Short Scale

The long and short scales are two powers of ten number naming systems that are consistent with each other for smaller numbers, but are contradictory for larger numbers.

Other numbering systems, particularly in East Asia and South Asia, 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 speaking countries, most Eurasian post-communist countries and Brazil.

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 (10⁶), whereas the short scale grows by multiples of one thousand (10³). For example, the short scale ''billion'' is one thousand million (10⁹), whereas in the long scale, ''billion'' is one million million (10¹²), making the word false friends 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 (SI) recommends using the metric prefixes to indicate magnitude.
For example, giga is always 10⁹, 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 10⁹. 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'' (10⁹); after a ''billion'' it is ''billiard'' (10¹⁵). 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. 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 (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 prefixes. For example, giga for 10⁹ and tera for 10¹² can give gigawatt (10⁹ W) and terawatt (10¹² 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 largely used the long scale,

whereas the United States used the short scale, so that the two systems were often referred to as ''British'' and ''American'' in the English language. 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 and Italy 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 10¹², being long-scale terms. Therefore, each of these words translates to the American English or post-1974 British English word: ''trillion'' (10¹² in the short scale), and not ''billion'' (10⁹ 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 10⁹, being short scale terms. Each of these words translates to the American English or post-1974 British English word ''billion'' (10⁹ in the short scale).

The term ''billion'' originally meant 10¹² when introduced. In long scale countries, ''milliard'' was defined to its current value of 10⁹, leaving ''billion'' at its original 10¹² 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 10⁹ and ''billion'' dropped altogether, with ''trillion'' redefined down to 10¹² and so on for the larger numbers. In many short scale countries, ''milliard'' was dropped altogether and ''billion'' was redefined down to 10⁹, 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 10⁹. 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., 10¹⁵), and the word ''trilliard'' (or similar) for one thousand long scale trillions (i.e., 10²¹), etc.

;Timeline

Current usage

Short scale users

English-speaking

Most English-language countries and regions use the short scale with 10⁹ 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 10⁹ being , except for a few countries like Saudi Arabia and the UAE which use the word بليون for 10⁹. For example:

*
*
*
*
*
*

Other short scale

Other countries also use a word similar to ''trillion'' to mean 10¹², etc. Whilst a few of these countries like English use a word similar to ''billion'' to mean 10⁹, most like Arabic have kept a traditionally long scale word similar to ''milliard'' for 10⁹. Some examples of short scale use, and the words used for 10⁹ and 10¹², are

* (Brazilian Portuguese: , )
* (, )
* (Hebrew: , )
* ( , )
* (, )

Long scale users

The long scale is used by most Continental European 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 10¹². Some use a word similar to ''milliard'' to mean 10⁹, while others use a word or phrase equivalent to ''thousand millions''.

Dutch-speaking

Most Dutch-language countries and regions use the long scale with 10⁹ = .

French-speaking

Most French-language countries and regions use the long scale with 10⁹ = , for example:

* (Canadian French) ''see'' Using both ''below''
*

German-speaking

German-language countries and regions use the long scale with 10⁹ = .

Portuguese-speaking

With the notable exception of Brazil, a short scale country, most Portuguese-language countries and regions use the long scale with 10⁹ = ''or'' .

Spanish-speaking

Most Spanish-language 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 10⁹ and 10¹², are:

* Esperanto (, )
* (Persian: میلیارد , بیلیون , تریلیون )
* (, )
* (, )
* (, ). There are ambiguities for numbers above 10¹².
* (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 (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