Decimal

The decimal numeral system (also called the base-ten positional numeral system and denary /ˈdiːnəri/^[1] or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (decimal fractions) of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.^[2]

For other uses, see Decimal (disambiguation).

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in $25.9703$ or $3,1415$ ).^[3] 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". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value.

The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form $a /10 n$ , where $a$ is an integer, and $n$ is a non-negative integer. Decimal fractions also result from the addition of an integer and a fractional part; the resulting sum sometimes is called a fractional number.

Decimals are commonly used to approximate real numbers. By increasing the number of digits after the decimal separator, one can make the approximation errors as small as one wants, when one has a method for computing the new digits.

Originally and in most uses, a decimal has only a finite number of digits after the decimal seperator. However, 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 usual decimals, with a finite number of non-zero digits 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 (e.g., $5.123144144144144... = 5.123 144$ ).^[4] An infinite decimal represents a rational number, the quotient of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer:
$a_{m}a_{m-1}\ldots a_{0}$

or a decimal mark separating two sequences of digits (such as "20.70828")

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;^[5] the decimal separator is the dot " $.$ " in many countries (mostly English-speaking),^[6] and a comma " $,$ " in other countries.^[3]

For representing a non-negative number, a decimal numeral consists of

If $m > 0$ , that is, if the first sequence contains at least two digits, it is generally assumed that the first digit $a m$ is not zero. 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 the final digit on the right of the decimal mark is zero—that is, if $b n = 0$ —it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; ^{[note 1]} for example, $15 = 15.0 = 15.00$ and $5.2 = 5.20 = 5.200$ .

For representing a negative number, a minus sign is placed before $a m$ .

The numeral $a_{m}a_{m-1}\ldots a_{0}.b_{1}b_{2}\ldots b_{n}$ represents the number

The integer part or integral part of a decimal numeral is the integer written to the left of the decimal separator (see also truncation). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the fractional part, which equals the difference between the numeral and its integer part.

When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example, $.1234$ , instead of $0.1234$ ). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation.

In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a positional numeral system.

Mesoamerican cultures such as the Maya used a base-20 system (perhaps based on using all twenty fingers and toes).

Pre-Columbian

The language in California and the Pamean languages^[37] in Mexico have octal (base-8) systems because the speakers count using the spaces between their fingers rather than the fingers themselves.^[38]

Yuki

The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were.^[40] Where this counting system is known, it is based on the "long hundred" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon's Introduction to Old Norse Archived 2016-04-15 at the Wayback Machine p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240. Goodare details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'.^[41]^[42]

[39]

Many or all of the originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.^[43]

Chumashan languages

Many languages use quinary (base-5) number systems, including Gumatj, Nunggubuyu,^[45] Kuurn Kopan Noot^[46] and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.

[44]

Some use duodecimal systems.^[47] So did some small communities in India and Nepal, as indicated by their languages.^[48]

Nigerians

The of Papua New Guinea is reported to have base-15 numbers.^[49] Ngui means 15, ngui ki means 15 × 2 = 30, and ngui ngui means 15 × 15 = 225.

Huli language

also known as Kakoli, is reported to have base-24 numbers.^[50] Tokapu means 24, tokapu talu means 24 × 2 = 48, and tokapu tokapu means 24 × 24 = 576.

Umbu-Ungu

is reported to have a base-32 number system with base-4 cycles.^[44]

Ngiti

The of Papua New Guinea is reported to have base-6 numerals.^[51] Mer means 6, mer an thef means 6 × 2 = 12, nif means 36, and nif thef means 36×2 = 72.