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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:

$${\displaystyle r=b_{k}b_{k-1}\ldots b_{0}.a_{1}a_{2}\ldots }$$

${\displaystyle b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots }$

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]

Integer and fractional parts

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

$${\displaystyle 0.a_{1}a_{2}\ldots =\sum _{i=1}^{\infty }{\frac {a_{i}}{10^{i}}},}$$

${\displaystyle 0,1),}$

${\displaystyle a_{i}}$

Finite decimal approximationsedit

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.

Assume ${\displaystyle x\geq 0}$. Then for every integer ${\displaystyle n\geq 1}$ there is a finite decimal ${\displaystyle r_{n}=a_{0}.a_{1}a_{2}\cdots a_{n}}$ such that:

$${\displaystyle r_{n}\leq x<r_{n}+{\frac {1}{10^{n}}}.}$$

Proof: Let ${\displaystyle r_{n}=\textstyle {\frac {p}{10^{n}}}}$, where ${\displaystyle p=\lfloor 10^{n}x\rfloor }$. Then ${\displaystyle p\leq 10^{n}x<p+1}$, and the result follows from dividing all sides by ${\displaystyle 10^{n}}$. (The fact that ${\displaystyle r_{n}}$ has a finite decimal representation is easily established.)

Non-uniqueness of decimal representation and notational conventionsedit

Some real numbers ${\displaystyle x}$ have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by 0.999... (where the infinite sequences of trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred. Moreover, in the standard decimal representation of ${\displaystyle x}$, an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if ${\displaystyle x}$ is an integer.

Certain procedures for constructing the decimal expansion of ${\displaystyle x}$ will avoid the problem of trailing 9's. For instance, the following algorithmic procedure will give the standard decimal representation: Given ${\displaystyle x\geq 0}$, we first define ${\displaystyle a_{0}}$ (the integer part of ${\displaystyle x}$) to be the largest integer such that ${\displaystyle a_{0}\leq x}$ (i.e., ${\displaystyle a_{0}=\lfloor x\rfloor }$). If ${\displaystyle x=a_{0}}$ the procedure terminates. Otherwise, for ${\textstyle (a_{i})_{i=0}^{k-1}}$ already found, we define ${\displaystyle a_{k}}$ inductively to be the largest integer such that: