View source for Decimal approximation of a real number

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

$$ 
a  =  \pm  \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} \dots ,
$$

where  $  \alpha _ {0} $
is a non-negative integer,  $  \alpha _ {n} $
is one of the digits  $  0 \dots 9 $
and  $  n = 1 , 2 , .  .  . $. 
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  $  a \geq  0 $; 
then the finite decimal fraction

$$ 
\underline{a _ {n} }  =  \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n}  $$

(or  $  \overline{ {a _ {n} }}\; = \alpha _ {0} . \alpha _ {1} \dots \alpha _ {n} + 10  ^ {-} n $) 
is said to be the lower (upper) decimal approximation of order  $  n $
of  $  a $. 
If  $  a < 0 $
and  $  a  ^  \prime  = - a $, 
then the lower  $  {\underline{a} {} _ {n} } $
and the upper  $  \overline{ {a _ {n} }}\; $
decimal approximations of order  $  n $
of  $  a $
are defined by

$$ 
\underline{a _ {n} }  =  - \overline{ {a _ {n}  ^  \prime  }}\; ,\ \ 
\overline{ {a _ {n} }}\;  =  - \underline{a _ {n}  ^  \prime  } .
$$

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

$$ 
\underline{a _ {n} }  \leq   \underline{a _ {n+} 1 }  \leq   a  \leq   {a _ {n+} 1 } bar  \leq   \overline{ {a _ {n} }}\; ,
$$

$$ 
\overline{ {a _ {n} }}\; - \underline{a _ {n} }  =  10  ^ {-} n .
$$

It follows that

$$ 
\lim\limits _ {n \rightarrow \infty } ( \underline{a _ {n} } \pm  \underline{b _ {n} } )  = \ 
a \pm  b,\ \ 
\lim\limits _ {n \rightarrow \infty }  \underline{a _ {n} } \underline{b _ {n} }  =  ab ,
$$

and if  $  b \neq 0 $, 
then  $  \lim\limits _ {n \rightarrow \infty }   {\underline{a} {} _ {n} } / {\underline{b} {} _ {n} } = a / b $, 
and upper approximations may be taken instead of lower.

Decimal approximations are used in practice for approximate calculations. The approximate values of the sums  $  a + b $, 
differences  $  a - b $, 
products  $  ab $, 
and quotients  $  a/b $
are given, respectively, by  $  {\underline{a} {} _ {n} } + {\underline{b} {} _ {n} } $, 
$  {\underline{a} {} _ {n} } - {\underline{b} {} _ {n} } $,

$$ 
\underline{( \underline{a _ {n} } \underline{b _ {n} } ) _ {n} } \ \ 
\textrm{ and } \  \underline{\left ( 
\frac{\underline{a _ {n} } }{\underline{b _ {n} } }
 \right ) _ {n} } .
$$

As a result of these operations on finite decimal fractions  $  {\underline{a} {} _ {n} } $
and  $  {\underline{b} {} _ {n} } $, 
which have at most  $  n $
significant figures to the right of the decimal point, one again obtains decimal fractions with at most  $  n $
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.