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This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.
The following table lists those names of large numbers that are found in many English dictionaries and thus have a claim to being "real words." The "Traditional British" values shown are unused in American English and are obsolete in British English, but their other-language variants are dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America; see Long and short scales.
Indian English does not use millions, but has its own system of large numbers including lakhs and crores.
English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.
Contents
Standard dictionary numbers
Name | Short scale (US, Eastern Europe, English Canadian, Australian, and modern British) |
Long scale (Western, Central Europe, older British, and French Canadian) |
Authorities | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
AHD4^{[1]} | CED^{[2]} | COD^{[3]} | OED2^{[4]} | OEDnew^{[5]} | RHD2^{[6]} | SOED3^{[7]} | W3^{[8]} | UM^{[9]} | |||
Million | 10^{6} | 10^{6} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
Milliard | 10^{9} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Billion | 10^{9} | 10^{12} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
Billiard | 10^{15} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Trillion | 10^{12} | 10^{18} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
Quadrillion | 10^{15} | 10^{24} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Quintillion | 10^{18} | 10^{30} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Sextillion | 10^{21} | 10^{36} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Septillion | 10^{24} | 10^{42} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Octillion | 10^{27} | 10^{48} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Nonillion | 10^{30} | 10^{54} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Decillion | 10^{33} | 10^{60} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Undecillion | 10^{36} | 10^{66} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Duodecillion | 10^{39} | 10^{72} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Tredecillion | 10^{42} | 10^{78} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Quattuordecillion | 10^{45} | 10^{84} | ✓ | ✓ | ✓ | ✓ | |||||
Quindecillion | 10^{48} | 10^{90} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Sexdecillion | 10^{51} | 10^{96} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Septendecillion | 10^{54} | 10^{102} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Octodecillion | 10^{57} | 10^{108} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Novemdecillion | 10^{60} | 10^{114} | ✓ | ✓ | ✓ | ✓ | ✓ | ||||
Vigintillion | 10^{63} | 10^{120} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | |
Centillion | 10^{303} | 10^{600} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
Apart from million, the words in this list ending with -illion are all derived by adding prefixes (bi-, tri-, etc., derived from Latin) to the stem -illion.^{[10]} Centillion^{[11]} appears to be the highest name ending in -"illion" that is included in these dictionaries. Trigintillion, often cited as a word in discussions of names of large numbers, is not included in any of them, nor are any of the names that can easily be created by extending the naming pattern (unvigintillion, duovigintillion, duoquinquagintillion, etc.).
Name | Value | Authorities | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
AHD4 | CED | COD | OED2 | OEDnew | RHD2 | SOED3 | W3 | UM | ||
Googol | 10^{100} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
Googolplex | 10^{googol} (10^{10100}) | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |
All of the dictionaries included googol and googolplex, generally crediting it to the Kasner and Newman book and to Kasner's nephew. None include any higher names in the googol family (googolduplex, etc.). The Oxford English Dictionary comments that googol and googolplex are "not in formal mathematical use".
Usage of names of large numbers
Some names of large numbers, such as million, billion, and trillion, have real referents in human experience, and are encountered in many contexts. At times, the names of large numbers have been forced into common usage as a result of hyperinflation. The highest numerical value banknote ever printed was a note for 1 sextillion pengő (10^{21} or 1 milliard bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (10^{14}) Zimbabwean dollar note, which at the time of printing was worth about US$30.^{[12]}
Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of the ways in which large numbers are named. Even well-established names like sextillion are rarely used, since in the context of science, including astronomy, where such large numbers often occur, they are nearly always written using scientific notation. In this notation, powers of ten are expressed as 10 with a numeric superscript, e.g. "The X-ray emission of the radio galaxy is 1.3×10^{45} joules." When a number such as 10^{45} needs to be referred to in words, it is simply read out as "ten to the forty-fifth". This is easier to say and less ambiguous than "quattuordecillion", which means something different in the long scale and the short scale.
When a number represents a quantity rather than a count, SI prefixes can be used—thus "femtosecond", not "one quadrillionth of a second"—although often powers of ten are used instead of some of the very high and very low prefixes. In some cases, specialized units are used, such as the astronomer's parsec and light year or the particle physicist's barn.
Nevertheless, large numbers have an intellectual fascination and are of mathematical interest, and giving them names is one of the ways in which people try to conceptualize and understand them.
One of the earliest examples of this is The Sand Reckoner, in which Archimedes gave a system for naming large numbers. To do this, he called the numbers up to a myriad myriad (10^{8}) "first numbers" and called 10^{8} itself the "unit of the second numbers". Multiples of this unit then became the second numbers, up to this unit taken a myriad myriad times, 10^{8}·10^{8}=10^{16}. This became the "unit of the third numbers", whose multiples were the third numbers, and so on. Archimedes continued naming numbers in this way up to a myriad myriad times the unit of the 10^{8}-th numbers, i.e. and embedded this construction within another copy of itself to produce names for numbers up to Archimedes then estimated the number of grains of sand that would be required to fill the known universe, and found that it was no more than "one thousand myriad of the eighth numbers" (10^{63}).
Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that really have no existence outside the imagination. One motivation for such a pursuit is that attributed to the inventor of the word googol, who was certain that any finite number "had to have a name". Another possible motivation is competition between students in computer programming courses, where a common exercise is that of writing a program to output numbers in the form of English words.
Most names proposed for large numbers belong to systematic schemes which are extensible. Thus, many names for large numbers are simply the result of following a naming system to its logical conclusion—or extending it further.
Origins of the "standard dictionary numbers"
The words bymillion and trimillion were first recorded in 1475 in a manuscript of Jehan Adam. Subsequently, Nicolas Chuquet wrote a book Triparty en la science des nombres which was not published during Chuquet's lifetime. However, most of it was copied by Estienne de La Roche for a portion of his 1520 book, L'arismetique. Chuquet's book contains a passage in which he shows a large number marked off into groups of six digits, with the comment:
Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers point tryllion Le quart quadrillion Le cinq^{e} quyllion Le six^{e} sixlion Le sept.^{e} septyllion Le huyt^{e} ottyllion Le neuf^{e} nonyllion et ainsi des ault'^{s} se plus oultre on vouloit preceder
(Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go).
Adam and Chuquet used the long scale of powers of a million; that is, Adam's bymillion (Chuquet's byllion) denoted 10^{12}, and Adam's trimillion (Chuquet's tryllion) denoted 10^{18}.
The googol family
The names googol and googolplex were invented by Edward Kasner's nephew Milton Sirotta and introduced in Kasner and Newman's 1940 book Mathematics and the Imagination^{[13]} in the following passage:
The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely 1 with one hundred zeroes after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex". A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. It was first suggested that a googolplex should be 1, followed by writing zeros until you got tired. This is a description of what would actually happen if one actually tried to write a googolplex, but different people get tired at different times and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance. The googolplex is, then, a specific finite number, equal to 1 with a googol zeros after it.
Value | Name | Authority |
---|---|---|
10^{100} | Googol | Kasner and Newman, dictionaries (see above) |
10^{googol} = 10^{10100} | Googolplex | Kasner and Newman, dictionaries (see above) |
Conway and Guy^{[14]} have suggested that N-plex be used as a name for 10^{N}. This gives rise to the name googolplexplex for 10^{googolplex} = 10^{1010100}. This number (ten to the power of a googolplex) is also known as a googolduplex and googolplexian.^{[15]} Conway and Guy^{[14]} have proposed that N-minex be used as a name for 10^{−N}, giving rise to the name googolminex for the reciprocal of a googolplex. None of these names are in wide use, nor are any currently found in dictionaries.
The names googol and googolplex inspired the name of the Internet company Google and its corporate headquarters, the Googleplex, respectively.
Extensions of the standard dictionary numbers
This section illustrates several systems for naming large numbers, and shows how they can be extended past vigintillion.
Traditional British usage assigned new names for each power of one million (the long scale): 1,000,000 = 1 million; 1,000,000^{2} = 1 billion; 1,000,000^{3} = 1 trillion; and so on. It was adapted from French usage, and is similar to the system that was documented or invented by Chuquet.
Traditional American usage (which was also adapted from French usage but at a later date), Canadian, and modern British usage assign new names for each power of one thousand (the short scale.) Thus, a billion is 1000 × 1000^{2} = 10^{9}; a trillion is 1000 × 1000^{3} = 10^{12}; and so forth. Due to its dominance in the financial world (and by the US dollar), this was adopted for official United Nations documents.
Traditional French usage has varied; in 1948, France, which had been using the short scale, reverted to the long scale.
The term milliard is unambiguous and always means 10^{9}. It is almost never seen in American usage and rarely in British usage, but frequently in continental European usage. The term is sometimes attributed to French mathematician Jacques Peletier du Mans circa 1550 (for this reason, the long scale is also known as the Chuquet-Peletier system), but the Oxford English Dictionary states that the term derives from post-Classical Latin term milliartum, which became milliare and then milliart and finally our modern term.
With regard to names ending in -illiard for numbers 10^{6n+3}, milliard is certainly in widespread use in languages other than English, but the degree of actual use of the larger terms is questionable. The terms "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish and "миллиард" in Russian are standard usage when discussing financial topics.
For additional details, see billion and long and short scales.
The naming procedure for large numbers is based on taking the number n occurring in 10^{3n+3} (short scale) or 10^{6n} (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion. In this way, numbers up to 10^{3·999+3} = 10^{3000} (short scale) or 10^{6·999} = 10^{5994} (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if n is 20 or smaller. For larger n (between 21 and 999), prefixes can be constructed based on a system described by John Horton Conway and Richard K. Guy:^{[14]}
Units | Tens | Hundreds | |
---|---|---|---|
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 | Quinqua | ^{NS} Quinquaginta | ^{NS} Quingenti |
6 | Se ^{(*)} | ^{N} Sexaginta | ^{N} Sescenti |
7 | Septe ^{(*)} | ^{N} Septuaginta | ^{N} Septingenti |
8 | Octo | ^{MX} Octoginta | ^{MX} Octingenti |
9 | Nove ^{(*)} | Nonaginta | Nongenti |
- ^{(*)} ^ When preceding a component marked ^{S} or ^{X}, “tre” changes to “tres” and “se” to “ses” or “sex”; similarly, when preceding a component marked ^{M} or ^{N}, “septe” and “nove” change to “septem” and “novem” or “septen” and “noven”.
Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 10^{6,000,258}, Conway and Guy have also proposed a consistent set of conventions which permit, in principle, the extension of this system to provide English names for any integer whatsoever.^{[14]}
The following table shows number names generated by the system described by Conway and Guy for the short and long scales.
Names of reciprocals of large numbers are not listed, as they are regularly formed by adding -th, e.g. quattuordecillionth, centillionth, etc.
Base -illion (short scale) |
Base -illion (long scale) |
Value | US, Canada and modern British (short scale) |
Traditional British (long scale) |
Traditional European (Peletier) (long scale) |
SI Symbol |
SI Prefix |
---|---|---|---|---|---|---|---|
1 | 1 | 10^{6} | Million | Million | Million | M | Mega- |
2 | 1 | 10^{9} | Billion | Thousand million | Milliard | G | Giga- |
3 | 2 | 10^{12} | Trillion | Billion | Billion | T | Tera- |
4 | 2 | 10^{15} | Quadrillion | Thousand billion | Billiard | P | Peta- |
5 | 3 | 10^{18} | Quintillion | Trillion | Trillion | E | Exa- |
6 | 3 | 10^{21} | Sextillion | Thousand trillion | Trilliard | Z | Zetta- |
7 | 4 | 10^{24} | Septillion | Quadrillion | Quadrillion | Y | Yotta- |
8 | 4 | 10^{27} | Octillion | Thousand quadrillion | Quadrilliard | ||
9 | 5 | 10^{30} | Nonillion | Quintillion | Quintillion | ||
10 | 5 | 10^{33} | Decillion | Thousand quintillion | Quintilliard | ||
11 | 6 | 10^{36} | Undecillion | Sextillion | Sextillion | ||
12 | 6 | 10^{39} | Duodecillion | Thousand sextillion | Sextilliard | ||
13 | 7 | 10^{42} | Tredecillion | Septillion | Septillion | ||
14 | 7 | 10^{45} | Quattuordecillion | Thousand septillion | Septilliard | ||
15 | 8 | 10^{48} | Quinquadecillion | Octillion | Octillion | ||
16 | 8 | 10^{51} | Sedecillion | Thousand octillion | Octilliard | ||
17 | 9 | 10^{54} | Septendecillion | Nonillion | Nonillion | ||
18 | 9 | 10^{57} | Octodecillion | Thousand nonillion | Nonilliard | ||
19 | 10 | 10^{60} | Novendecillion | Decillion | Decillion | ||
20 | 10 | 10^{63} | Vigintillion | Thousand decillion | Decilliard | ||
21 | 11 | 10^{66} | Unvigintillion | Undecillion | Undecillion | ||
22 | 11 | 10^{69} | Duovigintillion | Thousand undecillion | Undecilliard | ||
23 | 12 | 10^{72} | Tresvigintillion | Duodecillion | Duodecillion | ||
24 | 12 | 10^{75} | Quattuorvigintillion | Thousand duodecillion | Duodecilliard | ||
25 | 13 | 10^{78} | Quinquavigintillion | Tredecillion | Tredecillion | ||
26 | 13 | 10^{81} | Sesvigintillion | Thousand tredecillion | Tredecilliard | ||
27 | 14 | 10^{84} | Septemvigintillion | Quattuordecillion | Quattuordecillion | ||
28 | 14 | 10^{87} | Octovigintillion | Thousand quattuordecillion | Quattuordecilliard | ||
29 | 15 | 10^{90} | Novemvigintillion | Quindecillion | Quindecillion | ||
30 | 15 | 10^{93} | Trigintillion | Thousand quindecillion | Quindecilliard | ||
31 | 16 | 10^{96} | Untrigintillion | Sedecillion | Sedecillion | ||
32 | 16 | 10^{99} | Duotrigintillion | Thousand sedecillion | Sedecilliard | ||
33 | 17 | 10^{102} | Trestrigintillion | Septendecillion | Septendecillion | ||
34 | 17 | 10^{105} | Quattuortrigintillion | Thousand septendecillion | Septendecilliard | ||
35 | 18 | 10^{108} | Quinquatrigintillion | Octodecillion | Octodecillion | ||
36 | 18 | 10^{111} | Sestrigintillion | Thousand octodecillion | Octodecilliard | ||
37 | 19 | 10^{114} | Septentrigintillion | Novendecillion | Novendecillion | ||
38 | 19 | 10^{117} | Octotrigintillion | Thousand novendecillion | Novendecilliard | ||
39 | 20 | 10^{120} | Noventrigintillion | Vigintillion | Vigintillion | ||
40 | 20 | 10^{123} | Quadragintillion | Thousand vigintillion | Vigintilliard | ||
50 | 25 | 10^{153} | Quinquagintillion | Thousand quinquavigintillion | Quinquavigintilliard | ||
60 | 30 | 10^{183} | Sexagintillion | Thousand trigintillion | Trigintilliard | ||
70 | 35 | 10^{213} | Septuagintillion | Thousand quinquatrigintillion | Quinquatrigintilliard | ||
80 | 40 | 10^{243} | Octogintillion | Thousand quadragintillion | Quadragintilliard | ||
90 | 45 | 10^{273} | Nonagintillion | Thousand quinquaquadragintillion | Quinquaquadragintilliard | ||
100 | 50 | 10^{303} | Centillion | Thousand quinquagintillion | Quinquagintilliard | ||
101 | 51 | 10^{306} | Uncentillion | Unquinquagintillion | Unquinquagintillion | ||
110 | 55 | 10^{333} | Decicentillion | Thousand quinquaquinquagintillion | Quinquaquinquagintilliard | ||
111 | 56 | 10^{336} | Undecicentillion | Sesquinquagintillion | Sesquinquagintillion | ||
120 | 60 | 10^{363} | Viginticentillion | Thousand sexagintillion | Sexagintilliard | ||
121 | 61 | 10^{366} | Unviginticentillion | Unsexagintillion | Unsexagintillion | ||
130 | 65 | 10^{393} | Trigintacentillion | Thousand quinquasexagintillion | Quinquasexagintilliard | ||
140 | 70 | 10^{423} | Quadragintacentillion | Thousand septuagintillion | Septuagintilliard | ||
150 | 75 | 10^{453} | Quinquagintacentillion | Thousand quinquaseptuagintillion | Quinquaseptuagintilliard | ||
160 | 80 | 10^{483} | Sexagintacentillion | Thousand octogintillion | Octogintilliard | ||
170 | 85 | 10^{513} | Septuagintacentillion | Thousand quinquaoctogintillion | Quinquaoctogintilliard | ||
180 | 90 | 10^{543} | Octogintacentillion | Thousand nonagintillion | Nonagintilliard | ||
190 | 95 | 10^{573} | Nonagintacentillion | Thousand quinquanonagintillion | Quinquanonagintilliard | ||
200 | 100 | 10^{603} | Ducentillion | Thousand centillion | Centilliard | ||
300 | 150 | 10^{903} | Trecentillion | Thousand quinquagintacentillion | Quinquagintacentilliard | ||
400 | 200 | 10^{1203} | Quadringentillion | Thousand ducentillion | Ducentilliard | ||
500 | 250 | 10^{1503} | Quingentillion | Thousand quinquagintaducentillion | Quinquagintaducentilliard | ||
600 | 300 | 10^{1803} | Sescentillion | Thousand trecentillion | Trecentilliard | ||
700 | 350 | 10^{2103} | Septingentillion | Thousand quinquagintatrecentillion | Quinquagintatrecentilliard | ||
800 | 400 | 10^{2403} | Octingentillion | Thousand quadringentillion | Quadringentilliard | ||
900 | 450 | 10^{2703} | Nongentillion | Thousand quinquagintaquadringentillion | Quinquagintaquadringentilliard | ||
1000 | 500 | 10^{3003} | Millinillion | Thousand quingentillion | Quingentilliard |
Value | Name | Equivalent | ||
---|---|---|---|---|
US, Canadian and modern British (short scale) |
Traditional British (long scale) |
Traditional European (Peletier) (long scale) | ||
10^{100} | Googol | Ten duotrigintillion | Ten thousand sedecillion | Ten sedecilliard |
10^{10100} | Googolplex | n/a | n/a | n/a |
Binary prefixes
The International System of Quantities (ISQ) defines a series of prefixes denoting integer powers of 1024 between 1024^{1} and 1024^{8}.^{[16]}
Power | Value | ISQ symbol |
ISQ prefix |
---|---|---|---|
1 | 1024^{1} | Ki | Kibi- |
2 | 1024^{2} | Mi | Mebi- |
3 | 1024^{3} | Gi | Gibi- |
4 | 1024^{4} | Ti | Tebi- |
5 | 1024^{5} | Pi | Pebi- |
6 | 1024^{6} | Ei | Exbi- |
7 | 1024^{7} | Zi | Zebi- |
8 | 1024^{8} | Yi | Yobi- |
Other large numbers used in mathematics and physics
See also
- -yllion
- Asaṃkhyeya
- Chinese numerals
- Indefinite and fictitious numbers
- Indian numbering system
- Knuth's up-arrow notation
- Law of large numbers
- List of numbers
- Long and short scales
- History of large numbers
- Names of small numbers
- Nicolas Chuquet
- Number names
- Number prefix
- Orders of magnitude
- Orders of magnitude (data)
- Orders of magnitude (numbers)
- Powers of 10
- Metric prefix
References
- ^ The American Heritage® Dictionary of the English Language. American Heritage Dictionary (4th ed.). 2000. ISBN 0-395-82517-2. Archived from the original on 12 January 2007.
- ^ Collins English Dictionary, 11th Edition, HarperCollins Publishers.
- ^ Cambridge Dictionaries Online, Cambridge, UK: Cambridge University Press.
- ^ Oxford Dictionary, 2nd edition, Oxford, UK: Oxford University Press. ISBN 0-19-861186-2 (and addendums since publication in 1989.)
- ^ Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press. [1] (subscription required), checked April 2007
- ^ The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House.
- ^ Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press.
- ^ Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster.
- ^ "How Many? A Dictionary of Units of Measures". Russ Rowlett and the University of North Carolina at Chapel Hill. Retrieved 15 August 2009.
- ^ p. 316, The History of the English Language, Oliver Farrar Emerson, New York, London: Macmillan and Co., 1894.
- ^ Entry for centillion in the American Heritage Dictionary
- ^ "Zimbabwe rolls out Z$100tr note". BBC News. 16 January 2009. Retrieved 16 January 2009.
- ^ Kasner, Edward, and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
- ^ ^{a} ^{b} ^{c} ^{d} The Book of Numbers, J. H. Conway and R. K. Guy, New York: Springer-Verlag, 1996, pp. 15–16. ISBN 0-387-97993-X.
- ^ Bowers, Jonathan. "Infinity Scrapers". Polytope, 2010.
- ^ "IEC 80000-13:2008". International Organization for Standardization. Retrieved 21 July 2013.