Difference between revisions of "names of large numbers"

Revision as of 20:24, 30 September 2008

The standard dictionary numbers

Throughout this article, exponential or scientific notation is used. 10⁶ could also be written as the number 1 followed by six 0s, 1 000 000; 10⁹ could be written as 1 000 000 000, and so on.

Names of numbers larger than a quadrillion are almost never used, for reasons discussed further below. The following table lists those names of numbers which are found in many English dictionaries. The "Traditional British" values shown are becoming obsolete.

The entry "approximate frequency of use" shows how often the word is used compared to the word "trillion."^[1] Million is used thirty-nine times as often as trillion, while septillion is used only 1/200 as often.

Centillion AHD 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 (googolplexplex, etc.). The Shorter Oxford English Dictionary comments that googol and googolplex are "not in formal mathematical use".

• Vigintillion is used by H. P. Lovecraft in a Cthulhu story.

Dictionaries cited

• AHD4: American Heritage Dictionary, 4th edition, at http://www.bartleby.com/61
• COD: Cambridge Online Dictionaries, Cambridge, UK: Cambridge University Press ; at http://dictionary.cambridge.org
• Dcom: Dictionary.com
• OED2: Oxford English Dictionary, 2nd edition, Oxford, UK: Oxford University Press ; (and addendums since publication in 1989)
• OEDnew: Oxford English Dictionary, New Edition, Oxford, UK: Oxford University Press ; at http://www.oed.com (subscription required)
• RHD2: The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House
• SOED3: Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press
• W3: Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster

Template:unicode means the word is included in the dictionary

• ^  Frequency of usage determined by the number of Google hits on pages in English on the web, as of September, 2005.

Usage of names of large numbers

Some large numbers have real referents in human experience. Their names are real words, encountered in many contexts. For example, on one day in 2004, Google News showed 78 600 hits on "billion", starting with "Turkey Repays USD 1.6 Billion In Foreign Debt". It shows 9870 hits on "trillion", and 56 on "quadrillion": for example, "The US Department of Energy reports that in 2002, the United States economy consumed 97.6 quadrillion BTUs (quad BTUs)."

References to names of quantities larger than a quadrillion, however, are rare, and increasingly artificial; they tend to be limited to discussions of names of numbers, or mathematical concepts. For example, the first hit on "quintillion" is about a man who is trying to preserve the Etsako language by codifying it, and "went as far as providing numerals from one to quintillion". "Septillion" appears—as the name of a racing yacht.

Nevertheless, large numbers have an intellectual fascination and are of mathematical interest. Giving them names is one of the ways in which people try to conceptualize and understand them.

In The Sand Reckoner, Archimedes estimated the number of grains of sand that would be required to fill the known universe. To do this, he called a myriad myriad (10⁸) a "first number"; a myriad myriad first numbers as "second numbers" (10¹⁶), and so on up to "eighth numbers" (10⁶⁴). He concluded that it would take less than "one thousand myriad myriad eighth numbers" grains of sand to pack the universe solid with sand. This much sand would fill a volume larger than our galaxy, the Milky Way, but smaller than the group of galaxies it is part of, the Local Group.

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.

Names of numbers larger than a quadrillion, even if found in dictionaries, have a tenuous existence. Just as one can debate whether floccinaucinihilipilification is really an English word—it is in the Oxford English Dictionary, but no other—it is questionable how real the word "trigintillion" is. "Trigintillion" is only encountered in definitions, lists of names of large numbers, and "Sand Reckoner"-like discourses on the meaning of very large numbers.

Even well-established names like "sextillion" are rarely used, even in those contexts where they have a real meaning—science, astronomy, and engineering. In science, since the 1800s, numbers have been written using the familiar "scientific notation", in which powers of ten are expressed as a ten with a numeric superscript, e.g. "The X-ray emission of the radio galaxy is 1.3 × 10⁴⁵ ergs." When a number such as 10⁴⁵ needs to be referred to in words, it is simply read out: "ten to the forty-fifth." This is just as easy to say and easier to understand than "quattuordecillion". When a number represents a measurement rather than a count, SI prefixes are used; one says "femtosecond", not "one quadrillionth of a second." In some cases, specialized very large units are used, such as the astronomer's parsec and light year.

It should be noted, too, that 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.

Here we present some names that have been given to large numbers, and the context and authority for the names. These numbers are almost pure mathematical abstractions, not physical realities. The names for such numbers are very rarely used. They may have a claim staked out for them in reference books, but they remain more in the nature of curiosities, trivia, or mathematical recreation than genuine working English vocabulary.

Chuquet and the origins of the standard dictionary numbers

Nicolas Chuquet's book Triparty en la science des nombres was not published during his lifetime, but most of it was copied by Estienne de la Roche for a portion of his 1520 book, Larismetique. 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 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".
• The words "bymillions" and "trimillions" were first recorded in 1475 in a work by Jehan Adam.
• 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, did extend the naming scheme and invent the names for the higher powers.
• Notice that 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, a bimillion (Adam) 10¹², and a trimillion was 10¹⁸.

An Aide Memoire

An easy way to find out scientific notation for one of the above numbers is to take the Latin cardinal number indicated in the name (Such as 2 in 'bi'llion, 4 in 'quadri'llion, 18 in 'octodec'illion etc), then add one to it, and finally multiply the whole thing by 3. For example, in a trillion, the Latin numeral is 'tri', or 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 trillion in scientific notation: one trillion = 10¹².

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:

Words of wisdom are spoken by children at least as often as by scientists. 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 a 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, with so many zeros after the 1 that the number of zeros is a googol.

Extensions of the standard dictionary numbers

This table illustrates several systems for naming large numbers, and shows how they can be extended past decillion.

Traditional British usage assigned new names for each power of one million (the long scale). It was adapted from French usage, and is similar to the system that was documented or invented by Chuquet.

Traditional American usage (which, oddly enough, was also adapted from French usage but at a later date), and modern British usage, assigns new names for each power of one thousand (the short scale.) Thus, a "billion" is 10⁹, a "trillion" is 10¹², 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 10⁹. It is almost never seen in American usage, rarely in British usage, and frequently in European usage. The term is sometimes attributed to a French mathematician named Jacques Pelletier du Mans circa 1550 (therefore the long scale is also known as 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 to our modern term.

With regard to names ending in -illiard, milliard is certainly in widespread use in languages other than English, but the degree of actual use of the larger terms is questionable. For example, as of 2004, Google searches on French-language pages for "trillion", "trilliard", "quadrillion", "quadrilliard", and "quintillion" return 6630, 102, 312, 7, and 127 hits, respectively. However, one has to take into account that these large numbers are not needed too often and scientists almost always use exponents.

Names of reciprocals of large numbers do not need to be listed here, because they regularly formed by adding -th, e.g. quattuordecillionth, centillionth, etc. Googolminex has been proposed as a name for 1/googol but is not in any real use.

For additional details, see: billion and long scale.

Conway numbers

This system will itself become ambiguous for numbers much larger than this, with exponents of a size which the Romans rarely counted to, like 10 ^6,000,258. John Horton Conway has proposed, in The Book of Numbers, a consistent set of conventions which permit the system to provide "English names", in principle, for any integer whatever.

External links

• Large Numbers article by Robert Munafo
• Name of a number Automatic number-to-name converter by Landon Curt Noll
• Full list of large number names list sorted by 10ⁿ and by word length

References

• Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.
• Weisstein, Math World, Wolfram Research, http://mathworld.wolfram.com