Decimal

The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation. The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation. A decimal numeral, or just decimal, or casually decimal number, refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified for containing a decimal separator (for example the '.' in 10.00 or 3.14159). 'Decimal' may also refer specifically to the digits after the decimal separator, such as in '3.14 is the approximation of π to two decimals'. The numbers that may be represented in the decimal system are the decimal fractions, that is the fractions of the form a/10n, where a is an integer, and n is a non-negative integer. The decimal system has been extended to infinite decimals, for representing any real number, by using an infinite sequence of digits after the decimal separator (see Decimal representation). In this context, the decimal numerals with a finite number of non–zero places after the decimal separator are sometimes called terminating decimals. A repeating decimal is an infinite decimal that after some place repeats indefinitely the same sequence of digits (for example 5.123144144144144... = 5.123144). An infinite decimal represents a rational number if and only if it is a repeating decimal or has a finite number of nonzero digits. Many numeral systems of ancient civilisations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, and Chinese numerals. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the Hindu–Arabic numeral system for representing integers. This system has been extended to represent some non-integer numbers, called decimal fractions or decimal numbers for forming the decimal numeral system. For writing numbers, the decimal system uses ten decimal digits, a decimal mark, and, for negative numbers, a minus sign '−'. The decimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; the decimal separator is the dot '.' in many countries (including all English speaking ones), but may be a comma ',' in other countries (mainly in continental Europe). For representing a non-negative number, a decimal consists of It is generally assumed that, if m > 0, the first digit am is not zero, but, in some circumstances, it may be useful to have one or more 0's on the left. This does not change the value represented by the decimal. For example, 3.14 = 03.14 = 003.14. Similarly, if bn =0, it may be removed, and conversely, trailing zeros may be added without changing the represented number: for example, 15 = 15.0 = 15.00 and 5.2 = 5.20 = 5.200 . Sometimes the extra zeros are used for indicating the accuracy of a measurement. For example, 15.00 m may indicate that the measurement error is less than one centimeter (0.01 m), while 15 m may mean that the length is roughly fifteen meters, and that the error may exceed 10 cm. For representing a negative number, a minus sign is placed before am.