Long and short scales

The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes.

For whole numbers smaller than 1,000,000,000 (10⁹), such as one thousand or one million, the two scales are identical. For larger numbers, starting with 10⁹, the two systems differ. For identical names, the long scale proceeds by powers of one million, whereas the short scale proceeds by powers of one thousand. For example, in the short scale, "one billion" means one thousand millions (1,000,000,000), whereas in the long scale, it means one million millions (1,000,000,000,000). For interleaved values, the long scale system employs additional terms, typically substituting the word ending -ion for -iard.

Some languages, particularly in East Asia and South Asia, have large number naming systems that are different from both the long and short scales, such as Chinese, Japanese or Korean numerals, and the Indian numbering system. Much of the remainder of the world adopted either the Short Scale or the Long Scale for everyday counting powers of ten. Countries with usage of the long scale include most countries in continental Europe and most that are French-speaking, German-speaking, Spanish-speaking and Portuguese-speaking countries (except Brazil). Usage of the short scale is found in most English-speaking countries and most Arabic-speaking countries.

To avoid confusion resulting from the coexistence of the two terms, the International System of Units (SI) recommends using the metric prefix to indicate orders of magnitude.

Scales

In both short and long scale naming, names are given each multiplication step for increments of the base-10 exponent of three, i.e. for each integer ''n'' in the sequence of multipliers 10³ⁿ. For certain multipliers, including those for all numbers smaller than 10⁹, both systems use the same names. The differences arise from the assignment of identical names to specific values of ''n'', for numbers starting with 10⁹, for which ''n''=3. In the short scale system, the identical names are for ''n''=3, 4, 5, ..., while the long scale places them at ''n''=4, 6, 8, etc.

Short scale

In the short scale, ''billion'' means a thousand millions (''1,000,000,000'' which is 10⁹), ''trillion'' means one thousand billions (10¹²), and so on. Thus, an ''n''-illion equals 10^3''n''+3.

Long scale

In the long scale, ''billion'' means one million millions (10¹²) and ''trillion'' means one million billions (10¹⁸), and so on. Therefore, an ''n''-illion equals 10^6''n''. In some languages, the long scale uses additional names for the interleaving multipliers, replacing the ending ''-ion'' with ''-iard''; for example, the next multiplier after ''million'' is ''milliard'', after ''billion'' it is ''billiard''. Hence, an ''n''-iard equals 10^6''n''+3.

Comparison

The relationship between the numeric values and the corresponding names in the two scales can be described as:

The relationship between the names and the corresponding numeric values in the two scales can be described as:

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.

History

Although this situation has been developing since the 1200s, the first recorded use of the terms ''short scale'' (french: échelle courte) and ''long scale'' (french: échelle longue) was by the French mathematician Geneviève Guitel in 1975.

The short scale was never widespread before its universal 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 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.

;Timeline
}) and ''short scale'' (french: échelle courte) to refer to the two numbering systems.
, -
, valign="top", 1994
, The Italian Government confirmed their official usage of the long scale.

As large numbers in natural sciences are usually represented by metric prefixes, scientific notation or otherwise, the most commonplace occurrence of large numbers represented by long or short scale terms is in finance. The following table includes some historic examples related to hyper-inflation and other financial incidents.

;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)
*
* (English-speaking)
*
* (English-speaking)
* (English-speaking) ''see'' Using both ''below''
*
*
*
*
* (formerly ''Swaziland'')
*
*
*
*
* (English-speaking)
*
*
*
*
* (English-speaking)
* (English-speaking)
* (English-speaking, ga, billiún, ''trilliún'')
*
*
*
* (English-speaking)
*
*
*
* (English-speaking)
* (English-speaking; ms, bilion billion, trillion)
* (English-speaking; mt, biljun,
*
* (English speaking) ''see'' Using both ''below''
*
*
*
*
* (English-speaking)
*
*
*
*
* (English-speaking)
* (English-speaking)
*
*
*
*
*
*
*
* (English speaking) ''see'' Using both ''below''
*
* (English-speaking)
*
*
* (English-speaking)
* (English-speaking)
*
*
*
*
*
* (English-speaking)
* (''see also'' Wales ''below'')
*
*
* (English speaking) ''see'' Using both ''below''
* (English-speaking)
* (English-speaking)

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 traditional long scale word similar to ''milliard'' for 10⁹. Some examples of short scale use, and the words used for 10⁹ and 10¹², are

* (Dari: or , , Pashto: , بیلیون , )
* (, )
* ( , )
* (, ()
* ( , )
* (Brazilian Portuguese: , )
* ( ms, bilion, )
* ( , )
* (Greek: , , tr, milyar, )
* ( or , )
* ( , )
* (, )
* (Hebrew: , )
* ( Kazakh: , )
* ( Kyrgyz: , )
* (, )
* (, )
* (Romanian: , )
* (formerly ''Burma'') ( Burmese: , ; , )

* (Afrikaans speaking) ''see'' Using both ''below''
* (Spanish speaking) ''see'' Using both ''below''
* ( , )
* (Afrikaans speaking) ''see'' Using both ''below''
* ( Tajik: , )
* (, )
* ( Turkmen: , ; Russian: , )
* ( , )
* ( Uzbek: , ; Russian: , )
* (, ) (In some contexts a paraphrase is needed to resolve ambiguity, as the lenitive of both and is the same: .)

Long scale users

The traditional 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 = , for example:

*
*
*
*
*
*

French-speaking

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

*
*
*
* (Canadian French) ''see'' Using both ''below''
*
*
*
*
*
*
*
*
*
* (Côte d'Ivoire)
*
*
*
*
* (province of Canada, Canadian French) ''see'' Using both ''below''
*
* (French portion of St. Martin Island)
*
*
*

German-speaking

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

*
*
*
*

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'' , for example:

*
*
*
*
*
*
*
*

Spanish-speaking

Most Spanish-language countries and regions use the long scale, for example:

*
*
*
*
*
*
*
*
*
*
* ()
* ()
* ( or )
* ( or )
* ( or )
*
* ()
* ''see'' Using both ''below''
* ( or typ. )
*
*

Other long scale

Some examples of long scale use, and the words used for 10⁹ and 10¹², are

* ( ca, miliard or typ. , )
* ( bs, milijarda, ; hr, milijarda, , sr, милијарда , )
* (, )
* (, )
* (, )
* Esperanto (, )
* (, )
* ( fi, miljardi, ; Swedish: , )
* (, )
* (, or )
* (, )
* (Persian: میلیارد , بیلیون , تریلیون )
* (, )
* (French: , ; German: , ; lb, milliard, )
* (French: , ; mg, miliara, )
* (English speaking) ''see'' Using both ''below''
* ( Montenegrin: , )
* (Afrikaans speaking) ''see'' Using both ''below''
* ( , )
* (Bokmål: , ; Nynorsk: , )
* (, )
* (, ). There are ambiguities for numbers above 10¹².
* (Italian: , )
* ( , )
* (English speaking) ''see'' Using both ''below''
* (, )
* (, )
* (Afrikaans speaking) ''see'' Using both ''below''
* (, )
* (French: , ; German: , ; Italian: , ; Romansh: , )
* (English speaking) ''see'' Using both ''below''
* (Italian: , )

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. Examples include:

Notes on current usage

Short scale

Long scale

Both long and short scale

Neither long nor short scale

Alternative approaches

* In written communications, the simplest solution for moderately large numbers is to write the full amount, for example 1,000,000,000,000 rather than 1 trillion (short scale) or 1 billion (long scale).
* Combinations of the unambiguous word ''million'', for example: 10⁹ = "one thousand million"; 10¹² = "one million million".

* Scientific notation (also known as standard form or exponential notation, for example 1, 1, 1, 1, etc.), or its engineering notation variant (for example 1, 10, 100, 1, etc.), or the computing variant E notation (for example 1e9, 1e10, 1e11, 1e12, etc.). This is the most common practice among scientists and mathematicians, and is both unambiguous and convenient.
* SI prefixes in combination with SI units, for example, giga for 10⁹ and tera for 10¹² can give gigawatt (=10⁹ W) and terawatt (=10¹² W). The International System of Units (SI) is independent of whichever scale is being used. Use with non-SI units (e.g. "giga-dollars", "megabucks") is possible. k€ and M€ is frequently encountered, although the official scheme places the Euro sign in front of the value.

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