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# Positional numeral systems

(Redirected from Place-value systems of numeration)
Most positional numeral systems employ some number as a base ${\displaystyle b}$, usually an integer greater than 1 (though negative integers like –4 and imaginary numbers like ${\displaystyle 2i}$ can and have been used for this purpose). Then a digit ${\displaystyle d}$ placed at position 1 (two places left of the base point) means ${\displaystyle db}$, at position 2 it means ${\displaystyle db^{2}}$, at position 3 it means ${\displaystyle db^{3}}$, and so on so forth. Likewise, placed at position –1 (first place to the right of the base point) it means ${\displaystyle db^{-1}}$, at position –2 it means ${\displaystyle db^{-2}}$, at position –3 it means ${\displaystyle db^{-3}}$, etc. Position 0 of course works out to ${\displaystyle db^{0}=1d=d}$, and is therefore called "the one's place" regardless of what the base is.
If ${\displaystyle b}$ is a positive integer greater than 1, then base ${\displaystyle b}$ uses ${\displaystyle b}$ digits from 0 to ${\displaystyle b-1}$. For example, base 10 (decimal) uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Then, say, 500 means ${\displaystyle 5\times 10^{2}}$, 500.5 means ${\displaystyle 5\times 10^{2}+5\times 10^{-1}}$.