# Large numbers

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.[1] 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[2] puts all numbers into four 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 (106 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 [3] points out that one can always represent a very large finite number as an exponential power of smaller numbers. Examples of such notation in ancient literature abound. The best example in the Bible is probably Revelation 9:16 (KJV).[4]

## Current English notation

Confusingly, the English speaking world uses two systems for naming numbers larger than million using the name with ending -illion. The history of which system has been used in which countries at which times is complex.[5] 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 1012 and 1018, respectively.[5]

In both systems, a "million" means a thousand thousands or 106.

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 103n+3. 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,[5] introduces a new name for each group of six zeroes. If n represents the number corresponding to a Latinate root, then in the European system the number name that derives from this root represents the power 106n. 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.[5] 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.

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.

In technical or scientific writing, the problem does not arise because it is customary to use scientific notation (1012) or SI prefixes (tera-).

U.S. usage Value Former British usage
thousand 103 = 1,000 thousand
million 106 = 1,000,000 million
billion 109 = 1,000,000,000 thousand million or milliard
trillion 1012 = 1,000,000,000,000 billion
quadrillion 1015 = 1,000,000,000,000,000 thousand billion or (very rarely) billiard
quintillion 1018 = 1,000,000,000,000,000,000 trillion

Words for numbers above 1018 may be found in dictionaries or tables of number names, but are almost never used. In American English, billions are used for human population or for government budgets, and very rarely trillions are used for economic measures such as gross national product.

When engineers or scientists want to use large numbers, they use powers of ten notation:

• The speed of light is 3 x 108 meters per second

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. 10100, was popularized by a 1940 popular book on mathematics, Kasner and Newman's classic Mathematics and the Imagination.[6] It was invented by Kasner's nine-year-old nephew.[7] 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

1. Richard E. Crandall. "The Challenge of Large Numbers." Scientific American, February 1997. Retrieved May 23, 2007, from http://www.fortunecity.com/
2. Robert P. Munafo. "Large numbers." 1996-2006, released under Creative Commons Attribution 2.5 License. Retrieved May 23, 2007.
3. Matt Hudelson. "Large Numbers." Washington State University, October 2, 1997. Retrieved May 23, 2007.
4. 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 text reads "two myriads of myriads." A Greek myriad is ten thousand and was the Greek limit of named large numbers.
5. Russ Rowlett. "Names for Large Numbers." University of North Carolina at Chapel Hill, November 1, 2001. Retrieved May 23, 2007.
6. Edward Kasner and James Newman. Mathematics and the Imagination. Originally published 1940. Current edition: Dover Publications, 2001. ISBN 0486417034.
7. Frank Pilhofer. "The Googolplex." September 16, 2002. Retrieved May 23, 2007.