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Line 1: − + + + + + + + + + + + + + + + − + − Another system introduces a new name for each group of ''six'' zeroes. Thus, a billion means a million millions. This used to be British usage, ''but no longer is;'' newspapers and official government reports now use the same system as the U. S., and this system is the one now taught in British schools. Some older Britons remain attached to the older usage. + − + − + Line 34:
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Expansion and referencing
Confusingly, there are two systems for naming numbers using the ending ''-illion.'' The history of which system has been used in which countries at which times is complex.
'''Large numbers''' are [[numbers]] that are larger than those that one normally encounters in everyday life. A "large number" can also mean any number that most people consider too great to count.
== Concept ==
Since at least the rebuilding of human civilization after the [[Great Flood]], all societies that have worked with numbers at all, have of necessity set limits on the numbers that they would or could work with.<ref name=Crandall>Richard E. Crandall. "[http://www.fortunecity.com/emachines/e11/86/largeno.html The Challenge of Large Numbers]." ''Scientific American'', February 1997. Retrieved May 23, 2007, from http://www.fortunecity.com/</ref> The widely-held view that a society once existed that had only three words for numbers, those words translating as "one," "two," and "many," has no historical warrant. On the other hand, the maximum size of numbers that any society in recorded history has handled, has largely depended on their techniques for notation and/or storage of numbers.
Munafo<ref name=Munafo>Robert P. Munafo. "[http://home.earthlink.net/~mrob/pub/math/largenum.html Large numbers]." 1996-2006, released under [http://creativecommons.org/licenses/by/2.5/ Creative Commons Attribution 2.5 License]. Retrieved May 23, 2007.</ref> puts all numbers into three classes, in order of increasing size:
* Class 0: the smallest class, these numbers are easy to perceive directly. The largest class 0 number is six.
* Class 1: numbers that one cannot perceive directly, but that describe collections of objects that a person might still see with the naked eye. This class includes numbers from 7 to about one million (10<sup>6</sup> in either the American or British system; see below).
* Class 2: numbers that are too large for Class 1 but are still representable as exact values by [[place-value notation]] using the technology currently in existence. Today, therefore, Class 2 begins roughly at a million; how high this class reaches depends on the application and the requirements. Class 2 will likely expand with technological improvement.
* Class 3: all numbers beyond Class 2, representable only by approximation, using [[scientific notation]]. This class has no upper limit.
Hudelson <ref name=Hudelson>Matt Hudelson. "[http://www.sci.wsu.edu/math/faculty/hudelson/moser.html Large Numbers]." [[Washington State University]], October 2, 1997. Retrieved May 23, 2007.</ref> points out that one can always represent a very large finite number as an [[exponent]]ial [[power]] of smaller numbers. Examples of such notation in ancient literature abound. The best example in the [[Bible]] is probably {{Bible ref|book=Revelation|chap=9|verses=16|version=KJV}}.<ref>The [[King James Version]] renders the number as "two hundred thousand thousand," and most later [[English]] translations render it "two hundred million." The actual [[Greek language|Greek]] text reads "two myriads of myriads." A Greek myriad is ten thousand and was the Greek limit of named large numbers.</ref>
== Current English notation ==
Confusingly, the [[English]] speaking world uses two systems for naming numbers using the ending ''-illion.'' The history of which system has been used in which countries at which times is complex.<ref name=Rowlett>Russ Rowlett. "[http://www.unc.edu/~rowlett/units/large.html Names for Large Numbers]." [[University of North Carolina at Chapel Hill]], November 1, 2001. Retrieved May 23, 2007.</ref> The names for powers of a million, as used in the European system, trace to Nicholas Chuquet, who first proposed the names ''byllion'' and ''tryllion'' for the numbers 10<sup>12</sup> and 10<sup>18</sup>, respectively.<ref name=Rowlett/>
In both systems, a "million" means a thousand thousands or 10<sup>6</sup>.
In both systems, a "million" means a thousand thousands or 10<sup>6</sup>.
Above the "million," one system introduces a new name for each new group of ''three'' zeroes. In particular, a "billion" means a thousand millions. This system is used in the United States and has been for a long time.
Above the "million," the American system introduces a new name for each new group of ''three'' zeroes. In particular, a "billion" means a thousand millions. In general, if n represents the number corresponding to a [[Latin]] number from which the root of the name derives, then this Latinate name denotes the power 10<sup>3n+3</sup>. This system is used in the United States and has been for a long time.
The European system, also called the "British" system and used in the [[United Kingdom]] (and presumably in the [[Commonwealth of Nations]]) until at least 1974,<ref name=Rowlett/> introduces a new name for each group of ''six'' zeroes. If n represents the number corresponding to a [[Latin]]ate root, then in the European system the number name that derives from this root represents the power 10<sup>6n</sup>. Thus, a billion means a million millions. This used to be British usage, but in 1974, then-Prime Minister [[Harold Wilson]] declared the American system as official for government correspondence.<ref name=Rowlett/>. Newspapers and official government reports now use the same system as the [[United States]], and this system is the one now taught in British schools. Some older Britons remain attached to the older usage.
This system in which the word ''billion'' means 1,000,000,000,000, though not used England, is used in many European countries. For example, the series million, milliard, billion, billiard, trillion, trilliard, quadrillion, quadrilliard, quintillion, quintilliard (see below) is common use in Germany.
This system in which the word ''billion'' means 1,000,000,000,000, though not used England, is used in many European countries.
The word ''milliard,'' is unambiguous: it always means 1,000,000,000. It is rarely used in countries where a ''billion'' means the same thing. (Though rarely used, it is found in both British and American dictionaries and is thus a legitimate English word).
The word ''milliard,'' is unambiguous: it always means 1,000,000,000. It is rarely used in countries where a ''billion'' means the same thing. (Though rarely used, it is found in both British and American dictionaries and is thus a legitimate English word).
As a practical matter, it is important to be aware of the difference in meaning when reading old material, and to avoid using either set of names—especially the common but ambiguous ''billion''—in any context where misunderstanding can occur.
As a practical matter, it is important to be aware of the difference in meaning when reading old material, and to avoid using either set of names—especially the common but ambiguous ''billion''—in any context where misunderstanding can occur.
Words for numbers above 10<sup>18</sup> may be found in dictionaries or tables of number names, but are almost never used. For years, mathematicians and computer scientists have enjoyed a sort of game of inventing systems for naming large numbers, but there is no practical need to render such names as words, and it is very questionable whether any of them should be considered to be "real" words; they are seldom encountered outside of discussions of large numbers and number naming.
Words for numbers above 10<sup>18</sup> may be found in dictionaries or tables of number names, but are almost never used. For years, mathematicians and computer scientists have enjoyed a sort of game of inventing systems for naming large numbers, but there is no practical need to render such names as words, and it is very questionable whether any of them should be considered to be "real" words; they are seldom encountered outside of discussions of large numbers and number naming.
The name ''googol,'' for a 1 followed by a hundred zeroes, i.e. 10<sup>100</sup>, was popularized by a 1940 popular book on mathematics, Kasner and Newman's classic ''Mathematics and the Imagination.'' It was invented by Kasner's nine-year-old nephew. It and Kasner's own ''googolplex'' (a 1 followed by googol of zeroes) have entered the language, but are only used figuratively. The word ''googol'' was the inspiration for the name of the search service ''Google.''
The name ''googol,'' for a 1 followed by a hundred zeroes, i.e. 10<sup>100</sup>, was popularized by a 1940 popular book on mathematics, Kasner and Newman's classic ''Mathematics and the Imagination.'' It was invented by Kasner's nine-year-old nephew.<ref name=Pilhofer>Frank Pilhofer. "[http://www.fpx.de/fp/Fun/Googolplex/ The Googolplex]." September 16, 2002. Retrieved May 23, 2007.</ref> It and Kasner's own ''googolplex'' (a 1 followed by googol of zeroes) have entered the language, but are only used figuratively. The word ''googol'' was the inspiration for the name of the search service ''Google.''
== References ==
<references/>
[[Category:Mathematics]]