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

where k is a nonnegative integer and are integers in the range 0, ..., 9, which are called the digits of the representation.

This expression represents the infinite sum

The sequence of the —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.[1]

The integer , denoted by a0 in the remainder of this article, is called the integer part of r, and the sequence of the represents the number

which is called the fractional part of r.

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