Decimal expansion in the context of "Real number"

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.