\begin{definition}[Definition:Number-Naming System/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:
{{begin-axiom}}
{{axiom | lc= one billion:
        | m = = 1 \, 000 \, 000 \, 000
        | mm= = 10^9 = 10^{2 \times 3 + 3}
}}
{{axiom | lc= one trillion
        | m = = 1 \, 000 \, 000 \, 000 \, 000
        | mm= = 10^{12} = 10^{3 \times 3 + 3}
}}
{{axiom | lc= one quadrillion
        | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000
        | mm= = 10^{15} = 10^{4 \times 3 + 3}
}}
{{axiom | lc= one quintillion
        | m = = 1 \, 000 \, 000 \, 000 \, 000 \, 000 \, 000
        | mm= = 10^{18} = 10^{5 \times 3 + 3}
}}
{{end-axiom}}
Thus '''one $n$-illion''' equals $1000 \times 10^{3 n}$ or $10^{3 n + 3}$
\end{definition}