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A decimal representation of a Non-negative Real Number ''r'' is an expression of the form

: r=\sum_{i=0}^\infty rac{a_i}{10^i}

where a_0 is a nonnegative integer, and a_1,
a_2, \dots are integers satisfying 0\leq a_i\leq 9; this is usually written more briefly as

:r=a_0.a_1 a_2 a_3\dots.

That is to say, a_0 is the integer part of r, not necessarily between 0 and 9, and a_1, a_2, a_3,\dots are the digits forming the fractional part of r.


FINITE DECIMAL APPROXIMATIONS


Any real number can be approximated to any desired degree of accuracy by Rational Number s 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\cdots a_n such that

:r_n\leq x < r_n+ rac{1}{10^n}.\,

Proof:

Let r_n = p / 10^n, where p = \lfloor 10^nx floor.
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.)


MULTIPLE DECIMAL REPRESENTATIONS


Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.00000... as by 0.99999... (where for the sake of brevity the infinite sequences of digits 0 and 9, respectively, have been replaced by "..."). Conventionally, the version with zero digits is preferred; by omitting the infinite sequence of zero digits, removing any final zero digits and a possible final decimal point, a normalized finite decimal representation is obtained.


FINITE DECIMAL REPRESENTATIONS


The decimal expansion of non-negative real number ''x'' will end in zeros (or in nines) if, and only if, ''x'' is a rational number whose denominator is of the form 2''n''5''m'', where ''m'' and ''n'' are non-negative integers.

Proof:

If the decimal expansion of ''x'' will end in zeros, or x=\sum_{i=0}^n rac{a_i}{10^i}=\sum_{i=0}^n10^{n-i}a_i/10^n
for some ''n'',
then the denominator of ''x'' is of the form 10''n'' = 2''n''5''n''.

Conversely, if the denominator of ''x'' is of the form 2''n''5''m'',
x= rac{p}{2^n5^m}= rac{2^m5^np}{2^{n+m}5^{n+m}}=
rac{2^m5^np}{10^{n+m}}
for some ''p''.
While ''x'' is of the form p/10k,
p=\sum_{i=0}^{n}10^ia_i for some ''n''.
By x=\sum_{i=0}^n10^{n-i}a_i/10^n=\sum_{i=0}^n rac{a_i}{10^i},
''x'' will end in zeros.


INFINITE DECIMAL REPRESENTATIONS


Every real number except zero has a unique infinite decimal representation, that is, one in which not all of its digits become zero after a while. For example, the number 5/4 can be represented as 1.24999….


SEE ALSO