Difference between revisions of "Decimal approximation of a real number"
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| + | $#A+1 = 34 n = 0 | ||
| + | $#C+1 = 34 : ~/encyclopedia/old_files/data/D030/D.0300410 Decimal approximation of a real number | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
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| + | if TeX found to be correct. | ||
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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 | 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 | 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 | + | 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 | + | 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 | + | 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. | ||
Latest revision as of 17:32, 5 June 2020
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.
Decimal approximation of a real number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decimal_approximation_of_a_real_number&oldid=17107