Decimal representation explained

A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: $$r\; =\; b\_k\; b\_\backslash ldots\; b\_0.a\_1a\_2\backslash ldots$$Here is the decimal separator, is a nonnegative integer, and

b _{0,} \ldots, b _{k,} a _{1,} a _{2,\ldots} are digits, which are symbols representing integers in the range 0, ..., 9.

Commonly,

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

The decimal representation represents the infinite sum:$$r=\backslash sum\_^k\; b\_i\; 10^i\; +\; \backslash sum\_^\backslash infty\; \backslash frac.$$

Every nonnegative real number has at least one such representation; it has two such representations (with

b _{k ≠} 0 if k> 0 ) if and only if one has a trailing infinite sequence of, and the other has a trailing infinite sequence of . For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of are sometimes excluded.

Integer and fractional parts

The natural number $\backslash sum\_^k\; b\_i\; 10^i$, is called the integer part of, and is denoted by in the remainder of this article. The sequence of the

a _{i} represents the number $$0.a\_1a\_2\backslash ldots\; =\; \backslash sum\_^\backslash infty\; \backslash frac,$$which belongs to the interval [ 0 , 1 ), and is called the fractional part of (except when all a _{i} are equal to).

Finite decimal approximations

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

Assume

x \geq 0 . Then for every integer n\geq 1 there is a finite decimal r _{n=a} _{0.a} _{1a} _{2 …} a _{n} such that:

Proof:Let

r _{n} = style (p)/(10 ^{n}) , where p = \lfloor 10 ^{n} x\rfloor .Then p \leq 10 ^{nx} < p +1 , and the result follows from dividing all sides by 10 ^{n} .(The fact that r _{n} has a finite decimal representation is easily established.)

Non-uniqueness of decimal representation and notational conventions

See main article: 0.999.... Some real numbers

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 x , an infinite sequence of trailing 0's appearing after the decimal point is omitted, along with the decimal point itself if x is an integer.

Certain procedures for constructing the decimal expansion of

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