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A decimal expansion writes a nonnegative real number using decimal digits at successive powers of ten:Nonnegative indices describe the integer part, and negative indices describe the fractional part. For a negative real number, add a minus sign to the expansion of its positive magnitude. A terminating decimal has a zero tail; a general decimal expansion can require infinitely many fractional digits.

An infinite tail of nines can duplicate a terminating expansion. For example, the geometric series gives $0.999\dots ={\sum }_{k\ge 1}910^{-k}=1$. The usual uniqueness convention excludes a final infinite tail of nines, and redundant leading zeros are ignored.

For a nonnegative integer $M$, only finitely many nonnegative powers are needed: $M={\sum }_{j=0}^m{d}_{j}10^j$. This makes divisibility tests into polynomial calculations modulo the proposed divisor. Since $10\equiv -1(\mathrm{mod}11)$,Thus $M$ is divisible by $11$ exactly when its alternating digit sum is divisible by $11$. That sum need not be literally zero: for $209$, it is $9-0+2=11$, and $209=19\times 11$.