Definition:Number-Naming System/Long Scale

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Definition

The long 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 $1 \, 000 \, 000$, appended to the root -(i)llion, corresponding to the indices $2$, $3$, $4$, $5$, $\ldots$


Thus:

one billion:    \(\displaystyle = 1 \, 000 \, 000 \, 000 \, 000 \)    \(\displaystyle = 10^{12} = 10^{2 \times 6} \)             
one trillion    \(\displaystyle = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \)    \(\displaystyle = 10^{18} = 10^{3 \times 6} \)             
one quadrillion    \(\displaystyle = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \)    \(\displaystyle = 10^{24} = 10^{4 \times 6} \)             
one quintillion    \(\displaystyle = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000 \)    \(\displaystyle = 10^{30} = 10^{5 \times 6} \)             

Thus one $n$-illion equals $10^{6 n}$.


Additional terms are occasionally found to fill some of the gaps, but these are rare nowadays:

one milliard:    \(\displaystyle = 1 \, 000 \, 000 \, 000 \)    \(\displaystyle = 10^9 \)             
one billiard    \(\displaystyle = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \)    \(\displaystyle = 10^{15} \)             


Prefixes

The prefixes used in both the short scale and long scale number-naming systems are as follows:

bi-    \(\displaystyle 2 \)             
tri-    \(\displaystyle 3 \)             
quadri-    \(\displaystyle 4 \)             
quint-    \(\displaystyle 5 \)             
sext-    \(\displaystyle 6 \)             
sept-    \(\displaystyle 7 \)             
oct-    \(\displaystyle 8 \)             
non-    \(\displaystyle 9 \)             
dec-    \(\displaystyle 10 \)             
undec-    \(\displaystyle 11 \)             
duodec-    \(\displaystyle 12 \)             
tredec-    \(\displaystyle 13 \)             
quattuordec-    \(\displaystyle 14 \)             
quindec-    \(\displaystyle 15 \)             
sexdec-    \(\displaystyle 16 \)             
septendec-    \(\displaystyle 17 \)             
octodec-    \(\displaystyle 18 \)             
novemdec-    \(\displaystyle 19 \)             
vigint-    \(\displaystyle 20 \)             
cent-    \(\displaystyle 100 \)             


Also see


Historical Note

The long scale system of number representation was standard usage dates from around the $16$th century, when the words billion, trillion, and so on, were introduced by the French.

The words entered the English language in the $17$th century, and were adopted with their long scale interpretations.

Some time later, the French usage changed to the short scale system.

In the $19$th century, the United States adopted the short scale system from the French, while Britain 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.