The long and short scales are two
powers of ten number naming systems that are consistent with each other for smaller
numbers
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
, but are contradictory for larger numbers.
[
][
] Other numbering systems, particularly in
East Asia
East Asia is a geocultural region of Asia. It includes China, Japan, Mongolia, North Korea, South Korea, and Taiwan, plus two special administrative regions of China, Hong Kong and Macau. The economies of Economy of China, China, Economy of Ja ...
and
South Asia
South Asia is the southern Subregion#Asia, subregion of Asia that is defined in both geographical and Ethnicity, ethnic-Culture, cultural terms. South Asia, with a population of 2.04 billion, contains a quarter (25%) of the world's populatio ...
, have large number naming that differs from both the long and short scales. Such numbering systems include the
Indian numbering system and
Chinese,
Japanese, and
Korean numerals.
Much of the remainder of the world adopted either the short or long scale. Countries using the long scale include most countries in continental Europe and most that are
French-speaking,
German-speaking and
Spanish-speaking. Use of the short scale is found in most
English and
Arabic
Arabic (, , or , ) is a Central Semitic languages, Central Semitic language of the Afroasiatic languages, Afroasiatic language family spoken primarily in the Arab world. The International Organization for Standardization (ISO) assigns lang ...
speaking countries, most Eurasian post-communist countries and
Brazil
Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
.
For powers of ten less than 9 (one, ten, hundred, thousand, and million), the short and long scales are identical, but for larger powers of ten, the two systems differ in confusing ways. For identical names, the long scale grows by multiples of one million (10
6), whereas the short scale grows by multiples of one thousand (10
3). For example, the short scale ''billion'' is one thousand million (10
9), whereas in the long scale, ''billion'' is one million million (10
12), making the word
false friends
In linguistics, a false friend is a word in a different language that looks or sounds similar to a word in a given language, but differs significantly in meaning. Examples of false friends include English language, English ''embarrassed'' an ...
between long and short scale languages. The long scale system includes additional names for interleaved values, typically replacing the word ending "-ion" with "-iard".
To avoid confusion, the
International System of Units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(SI) recommends using the
metric prefix
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pr ...
es to indicate magnitude.
For example,
giga
Giga- ( or ) is a metric prefix, unit prefix in the metric system denoting a factor of a Long and short scales, short-scale billion or long-scale milliard (109 or 1,000,000,000). It has the symbol G.
''Giga-'' is derived from the Ancient Greek, ...
is always 10
9, which is billion in short scale but milliard in long scale.
Definition
In both scales, names are given to orders of magnitude at increments of 1000. Both systems use the same names for magnitudes less than 10
9. Differences arise from the use of identical names for larger magnitudes. For the same magnitude name (''n''-illion), the value is 10
3''n''+3 in the short scale but 10
6''n'' in the long scale for positive integers ''n''.
In some languages, the long scale uses additional names for the intermediate multipliers, replacing the ending ''-ion'' with ''-iard''; for example, the next multiplier after ''million'' is ''milliard'' (10
9); after a ''billion'' it is ''billiard'' (10
15). Hence, a long scale ''n''-illiard equals 10
6''n''+3.
The following table shows the size of first few short and long scale magnitudes. Notice how billion and trillion are in both scales but have different sizes.
Comparison
The following tables show the corresponding names and values of the two scales.
Note that instead of using an intermediate long scale word (illiard), a quantity is sometimes specified in terms of the smaller illion word. For example, "thousand billion" instead of "billiard".
The different sizes of the same name of the two scales can be described as:
Avoiding confusion
One way to avoid confusion between the two scales is to use
positional notation
Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a posit ...
. For example, 1,000,000,000,000 rather than 1 trillion (short scale) or 1 billion (long scale). This method becomes unwieldy for very large numbers.
Combinations of the unambiguous words: ten, hundred, thousand, and million. For example: one thousand million and one million million.
Scientific notation
Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form, since to do so would require writing out an inconveniently long string of digits. It may be referred to as scientif ...
(for example 1), or its
engineering notation variant (for example 10), or the computing variant
E notation (for example
1e10
). This is the most common practice among scientists and mathematicians.
SI metric prefix
A metric prefix is a unit prefix that precedes a basic unit of measure to indicate a multiple or submultiple of the unit. All metric prefixes used today are decadic. Each prefix has a unique symbol that is prepended to any unit symbol. The pr ...
es. For example,
giga
Giga- ( or ) is a metric prefix, unit prefix in the metric system denoting a factor of a Long and short scales, short-scale billion or long-scale milliard (109 or 1,000,000,000). It has the symbol G.
''Giga-'' is derived from the Ancient Greek, ...
for 10
9 and
tera for 10
12 can give
gigawatt (10
9 W) and
terawatt (10
12 W).
Use with non-SI units is unambiguous. For example, giga-dollars, megabucks, k€, and M€.
History
Although this situation has been developing since the 1200s, the first recorded use of the terms ''short scale'' () and ''long scale'' () was by the French mathematician
Geneviève Guitel in 1975.
The short scale was never widespread before its general adoption in the United States. It has been taught in American schools since the early 1800s.
It has since become common in other
English-speaking nations and several other countries. For most of the 19th and early 20th centuries, the
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Northwestern Europe, off the coast of European mainland, the continental mainland. It comprises England, Scotlan ...
largely used the long scale,
[
] whereas the
United States
The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 U.S. state, states and a federal capital district, Washington, D.C. The 48 ...
used the short scale,
so that the two systems were often referred to as ''British'' and ''American'' in the
English language
English is a West Germanic language that developed in early medieval England and has since become a English as a lingua franca, global lingua franca. The namesake of the language is the Angles (tribe), Angles, one of the Germanic peoples th ...
. After several decades of increasing informal British usage of the short scale, in 1974 the government of the UK adopted it,
[
] and it is used for all official purposes.
[
][
][
][
][
] The British usage and American usage are now identical.
The existence of the different scales means that care must be taken when comparing large numbers between languages or countries, or when interpreting old documents in countries where the dominant scale has changed over time. For example, British English, French, and Italian historical documents can refer to either the short or long scale, depending on the date of the document, since each of the three countries has used both systems at various times in its history. Today, the United Kingdom officially uses the short scale, but
France
France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan ...
and
Italy
Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...
use the long scale.
The pre-1974 former British English word ''billion'', post-1961 current French word ''billion'', post-1994 current Italian word ''bilione'', Spanish ''billón'', German ''Billion'', Dutch ''biljoen'', Danish ''billion'', Swedish ''biljon'', Finnish ''biljoona'', Slovenian ''bilijon'', Polish ''bilion'', and European Portuguese word ''bilião'' (with a different spelling to the Brazilian Portuguese variant, but in Brazil referring to short scale) all refer to 10
12, being long-scale terms. Therefore, each of these words translates to the American English or post-1974 British English word: ''trillion'' (10
12 in the short scale), and not ''billion'' (10
9 in the short scale).
On the other hand, the pre-1961 former French word ''billion'', pre-1994 former Italian word ''bilione'', Brazilian Portuguese word ''bilhão'', and Welsh word ''biliwn'' all refer to 10
9, being short scale terms. Each of these words translates to the American English or post-1974 British English word ''billion'' (10
9 in the short scale).
The term ''billion'' originally meant 10
12 when introduced.
In long scale countries, ''milliard'' was defined to its current value of 10
9, leaving ''billion'' at its original 10
12 value and so on for the larger numbers.
Some of these countries, but not all, introduced new words ''billiard'', ''trilliard'', etc. as intermediate terms.
In some short scale countries, ''milliard'' was defined to 10
9 and ''billion'' dropped altogether, with ''trillion'' redefined down to 10
12 and so on for the larger numbers.
In many short scale countries, ''milliard'' was dropped altogether and ''billion'' was redefined down to 10
9, adjusting downwards the value of ''trillion'' and all the larger numbers.
The root ''mil'' in ''million'' does not refer to the numeral, ''1''. The word, ''million'', derives from the Old French, ''milion'', from the earlier Old Italian, ''milione'', an intensification of the Latin word, ''mille'', a thousand. That is, a ''million'' is a ''big thousand'', much as a ''
great gross'' is a dozen gross or 12 × 144 =
1728.
[
]
The word ''milliard'', or its translation, is found in many European languages and is used in those languages for 10
9. However, it is not found in American English, which uses ''billion'', and not used in British English, which preferred to use ''thousand million'' before the current usage of ''billion''. The financial term ''yard'', which derives from ''milliard'', is used on financial markets, as, unlike the term ''billion'', it is internationally unambiguous and phonetically distinct from ''million''. Likewise, many long scale countries use the word ''billiard'' (or similar) for one thousand long scale billions (i.e., 10
15), and the word ''trilliard'' (or similar) for one thousand long scale trillions (i.e., 10
21), etc.
[
][
][
][
][
]
;Timeline
Current usage
Short scale users
English-speaking
Most English-language countries and regions use the short scale with 10
9 being ''billion''. For example:
*
* (English-speaking) ''see''
Using both ''below''
* (English-speaking, , ''trilliún'')
*
*
Arabic-speaking
Most
Arabic-language countries and regions use the short scale with 10
9 being , except for a few countries like Saudi Arabia and the UAE which use the word بليون for 10
9. For example:
*
*
*
*
*
*
Other short scale
Other countries also use a word similar to ''trillion'' to mean 10
12, etc. Whilst a few of these countries like English use a word similar to ''billion'' to mean 10
9, most like Arabic have kept a traditionally long scale word similar to ''milliard'' for 10
9. Some examples of short scale use, and the words used for 10
9 and 10
12, are
* (
Brazilian Portuguese
Brazilian Portuguese (; ; also known as pt-BR) is the set of Variety (linguistics), varieties of Portuguese language native to Brazil. It is spoken by almost all of the 203 million inhabitants of Brazil and widely across the Brazilian diaspora ...
: , )
* (, )
* (
Hebrew
Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...
: , )
* ( , )
* (, )
Long scale users
The long scale is used by most
Continental Europe
Continental Europe or mainland Europe is the contiguous mainland of Europe, excluding its surrounding islands. It can also be referred to ambiguously as the European continent, – which can conversely mean the whole of Europe – and, by som ...
an countries and by most other countries whose languages derive from Continental Europe (with the notable exceptions of Albania, Greece, Romania,
and Brazil). These countries use a word similar to ''billion'' to mean 10
12. Some use a word similar to ''milliard'' to mean 10
9, while others use a word or phrase equivalent to ''thousand millions''.
Dutch-speaking
Most
Dutch-language
Dutch ( ) is a West Germanic languages, West Germanic language of the Indo-European language family, spoken by about 25 million people as a first language and 5 million as a second language and is the List of languages by total number of speak ...
countries and regions use the long scale with 10
9 = .
French-speaking
Most
French-language countries and regions use the long scale with 10
9 = , for example:
* (
Canadian French
Canadian French (, ) is the French language as it is spoken in Canada. It includes multiple varieties, the most prominent of which is Québécois (Quebec French). Formerly ''Canadian French'' referred solely to Quebec French and the closely re ...
) ''see''
Using both ''below''
*
German-speaking
German-language
German (, ) is a West Germanic language in the Indo-European language family, mainly spoken in Western and Central Europe. It is the majority and official (or co-official) language in Germany, Austria, Switzerland, and Liechtenstein. It is a ...
countries and regions use the long scale with 10
9 = .
Portuguese-speaking
With the notable exception of Brazil, a short scale country, most
Portuguese-language countries and regions use the long scale with 10
9 = ''or'' .
Spanish-speaking
Most
Spanish-language
Spanish () or Castilian () is a Romance languages, Romance language of the Indo-European languages, Indo-European language family that evolved from the Vulgar Latin spoken on the Iberian Peninsula of Europe. Today, it is a world language, gl ...
countries and regions use the long scale, for example:
*
* ( or )
* ( or typ. )
Other long scale
Some examples of long scale use, and the words used for 10
9 and 10
12, are:
*
Esperanto
Esperanto (, ) is the world's most widely spoken Constructed language, constructed international auxiliary language. Created by L. L. Zamenhof in 1887 to be 'the International Language' (), it is intended to be a universal second language for ...
(, )
* (
Persian:
میلیارد ,
بیلیون ,
تریلیون )
* (, )
* (, )
*
(, ). There are ambiguities for numbers above 10
12.
* (French: , ;
German: , ; Italian: , ;
Romansh: ,
)
Using both
Some countries use either the short or long scales, depending on the internal language being used or the context.
Using neither
The following countries use naming systems for large numbers that are not etymologically related to the short and long scales:
By continent
The long and short scales are both present on most continents, with usage dependent on the language used. For example:
Notes on current usage
Short scale
Long scale
Both long and short scale
Neither long nor short scale
See also
*
Googol
A googol is the large number 10100 or ten to the power of one hundred. In decimal notation, it is written as the digit 1 followed by one hundred zeros: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, ...
(number)
*
Googolplex (number)
*
Names of large numbers
*
Names of small numbers
*
Orders of magnitude (numbers)
*
Hindu units of time which displays some similar issues
*
Indian numbering system
References
External links
BBC News article: "Is trillion the new billion?"Live-Counter.com: How to visualize large numbers:"
{{DEFAULTSORT:Long And Short Scales
Numerals
Numeral systems