Decimal

The decimal numeral system is positional notation in base ten.

A decimal representation is positional notation with radix ten and numeral digits zero through nine. For an integer, the nonnegative part is a finite decimal expansion; a negative integer has a separate minus sign. For example, $205=2\cdot 100+0\cdot 10+5$, whereas $250=2\cdot 100+5\cdot 10+0$. The numeral digits alone do not determine the value: their positions do.

Fractional positions extend the same place value rule to the right of the decimal point, with successive weights $1/10,1/100,1/1000,\dots$. ^[ref] For example,The matching binary representation is $11001101.011_2$, since its fractional part is $0\cdot 2^{-1}+1\cdot 2^{-2}+1\cdot 2^{-3}=3/8$. These representations terminate in both bases; finiteness in one radix does not imply finiteness in another.

Leading zeros in the integer part and trailing zeros in the fractional part leave the value unchanged: $00205.3750=205.375$. A zero appended to the right of an integer string has a different effect: $2050=10\cdot 205$. Keeping the decimal point fixed prevents confusing these two operations.

Every integer has a finite decimal expansion, but a noninteger rational number may need infinitely many fractional numeral digits, as in $1/3=0.333\dots$. A terminating decimal has a zero tail by convention. Infinite decimal strings have a further ambiguity that finite canonical integer strings do not: $0.999\dots =1$, because the geometric series ${\sum }_{j=1}^{\infty }9\cdot 10^{-j}$ sums to one. An infinite decimal tail denotes a limit, not a sufficiently long rounded approximation.

A decimal representation with only finitely many nonzero fractional digits. A reduced fraction has this property exactly when its denominator has no prime factors other than two and five. The zero tail is distinguished from the genuine repeating block measured by decimal period from multiplicative order.