Positional numeral systems

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Positional numeral systems, also called place-value numeral systems or place-value systems of numeration, are systems in which the placement of a digit in connection to its intrinsic value determines its actual meaning in a numeral string (whereas in most ancient numeral systems, each digit had a value that did not change regardless of its placement in the numeral string).

Most positional numeral systems employ some number as a base $b$, usually an integer greater than 1 (though negative integers like –4 and imaginary numbers like $2i$ can and have been used for this purpose). Then a digit $d$ placed at position 1 (two places left of the base point) means $db$, at position 2 it means $db^{2}$, at position 3 it means $db^{3}$, and so on so forth. Likewise, placed at position –1 (first place to the right of the base point) it means $db^{-1}$, at position –2 it means $db^{-2}$, at position –3 it means $db^{-3}$, etc. Position 0 of course works out to $db^{0}=1d=d$, and is therefore called "the one's place" regardless of what the base is.

If $b$ is a positive integer greater than 1, then base $b$ uses $b$ digits from 0 to $b-1$. For example, base 10 (decimal) uses 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Then, say, 500 means $5\times 10^{2}$, 500.5 means $5\times 10^{2}+5\times 10^{-1}$.