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.
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:
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]
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:
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.
Categories: [Mathematics]