Many languages have words expressing indefinite and fictitious numbers—inexact terms of indefinite size, used for comic effect, for exaggeration, as placeholder names, or when precision is unnecessary or undesirable. One technical term for such words is "non-numerical vague quantifier". Such words designed to indicate large quantities can be called "indefinite hyperbolic numerals".
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General placeholder names
English has many words whose definition includes an indefinite quantity, such as "lots", "many", "several", "a lot", and "some". These placeholders can and often do have a generally equivalent numerical counterpart, e.g., "a couple" meaning two (2) or "a few" meaning approximately 3 to 8. Other placeholders can quantify items by describing how many fit into an approximately-specified volume; e.g., "a handful" represents more peas than grapes.
Specific numbers used as indefinite
In various Middle Eastern traditions, the number 40 is used to express a large but unspecific number, as in the Hebrew Bible's "forty days and forty nights", the Forty Martyrs of Sebaste, the story Ali Baba and the Forty Thieves, etc. This usage is sometimes found in English as well.
In Latin, sescenti (literally 600) was used to mean a very large number, perhaps from the size of a Roman cohort.
In Arabic, 1001 is used similarly, as in the The Book of One Thousand and One Nights (lit. "a thousand nights and one night"). Many modern English book titles use this convention as well: 1,001 Uses for ....
In Japanese, 八千, literally 8000, is used: 八千草 (lit. 8000 herbs) means a variety of herbs and 八千代 ( lit. 8000 generations) means eternity.
The number 10,000 is used to express an even larger approximate number, as in Hebrew רבבה rebâbâh, rendered into Greek as μυριάδες, and to English myriad. Similar usage is found in the East Asian 萬 or 万 (lit. 10,000), and the South Asian lakh (lit. 100,000).
In Chinese, 十萬八千里, literally 108,000 li, means a great distance.
Other specific numbers are occasionally used as indefinite as well. English does this with count nouns that refer to numbers: a dozen/dozens, a score/scores, a hundred/hundreds, and similarly thousand, million, billion. Unlike cardinal numbers, these can be pluralized, in which case they require of before the noun (millions of dollars, but five million dollars), and require the indefinite article "a" in the singular (a million letters (indefinite) but one million letters (definite)).
Umpteen
Umpteen, umteen, or umpty is an unspecified but large number, used in a humorous fashion or to imply that it is not worth the effort to pin down the actual figure. Despite the -teen ending, which would seem to indicate that it lies between 12 and 20, umpteen can be much larger.
"Umpty" is first attested in 1905, in the expression "umpty-seven", implying that it is a multiple of ten. Ump(ty) came from a verbalization of a dash in Morse code.
"Umpteen", adding the ending -teen, as in "thirteen", is first attested in 1918, and has become by far the most common form.
-illion
Words with the suffix -illion (e.g., zillion, gazillion, kazillion and bajillion) are often used as informal names for unspecified large numbers by analogy to names of large numbers such as million (106), billion (109) and trillion (1012). These words are intended to denote a number that is large enough to be unfathomable.
These terms are often used as hyperbole or for comic effect, or in loose conversation to present an unguessably large number. Since these are undefined, they have no mathematical validity and no accepted order.
The "-illion" concept is so well established that it is the basis of a joke, in which a speaker misunderstands the word Brazilian (a person from the nation of Brazil) as an enormous number called "brazillion".
Many similar words are used, such as bazillion, dillion, gadzillion, gagillion, gajillion, godzillion, grillion, hojillion, jillion kabillion, kajillion, katrillion, killion, robillion, skillion, squillion, and umptillion.
These words can be transformed into ordinal numbers or fractions by the usual pattern of appending the suffix -th, e.g., "I asked her for the jillionth time." Also, the suffix can be replaced with "-illionaire" to describe wealthy people.
These terms are sometimes used in technical jargon to represent some arbitrary yet common power of a base radix, typically when the number of trailing zeroes is not significant for the concept being discussed. For example, in computer architecture, if memory region 1 spans the hexadecimal range 20000000-23FFFFFF (in bytes) and memory region 2 spans the hexadecimal range 24000000-27FFFFFF, then it could be said that the base addresses of the memory regions are located at "20 bazillion" and "24 bazillion".
Sagan's number
Sagan's number is the number of stars in the observable universe. It is named in honor of Carl Sagan. This number is reasonably well defined, because it is known what stars are and what the observable universe is, but its value is highly uncertain.
Sagan's number is to be distinguished from the sagan unit or the humorous use of the term "sagan" to denote any large quantity—specifically, any number in the billions, due to Sagan's association with the phrase "billions and billions".