# Decimal fraction

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An arithmetical fraction with an integral power of 10 as its denominator. The following notation has been accepted for a decimal fraction:

$$\tag{1 } a _ {k} \dots a _ {0} . b _ {1} \dots b _ {l} ,$$

where $0 \leq a _ {i} , b _ {j} < 10$ are integers and if $k \neq 0$ then $a _ {k}$ is also non-zero.

Formula (1) expresses the number

$$a _ {k} 10 ^ {k} + \dots + a _ {1} 10 + a _ {0} + \frac{b _ {1} }{10 } + \dots + \frac{b _ {l} }{10 ^ {l} } .$$

For example,

$$\frac{3}{10} = 0.3 ; \ \frac{3524}{100} = 35.24 ; \ \ \frac{15}{1000} = 0.015 .$$

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

$$\tag{2 } a _ {0} . b _ {1} b _ {2} \dots ,$$

where $a _ {0}$ is an integer, while each one of the numbers $b _ {j}$, $j = 1 , 2 \dots$ assumes one of the values $0 \dots 9$. Any real number $\alpha$ is the sum of such a series, i.e.

$$\alpha = a _ {0} + \sum _ {k = 1 } ^ \infty \frac{b _ {k} }{10 ^ {k} } .$$

The partial sums of the series (2) are finite decimal fractions $a _ {0} . b _ {1} \dots b _ {n}$, which are approximate values of the number $\alpha$ smaller than $\alpha$; the numbers

$$a _ {0} . b _ {1} \dots b _ {n} + \frac{1}{10 ^ {n} }$$

are the respective approximate values larger than $\alpha$. If there exists integers $n$ and $m$ such that for all $i > n$ the equalities

$$b _ {i} = b _ {i+} m ,$$

are valid, the infinite decimal fraction is said to be periodic. Any finite decimal fraction may be regarded as an infinite periodic fraction with $b _ {i} = 0$ for $i> n$. If $\alpha$ is a rational number, the corresponding fraction (2) will be periodic. If $\alpha$ is irrational, the fraction (2) cannot be periodic.

How to Cite This Entry:
Decimal fraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Decimal_fraction&oldid=46595
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article