; the second word has 3868 letters.

Louis Epstein²⁵ takes Borgmann's milli-millillion and Ondrejka's milli-millimillillion as inspiration and forms names like kilillion=1000000¹⁰⁰⁰=10⁶⁰⁰⁰, megillion=1000000^1000000=10^6000000, gigillion=1000000¹⁰⁹=10^6×109, and so on. The idea is to use SI-prefix-like prefixes to indicate the power of a million, thus "meg-" prefix for a million to the millionth power. For arbitrary powers of a million he uses strings of Greek prefixes in reverse order to the digits of the exponent in question (for example, he states "The 1048576^th power of a million is a sexseptaginquinhectooctoquatriginkilmegillion") and gets as high as (10⁶)¹⁰²⁴ = 10^6×1024 before having to put hyphens into his names. For example, he gives the name yottillio-illion to 10^6×106×1024=(10⁶)^(106)1024. He adds hyphens and eventually apostrophes to get up numbers a fair bit bigger than anything else discussed so far.

(Sources:
http://www.wikipedia.org/wiki/List_of_numbers (Another version that goes up to 10¹⁸⁰)
http://www30.brinkster.com/manfear/office/large.html (Also gives names in German and proposes names based on Greek)
http://g42.org/MiscInfo/numbers.html (Much of the same info as on the previous page)
http://www.unc.edu/%7Erowlett/units/large.html (Much of the same info as on the previous page)
http://mathforum.org/library/drmath/view/59155.html (An introduction for kids)
http://www.grammarstation.com/KnowYourMath/numbers_symbols.html (An introduction for kids)
http://www.ex.ac.uk/cimt/dictunit/notesp.htm (Described the SI prefixes for powers of 1024, but now offline)
http://mathworld.wolfram.com/LargeNumber.html
http://journals.iranscience.net:800/www.newscientist.com/www.newscientist.com/lastword/article.jsp@id=lw77 (comments on uses of zetta- and yotta-)
http://www.lewrockwell.com/orig/kinsella6.html (introduction to the SI units)
http://www.plexos.com/256_bit_CPUs_should_be_enough.htm (xona)
http://jimvb.home.mindspring.com/unitsystem.htm (xona)
http://www.omniglot.com/writing/latin.htm (Latin alphabet) )

SI Prefixes

The original SI (systeme international, or "Metric" system) in 1793 had only the prefixes kilo, hecto, deca, deci, centi, and milli. In 1795 myria was added but later became deprecated (no longer officially allowed). mega dates from the late 1800's and was officially adopted in France in 1919. During the 1900's kilomega and megamega were used but it was eventually decided these needed their own prefixes; giga, nano and tera, pico were adopted in 1960, femto and atto in 1964, peta and exa in 1975, and zetta, zepto and yotta, yocto in 1991. These extensions have been added mainly for the convenience of scientists. For example, in 1993 some researchers had to refer to units of 10^-21 volts but they didn't yet know about the prefix zepto so they called it "milliattovolt". There have always been fields where very small or very large values are expressed simply as big exponents of 10, or where a field-specific and somewhat arbitrary unit such as the parsec or electron-volt is used. The standard unit of atomic mass (1/12 the mass of a Carbon atom, or roughly the mass of a proton or a neutron) is 1.66×10^-27 kg or 1.66 yg (1.66 yoctograms). Quasars have been discovered that are about 125 yottameters away. When volumes or weights are involved the units are even more often found to be insufficient — the Earth weighs about 6000 Yg (6000 yottagrams), and the mass of an electron is about 0.000 91 yg (yoctograms). By comparison, the diameter of the Earth and of an atom (both one-dimensional measurements) are both easily handled with the older kilo- and pico- prefixes.

All of the "SI prefixes" in italics are unofficial. The following prefixes can be dismissed immediately: hepa, ento, otta, fito, nea, syto, dea, tredo, una and revo²³. In recent times these have been perpetuated by Internet rumors (compare two posts by Alex Lopez-Ortiz ²⁴,²¹), but they may well go back before widespread use of the Internet²².

I have heard that the official SI prefix names "peta" and "exa" are derived from "pente" and "hexa" (five and six in Greek), although the official BIPM (Bureau International des Poids at Mesures) website does not give an explanation.

"Zetta" and "yotta" are derived from "septo" and "octo", which are quasi- number names. Quoting BIPM:¹⁹

The names zepto and zetta are derived from septo suggesting the number seven (the seventh power of 10³) and the letter "z" is substituted for the letter "s" to avoid the duplicate use of the letter "s" as a symbol. The names yocto and yotta are derived from octo, suggesting the number eight (the eighth power of 10³); the letter "y" is added to avoid the use of the letter "o" as a symbol because it may be confused with the number zero.

If BIPM decides to adopt further prefixes for 10²⁷ and 10³⁰ and their reciprocals 10^-27 and 10^-30, they will probably adopt something vaguely resembling names for nine and ten for a similar reason — perhaps something like novetta, novemo, decetta and decemo. If so they would almost certainly be assigned two-letter abbreviations such as "No-", "De-", "no-" and "de-" because N-, n- and d- are already used for other prefixes.

Ad-Hoc SI Prefix Proposals

Oxford professor Jeff K. Aronson has suggested extending beyond zetta, zepto and yotta, yocto with xenta, xenno, wekta, weko, vendeka, vendeko, udeka, udeko based on the idea that the 'Z' and 'Y' prefixes would continue backwards through the English or modern Latin alphabet. Using a similar idea, Jim Blowers²⁰ says:

The pattern here is that we go backwards from the beginning of the alphabet [shouldn't this be "end of the alphabet"? -ed], starting with z and y, and we follow it up with an alteration of the Greek or Latin for the next number. According to this pattern, the next ending should be xona-, since x comes before y in the alphabet, and 9 is noni- in Latin. Similarly, 10³⁰ should be weka-, since w precedes x and 10 is deka in Greek.

He goes on to list a large number of prefixes starting with Xona-, Weka-, Vunda-, Uda-, Treda-, Sorta-, ... One-letter abbreviations are used if unambiguous, otherwise another letter is added, e.g. TD- for the Treda- prefix. He goes as far as 10⁶³ using an L- prefix, based on the 26-letter "Latin" alphabet used in several European languages including English, although the original classical Latin language had no U or W.

For completeness I'll also mention that in 1993, as a joke that was reportedly well received on USENET, Morgan Burke proposed harpo for 10^-27 and groucho for 10^-30 (and therefore harpi for 10²⁷ and grouchi for 10³⁰).

The Knuth -yllion Notation

Donald Knuth invented a system that extends much further than the standard Latin-based system. In the essay Supernatural Numbers[32] he wrote:

When we stop to examine our conventional numbers, it is immediately apparent that these names are "Menschenwerk"; they could have been designed much better. For example, it would be better to forget about thousands entirely, and to make a myriad (10⁴) the next unit after hundreds.

So in this system the word "thousand" is not used, and instead everything up to 9999 is named using the traditional names for numbers up to 99 plus "hundred", and no comma is used. For example:

127 = One hundred twenty-seven
1000 = Ten hundred
1356 = Thirteen hundred fifty-six
3000 = Thirty hundred
4192 = Forty-one hundred ninety-two

10⁴ is called "myriad", a name that originally comes from ancient Egypt. It is written 1,0000 — note that the comma is added to separate the lowest four digits, not three. Numbers up to 9999,9999 are named like so:

1,2345 = One myriad twenty-three hundred forty-five
10,0000 = Ten myriad
26,0044 = Twenty-six myriad forty-four
100,0000 = One hundred myriad
1000,0000 = Ten hundred myriad
1400,2054 = Fourteen hundred myriad twenty hundred fifty-four
4309,8127 = Forty-three hundred nine myriad eighty-one hundred twenty-seven

10⁸ is called "myllion" (pronounced "mile-yun") and is written 1;0000,0000. Notice a new punctuation mark is used to represent "myllion". Numbers up to 9999,9999;9999,9999 are named as in these examples:

9;0000,0000 = Nine myllion
100;0001,0000 = One hundred myllion one myriad
2000;0000,1234 = Twenty hundred myllion twelve hundred thirty four
4,0006;5020,0100 = Four myriad six myllion fifty hundred twenty myriad one hundred

Then 10¹⁶ is called "byllion", and a new punctuation mark is used. Knuth points out the advantage of avoiding the long scale vs. short scale confusion. Notice each punctuation mark can be read exactly when it appears so it's easy to read off these numbers in words:

1844:6744,0737;0955,1616 = Eighteen hundred forty-four byllion sixty-seven hundred forty-four myriad seven hundred thirty-seven myllion nine hundred fifty-five myriad sixteen hundred sixteen

Each new number name is the square of the previous one — therefore, each new name allows us to name numbers with twice as many digits. This gives us a lot more mileage out of each name. Knuth continues borrowing the traditional names changing "illion" to "yllion" on each one. "vigintyllion" ends up being 10^4194304, a bit beyond the upper limit of class-2 numbers.

In the same article [32], Knuth reports that Hsu Yo (living near the end of the Han dynasty) used the names wan=10⁴, i=10⁸, chao=10¹⁶ and ching=10³² as part of a nomenclature system for large numbers. The names descended into the present-day Chinese wàn, yì, zhào and jing respectively. Usage of the names zhào and jing for 10¹⁶ and 10³² respectively is the "higher degree system" reported by [33], but this usage did not continue into the present (see Wikipedia's Chinese numerals article). The ancient usage corresponds directly to myirad, myllion, byllion and tryllion in Knuth's system, including the ordering of words to make the names of arbitrary large numbers. A specific example showing the recursive grouping, with Chinese spelling, phonetic pronunciation and translation into more familiar numeric notation is shown in [33] figure 21.41 (page 278). The Chinese names continue with gai which would be 10⁶⁴, all the way up to zài=10⁴⁰⁹⁶ (which is Knuth's decyllion), but usage of the larger ones has only ever been "theoretical" — no actual usage is known.

Class-3 Numbers

Class-3 numbers are those that can be represented inexactly using scientific notation, to within a given percentage of error. Numbers about the size of Googolplex are class-3 numbers, although Googolplex itself can be represented exactly. Class-3 numbers include (almost) all combinatoric enumerations of physical systems (i.e. the number of possible states of a system containing N particles, where N is a class-2 number, see googolplex notes). The limit of class-3 numbers depends on the limit of class-2 numbers and the base, for this discussion let's say that class-3 goes from 10¹⁰⁶ to 10^101000000.

Class-3 numbers are the largest which can effectively be compared to see if they are of comparable magnitude. For example, the following two numbers are class-3 (and are at the low end, as class-3 numbers go) :

A = 2^{79641170620168673833}

B = 3^{50247984153525417450}

Which is larger?

We cannot compute the exact values of these two numbers and compare directly — they have way too many digits to store the values on a computer. That is the nature of class-3 numbers. However, we can represent both in scientific notation with 10 digits of accuracy. This is accomplished in much the same way that your computer or a scientific calculator would do it. Starting with the logarithm of 2 (or 3), multiply by the exponent, then divide by the logarithm of 10, separating the integer from the fractional part, and use the fractional part to determine the first few digits of the answer. In this case we get:

A = 5.0760252191 × 10^{23974381246463762439}

B = 5.0760252191 × 10^{23974381246463762439}

Now you begin to see the problem. Using 10 decimal places, both values seem to be the same. (We know they are not, because one is a power of 2 and must be even, and the other, being a power of 3 is odd). As it turns out, you need at least 20 decimal places to see that B is slightly larger.

Class-4 Numbers

Now we move on to Class-4 numbers and higher classes. You have probably seen a pattern here, and we'll just continue the pattern:

class-4 numbers are those numbers that are larger than class-3, and whose logarithm can be represented as a class-3 number.

Now we have another problem as before. Which of the following class-4 numbers is larger?

C = 2²²⁸³
D = 3³³⁵²

as before we take the logarithm of both but this time we must do it twice, and we find

ln(ln(C)) = ln(ln(2)) + [ln(2) * 9671406556917033397649408]
= 6703708186976009930559261.24579...
ln(ln(D)) = ln(ln(3)) + [ln(3) * 6461081889226673298932241]
= 7098223961595389530659098.10481...

so D is larger.

Skewes' Numbers

These numbers occur in the study of prime numbers, and particularly the frequency of occurrence of prime numbers. Gauss' well-known estimate of the number of prime numbers less than N is

For all values of n up to 10²² (which is as far as we've been able to compute so far) Li(n) is an overestimate. Littlewood showed that above some value of n it becomes an underestimate, then at an even higher value of n it becomes an overestimate again and so on. In 1933 Skewes showed that (if the Riemann Hypothesis is true) the first crossing cannot be greater than e^ee79. This is the first or "Riemann true" Skewes' Number; it is class-4. Converted to base 10, the value is normally approximated as 10^{1010^34}; a more accurate approximation is 10^{108.852142×1033} or 10^{108852142197543270606106100452735038.55}.

Since then, others have improved the estimate dramatically. Conway and Guy (The Book of Numbers, page 61) cite the result of Lehman, who in 1966 gave an upper bound of about 10¹¹⁶⁷. According to Eric W. Weisstein and Wikipedia, in 1987 H. J. J. te Riele reduced the upper bound of the first crossing to e^e(27/4), a class 2 number approximately equal to 8.185×10³⁷⁰. In 2000 Bays and Hudson found an actual crossover point using numerical techniques — around 1.39822×10³¹⁶. Most recently, in 2005 Patrick Demichel found a smaller crossover point near 1.397162914×10³¹⁶. In any case, the original Skewes' Number is now just an interesting part of history.

In 1955 Skewes also defined an upper bound if the Riemann Hypothesis is false: 10^10101000. This is the "second Skewes' Number"; it is much larger, but still class-4.

Class 5 and Higher

In a similar way, we can define higher classes:

class-5 numbers are those numbers that are larger than class-4, and whose logarithm can be represented as a class-4 number.

class-N numbers are those numbers that are larger than class N-1, and whose logarithm can be represented as a class N-1 number.

but as it turns out, these higher classes aren't too useful for representing the large numbers of abstract mathematics. Once we get into the really big numbers like the ones discussed below, exponents are so unwieldy that they are no longer used directly — instead faster-growing functions like the hyper4 function are used.

Here is a summary of what has been covered so far:

First page . . . Forward to page 3 . . . Last page (page 7)

This work is licensed under a Creative Commons Attribution 2.5 License. Details here s.13