Definition:Number-Naming System

Definition

There are various number-naming systems for naming large numbers (that is: greater than $1 \, 000 \, 000$).

Short Scale

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 $10^{3 n + 3}$

Long Scale

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

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

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-		\(\ds 100 \)