Names of large numbers

Usage: • Short scale: US, English Canada, modern British, Australia, and Eastern Europe • Long scale: French Canada, older British, Western & Central Europe Apart from million, the words in this list ending with '' are all derived by adding prefixes (bi-, tri-, etc., derived from Latin) to the stem . Centillion appears to be the highest name ending in 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.). All of the dictionaries included googol and googolplex, generally crediting it to the Kasner and Newman book and to Kasner's nephew (see below). 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, particularly in finance and economics. 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 (1021 or 1 milliárd bilpengő as printed) printed in Hungary in 1946. In 2009, Zimbabwe printed a 100 trillion (1014) Zimbabwean dollar note, which at the time of printing was worth about US$30. A Kremlin spokesperson, Dmitry Peskov, stated that this value was symbolic. Names of larger numbers, however, have a tenuous, artificial existence, rarely found outside definitions, lists, and discussions of how 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 ." When a number such as 1045 needs to be referred to in words, it is simply read out as "ten to the forty-fifth" or "ten to the forty-five". 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 way 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 (108) "first numbers" and called 108 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, 108·108=1016. 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 108-th numbers, i.e. (10^8)^{(10^8)}=10^{8\cdot 10^8}, and embedded this construction within another copy of itself to produce names for numbers up to ((10^8)^{(10^8)})^{(10^8)}=10^{8\cdot 10^{16}}. 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" (1063). == 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 cinqe quyllion Le sixe sixlion Le sept.e septyllion Le huyte ottyllion Le neufe 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 1012, and Adam's trimillion (Chuquet's tryllion) denoted 1018. == 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 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): ; ; ; 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 ; a trillion is ; and so forth. Due to its dominance in the financial world (along with the US dollar), this was adopted for official United Nations documents. Traditional French usage has varied; in 1948, France, which had originally popularized the short scale worldwide, reverted to the long scale. The term milliard is unambiguous and always means 109. It is seldom 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 (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. Concerning names ending in -illiard for numbers 106n+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 "milliardo" in Italian, "Milliarde" in German, "miljard" in Dutch, "milyar" in Turkish, and "миллиард", milliard (transliterated) in Russian, are standard usage when discussing financial topics. The naming procedure for large numbers is based on taking the number n occurring in 103n+3 (short scale) or 106n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix ''. If the final root is multisyllabic and ends in a vowel, that vowel is removed; for example, centi + illion = centillion, not centiillion. Monosyllabic final roots ending in vowels only occur when the naming system does not apply, for very small numbers (million, billion, and sextillion), so they do not have defined behavior. The number "0" is skipped, i.e. it produces the empty string - 103 results in the root for units 3 followed by the root for hundreds 1. In this way, numbers up to 103·999+3 = 103000 (short scale) or 106·999 = 105994 (long scale) may be named. The choice of roots and the concatenation procedure is that of the standard dictionary numbers if n is 9 or smaller. For larger n'' (between 10 and 999), prefixes can be constructed based on a system described by Conway and Guy. Since the system of using Latin prefixes will become ambiguous for numbers with exponents of a size which the Romans rarely counted to, like 106,000,258, Conway and Guy co-devised with Allan Wechsler the following set of consistent conventions that permit, in principle, the extension of this system indefinitely to provide English short-scale names for any integer whatsoever. The name of a number 103n+3, where n is greater than or equal to 1000, is formed by concatenating the names of the numbers of the form 103m+3, where m represents each group of comma-separated digits of n, with each but the last "" trimmed to "", or, in the case of m = 0, either "-nilli-" or "-nillion". For example, 103,000,012, the 1,000,003rd "" number, equals one "millinillitrillion"; 1033,002,010,111, the 11,000,670,036th "" number, equals one "undecillinilliseptuagintasescentillisestrigintillion"; and 1029,629,629,633, the 9,876,543,210th "" number, equals one "nonilliseseptuagintaoctingentillitresquadragintaquingentillideciducentillion". The following table shows number names generated by the system described by Conway and Guy for the short and long scales. Note that since the scale begins at n=10, the short scale begins at 1033 and the long scale begins at 1060; below that, one is supposed to consult the dictionary.{{Cite web|author=Fish|url=https://kyodaisuu.github.io/illion/conway.html|title=Conway's illion converter == Unit prefixes ==

The following table lists the unit prefixes for powers of 1000 and 1024 according to the International System of Quantities (ISQ). == Other named large numbers used in mathematics, physics and chemistry ==

• Avogadro number – Number of particles in one mole, • Eddington number – Number of protons in the observable universe, about • Shannon number – Lower bound on the game-tree complexity of chess, about 10120 • Skewes's number – Large upper bound related to the prime-counting function, about • Moser's number • Graham's number – Upper bound on the answer to a problem in Ramsey theory • TREE(3) • SSCG(3) • Rayo's number – Claimed to be the largest named number == See also ==