Decimal fraction

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.