# Glossary

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.