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.^[1][2]

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 the Indian numbering system and the Chinese, Japanese, or Korean numerals.^[1][2]

Much of the remainder of the world adopted either the short scale or the long scale for everyday counting powers of ten. Countries with the usage of the long scale include most countries in continental Europe and most that are French-speaking, German-speaking and Spanish-speaking.^[3] Usage of the short scale is found in Brazil and in most English-speaking and Arabic-speaking countries.

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, on the short scale, "one billion" means one thousand million (1,000,000,000), whereas in the long scale, it means one million million (1,000,000,000,000). The long scale system employs additional terms for interleaved values, typically replacing the word ending -ion by -iard.

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, associated with physical quantities.

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. Both systems use the same names for certain multipliers, including those for all numbers smaller than 10⁹. 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.

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.^[5]

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.^{[6][7][8][9][10]}

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.^[1][2]

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.^[5] 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,^[4][11] whereas the United States used the short scale,^[11] 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,^[12] and it is used for all official purposes.^{[13][14][15][16][17][18]} 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.^[5]

• 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.^[5] Some of these countries, but not all, introduced new words billiard, trilliard, etc. as intermediate terms.^{[6][7][8][9][10]}
• 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.^[5]
• 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

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

• 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".^[66]
• Scientific notation (also known as standard form or exponential notation, for example 1×10⁹, 1×10¹⁰, 1×10¹¹, 1×10¹², etc.), or its engineering notation variant (for example 1×10⁹, 10×10⁹, 100×10⁹, 1×10¹², 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.^[34] Use with non-SI units (e.g. "giga-dollars", "megabucks") is possible. k€ and M€ is frequently encountered.