A decimal is a numeral in the decimal number system, which is the place-value system most commonly used for representing real numbers. In this system numbers are expressed as sequences of Arabic numerals 0 to 9, in which each successive digit to the left or right of the decimal point indicates a multiple of successive powers of 10; for example, the number represented by the decimal 123.45 is the sum
\(1\times10^2+2\times10^1+3\times10^0+4\times10^{-1}+5\times10^{-2}\)
\(=1\times100+2\times10+3\times1+4\times\frac1{10}+5\times\frac1{100}\)
The digits after the decimal point can be terminating or non-terminating. A terminating decimal is a decimal that contains a finite number of digits, as shown in the example above. A decimal is non-terminating, if it has an infinite number of digits after the decimal point. Non-terminating decimals may be recurring, that is, contain a pattern of digits that repeats indefinitely after a certain number of places. For example, the fraction \(\frac13\), written in the decimal number system, results in an infinite sequence of 3s after the decimal point. This can be represented by a dot above the recurring decimal.
\(\frac13=0.333333\dots=0.\dot3\)
Similarly, the fraction \(\frac17\) results in a recurring group of digits, which is represented by a bar above the whole group of repeating digits
\(\frac17=0.142857142857142857\dots=0.\overline{142857}\)
Non-terminating decimals may also be non-recurring, that is the digits after the decimal point never repeat in a pattern. This is the case for irrational number, such as pi, e, or \(\sqrt[{}]2\). For example,
\(\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\)
Irrational numbers can only be approximated in the decimal number system.