Decimal approximation of a real number

An approximate representation of a real number by a finite 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.