Decimal representation

A decimal representation of a non-negative real number r is an expression in the form of a sequence of decimal digits traditionally written with a single separator

$r=b_{k}b_{k-1}\ldots b_{0}.a_{1}a_{2}\ldots \,,$ where k is a nonnegative integer and $b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots$ are integers in the range 0, ..., 9, which are called the digits of the representation.

This expression represents the infinite sum

$r=\sum _{i=0}^{k}b_{i}10^{i}+\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}}.$ The sequence of the $a_{i}$ —the digits after the dot—may be finite, in which case the lacking digits are assumed to be 0.

Every nonnegative real number has at least one such representation; it has two such representations if and only one has a trailing infinite sequence of zeros, and the other has a trailing infinite sequence of nines. Some authors forbid decimal representations with a trailing infinite sequence of nines because this allows a one-to-one correspondence between nonnegative real numbers and decimal representations.

The integer $\sum _{i=0}^{k}b_{i}10^{i}$ , denoted by a0 in the remainder of this article, is called the integer part of r, and the sequence of the $a_{i}$ represents the number

$0.a_{1}a_{2}\ldots =\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}},$ which is called the fractional part of r.