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Introduction

Some large numbers have real referents in human experience. Their names are real words, encountered in many contexts. For example, today, Google News shows 78,600 hits on "billion," starting with "Turkey Repays $1.6 Billion In Foreign Debt." It shows 9870 hits on "trillion," and 56 on "quadrillion:" "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."

It can be argued that a number such as a "trigintillion" has no existence in the physical world. There simply is not a trigintillion of anything, and no practical need for such a name.

Nevertheless, large numbers have an intellectual fascination and are of mathematical interest. Giving them names 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 (10) a "first number;" a myriad myriad first numbers as "second numbers" (10), and so on up to "eighth numbers" (10). He concluded that it would take less than "one thousand myriad myriad eighth numbers" grains of sand to pack the universe solid with sand.

Since then, many others have engaged in the pursuit of conceptualizing and naming numbers that really have no existence outside of the imagination. Such names, 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 × 10 ergs." When a number such as 10 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 to American and British audiences—assuming that there are any listeners who understand this number without consulting a dictionary). 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 or 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. Glancing at the table for the Googol family, it is clear what the value of a "Googolquattuordeciplex" would be if there were such a word, but that does not mean that such a word exists, nor that there is any value in coining it.

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.

Dictionary numbers

These names of numbers are found in many English dictionaries and as such have a special claim to being "real words" even though those above quadrillion are very rarely used.

Note the oddity that two of the dictionaries consulted list "decillion" and "vigintillion," but none of the numbers in between (undecillion, duodecillion, etc.)

Vigintillion 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 Kasners' 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."

Dictionaries cited:
AHD4: American Heritage Dictionary, 4th edition, at http://www.bartleby.com/61
ShorterOED3-1993: Shorter Oxford English Dictionary, 3rd edition, 1993, Oxford: Clarendon Press
RHD2-1987: The Random House Dictionary, 2nd Unabridged Edition, 1987, Random House
W3-1993: Webster's Third New International Dictionary, Unabridged, 1993, Merriam-Webster

The Googol family

The googol and googolplex were introduced in Kasner and Newman's 1940 book, Mathematics and the Imagination, thus:

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 as 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.

Chuquet and the origins of the 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 quyllion Le six sixlion Le sept. septyllion Le huyt ottyllion Le neuf nonyllion et ainsi des ault' 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).

Because of this passage, 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.

Both Adam and Chuquet applied the names to powers of a million; that is, a bimillion (Adam) or byllion (Chuquet) was 10, and a trimillion (Adam) or tryllion (Chuquet) was 10.

Extensions of the standard dictionary numbers

Chuquet-ized Numbering

This table contains a list of numbers that extend to very large sums. Note that there is no standard way of naming after centilliard or centillion.

Commercial Product using Yoctillion

See Also

• List of numbers
• Large numbers
• Chuquet
• Pelletier
• Knuth

References

• Kasner, Edward and James Newman, Mathematics and the Imagination, 1940, Simon and Schuster, New York.

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