Decimal representation

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A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: $${\displaystyle r=b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots }$$ Here . is the decimal separator, k is a nonnegative integer, and ${\displaystyle b_{0},\cdots ,b_{k},a_{1},a_{2},\cdots }$ are digits, which are symbols representing integers in the range 0, ..., 9.

Commonly, ${\displaystyle b_{k}\neq 0}$ if ${\displaystyle k\geq 1.}$ The sequence of the ${\displaystyle a_{i}}$—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all ${\displaystyle a_{i}}$ are 0, the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number.

The decimal representation represents the infinite sum: $${\displaystyle r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.}$$

Every nonnegative real number has at least one such representation; it has two such representations (with ${\displaystyle b_{k}\neq 0}$ if ${\displaystyle k>0}$) if and only if one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of 9 are sometimes excluded.^[1]