Long and short scales

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). The long scale system includes additional names for interleaved values, typically replacing the word ending "-ion" by "-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.

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 10³ⁿ⁺³ in the short scale but 10⁶ⁿ 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 10⁶ⁿ⁺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.

The following tables shows 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 names of the two scales can be described as:

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 such as ten, hundred, thousand and million. For example: one thousand million and one million million.

Scientific notation (for example 1×10¹⁰), or its engineering notation variant (for example 10×10⁹), 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€.

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 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

Most English-language countries and regions use the short scale with 10⁹ being billion. For example:

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

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

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.

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

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

•  Canada (Canadian French) see Using both below
•  France

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

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

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

•  Argentina
•  Mexico (mil millones or millardo)
•  Spain (millardo or typ. mil millones)

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

•  Iran (Persian: میلیارد miliyard, بیلیون billion, تریلیون trillion)
•  Italy (miliardo, bilione)
•  Poland (miliard, bilion)
•  Switzerland (French: milliard, billion; German: Milliarde, Billion; Italian: miliardo, bilione; Romansh: milliarda, billiun)

Some countries use either the short or long scales, depending on the internal language being used or the context.

The following countries use naming systems for large numbers that are not etymologically related to the short and long scales:

The long and short scales are both present on most continents, with usage dependent on the language used. For example:

• BBC News article: "Is trillion the new billion?"
• Live-Counter.com: How to visualize large numbers:"