Million Billion Trillion
Million Billion Trillion
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 which are found in many English dictionaries and thus have a special claim to being "real words". The "Traditional British" values shown are unused in American English and are obsolete in British English, but are dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America; see Long and short scales.
The "standard dictionary numbers"
Apart from million , the words in this list ending with - illion are all derived by adding prefixes ( bi -, tri -, etc.) to the stem - illion . centillion 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.).
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 excessive inflation. 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 only worth about 30 US dollars.
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 contexts of science, astronomy, and engineering, where large numbers often occur, numbers are usually 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 ergs." When a number such as 10 45 needs to be referred to in words, it is simply read out: "ten to the forty-fifth." This is just as easy to say, easier to understand, 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 first 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 of 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 poit 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).
Chuquet is sometimes credited with inventing the names million , billion , trillion , quadrillion , and so forth. This is an oversimplification.
Million was certainly not invented by Adam or Chuquet. Milion is an Old French word thought to derive from Old Italian milione , an intensification of mille , a thousand. That is, a million is a big thousand , much as 1728 is a great gross .
From the way in which Adam and Chuquet use the words, it can be inferred that they were recording usage rather than inventing it. One obvious possibility is that words similar to billion and trillion were already in use and well-known, but that Chuquet, an expert in exponentiation, extended the naming scheme and invented the names for the higher powers.
Chuquet's names are only similar to, not identical to, the modern ones.
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 .
An aide-memoire
An easy way to find the value of the above numbers in the short scale is to take the number indicated by the prefix (such as 2 in bi llion, 4 in quadri llion, 18 in octodec illion, etc.), add one to it, and multiply that result by 3. For example, in a trillion, the prefix is tri , meaning 3. Adding 1 to it gives 4. Now multiplying 4 by 3 gives us 12, which is the power to which 10 is to be raised to express a short-scale trillion in scientific notation: one trillion = 10 12 .
In the long scales, this is done simply by multiplying the number from the prefix by 6. For example, in a billion, the prefix is bi , meaning 2. Multiplying 2 by 6 gives us 12, which is the power to which 10 is to be raised to express a long-scale billion in scientific notation: one billion = 10 12 . The intermediate values (billiard, trilliard, etc.) can be converted in a similar fashion, by adding ½ to the number from the prefix and then multiplying by six. For example, in a septilliard, the prefix is sept meaning 7. Multiplying 7½ by 6 yields 45, and one septilliard equals 10 45 . Doubling the prefix and adding one then multiplying the result by three would give the same result.
These mechanisms are illustrated in the table in long and short scales.
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 , 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
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