Decimal fraction
An arithmetical fraction with an integral power of 10 as its denominator. The following notation has been accepted for a decimal fraction:
![]() | (1) |
where are integers and if
then
is also non-zero.
Formula (1) expresses the number
![]() |
For example,
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The digits to the right of the decimal point are known as the decimal digits. If a decimal fraction contains no integer part, i.e. its absolute value is smaller than one, a zero is placed to the left of the decimal point.
An infinite decimal fraction is an infinite sequence of digits such as
![]() | (2) |
where is an integer, while each one of the numbers
,
assumes one of the values
. Any real number
is the sum of such a series, i.e.
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The partial sums of the series (2) are finite decimal fractions , which are approximate values of the number
smaller than
; the numbers
![]() |
are the respective approximate values larger than . If there exists integers
and
such that for all
the equalities
![]() |
are valid, the infinite decimal fraction is said to be periodic. Any finite decimal fraction may be regarded as an infinite periodic fraction with for
. If
is a rational number, the corresponding fraction (2) will be periodic. If
is irrational, the fraction (2) cannot be periodic.
Decimal fraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decimal_fraction&oldid=46595