= Decimal representation
{title2=$N=\sum_{j=0}^n d_j10^j$}
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, $205=2\cdot100+0\cdot10+5$, whereas $250=2\cdot100+5\cdot10+0$. 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 $1/10,1/100,1/1000,\ldots$. <Prealgebra 2e by Lynn Marecek and others/5.1 Decimals>[Prealgebra 2e, §5.1, Name Decimals, Tables 5.1–5.2 and Figure 5.2: whole-number and fractional place values]{ref} For example,
$$
205.375=205+\frac3{10}+\frac7{100}+\frac5{1000}=205+\frac38.
$$
The matching <binary representation> is $11001101.011_2$, since its fractional part is $0\cdot2^{-1}+1\cdot2^{-2}+1\cdot2^{-3}=3/8$. 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: $2050=10\cdot205$. 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 $1/3=0.333\ldots$. A <terminating decimal> has a zero tail by convention. Infinite decimal strings have a further ambiguity that finite canonical integer strings do not: $0.999\ldots=1$, because the <geometric series> $\sum_{j=1}^{\infty}9\cdot10^{-j}$ sums to one. An infinite decimal tail denotes a <limit>, not a sufficiently long rounded approximation.
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