Decimal approximation of a real number

An approximate representation of a real number by a finite decimal fraction. Any real number can be written in the form of an infinite decimal fraction

where is a non-negative integer, is one of the digits and . If one excludes infinite periodic decimal fractions with periods exclusively consisting of nines, one can write any real number in a unique manner as an infinite decimal fraction. Select such a notation for numbers and let ; then the finite decimal fraction

(or ) is said to be the lower (upper) decimal approximation of order of . If and , then the lower and the upper decimal approximations of order of are defined by

The following relations are valid for a decimal approximation of a real number

It follows that

and if , then , and upper approximations may be taken instead of lower.

Decimal approximations are used in practice for approximate calculations. The approximate values of the sums , differences , products , and quotients are given, respectively, by , ,

As a result of these operations on finite decimal fractions and , which have at most significant figures to the right of the decimal point, one again obtains decimal fractions with at most significant figures to the right of the decimal point. The sought-for result may be obtained to any desired degree of accuracy using these fractions.