Definition:Number-Naming System/Short Scale
Definition
The short scale system is the number-naming system which uses:
- the word million for $10^6 = 1 \, 000 \, 000$
- the Latin-derived prefixes bi-, tri-, quadri-, quint-, etc. for each further multiple of $1 \, 000$, appended to the root -(i)llion, corresponding to the indices $2$, $3$, $4$, $5$, $\ldots$
Thus:
| one billion: | \(\ds = 1 \, 000 \, 000 \, 000 \) | \(\ds = 10^9 = 10^{2 \times 3 + 3} \) | |||||||
| one trillion | \(\ds = 1 \, 000 \, 000 \, 000 \, 000 \) | \(\ds = 10^{12} = 10^{3 \times 3 + 3} \) | |||||||
| one quadrillion | \(\ds = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \) | \(\ds = 10^{15} = 10^{4 \times 3 + 3} \) | |||||||
| one quintillion | \(\ds = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \) | \(\ds = 10^{18} = 10^{5 \times 3 + 3} \) |
Thus one $n$-illion equals $1000 \times 10^{3 n}$ or $10^{3 n + 3}$
Prefixes
The prefixes used in both the short scale and long scale number-naming systems are as follows:
| bi- | \(\ds 2 \) | ||||||||
| tri- | \(\ds 3 \) | ||||||||
| quadri- | \(\ds 4 \) | ||||||||
| quint- | \(\ds 5 \) | ||||||||
| sext- | \(\ds 6 \) | ||||||||
| sept- | \(\ds 7 \) | ||||||||
| oct- | \(\ds 8 \) | ||||||||
| non- | \(\ds 9 \) | ||||||||
| dec- | \(\ds 10 \) | ||||||||
| undec- | \(\ds 11 \) | ||||||||
| duodec- | \(\ds 12 \) | ||||||||
| tredec- | \(\ds 13 \) | ||||||||
| quattuordec- | \(\ds 14 \) | ||||||||
| quindec- | \(\ds 15 \) | ||||||||
| sexdec- | \(\ds 16 \) | ||||||||
| septendec- | \(\ds 17 \) | ||||||||
| octodec- | \(\ds 18 \) | ||||||||
| novemdec- | \(\ds 19 \) | ||||||||
| vigint- | \(\ds 20 \) | ||||||||
| cent- | \(\ds 100 \) |
Further prefixes can be created by combining them in various combinations to indicate addition or multiplication of indices, for example:
- one primo-vigesimo-centillion: $10^{366} = 10^{3 \times 121 + 3}$
- one milli-millillion: $10^{3 \, 000 \, 003} = 10^{3 \times \ 1, 000 \, 000 + 3}$
Also see
Historical Note
The short scale system of number representation was adopted by the French some time in the $17$th or $18$th century.
In the $19$th century it was also adopted from them by the United States.
Britain at this time continued with the long scale system.
It was not until $1974$ that the United Kingdom officially accepted the short scale system, although use of the long scale system can still be encountered.
The other nations of the world vary in their usage of long scale or short scale.