The decimal numeral system (also called the base-ten positional numeral system and denary 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.[1]
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 or).[2] Decimal may also refer specifically to the digits after the decimal separator, such as in " is the approximation of 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, where is an integer, and 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 separator. 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.,).[3] 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.
Many numeral systems of ancient civilizations 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 firstly the Egyptian numerals, then the 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;[4] the decimal separator is the dot "" in many countries (mostly English-speaking),[5] and a comma "" in other countries.
For representing a non-negative number, a decimal numeral consists of
amam-1\ldotsa0
amam-1\ldotsa0.b1b2\ldotsbn
For representing a negative number, a minus sign is placed before .
The numeral
amam-1\ldotsa0.b1b2\ldotsbn
| m+a | |
| a | |
| m-1 |
10m-1+ … +a0
| |||||
| 10 | + |
| b2 | + … + | |
| 102 |
| bn | |
| 10n |
When the integral part of a numeral is zero, it may occur, typically in computing, that the integer part is not written (for example,, instead of). 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.
Decimal fractions (sometimes called decimal numbers, especially in contexts involving explicit fractions) are the rational numbers that may be expressed as a fraction whose denominator is a power of ten.[6] For example, the decimal expressions
0.8,14.89,0.00079,1.618,3.14159
More generally, a decimal with digits after the separator (a point or comma) represents the fraction with denominator, whose numerator is the integer obtained by removing the separator.
It follows that a number is a decimal fraction if and only if it has a finite decimal representation.
Expressed as fully reduced fractions, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are
1=20 ⋅ 50,2=21 ⋅ 50,4=22 ⋅ 50,5=20 ⋅ 51,8=23 ⋅ 50,10=21 ⋅ 51,16=24 ⋅ 50,20=22 ⋅ 51,25=20 ⋅ 52,\ldots
Decimal numerals do not allow an exact representation for all real numbers. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates, being less than 10−5 off; so decimals are widely used in science, engineering and everyday life.
More precisely, for every real number and every positive integer, there are two decimals and with at most digits after the decimal mark such that and .
Numbers are very often obtained as the result of measurement. As measurements are subject to measurement uncertainty with a known upper bound, the result of a measurement is well-represented by a decimal with digits after the decimal mark, as soon as the absolute measurement error is bounded from above by . In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also significant figures).
See main article: Decimal representation.
For a real number and an integer, let denote the (finite) decimal expansion of the greatest number that is not greater than that has exactly digits after the decimal mark. Let denote the last digit of . It is straightforward to see that may be obtained by appending to the right of . This way one has
,
and the difference of and amounts to
\left\vert\left[x\right]n-\left[x\right]n-1
| -n | |
| \right\vert=d | |
| n ⋅ 10 |
<10-n+1
which is either 0, if, or gets arbitrarily small as tends to infinity. According to the definition of a limit, is the limit of when tends to infinity. This is written asor
,which is called an infinite decimal expansion of .
Conversely, for any integer and any sequence of digits the (infinite) expression is an infinite decimal expansion of a real number . This expansion is unique if neither all are equal to 9 nor all are equal to 0 for large enough (for all greater than some natural number).
If all for equal to 9 and, the limit of the sequence is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.:, by, and replacing all subsequent 9s by 0s (see 0.999...).
Any such decimal fraction, i.e.: for, may be converted to its equivalent infinite decimal expansion by replacing by and replacing all subsequent 0s by 9s (see 0.999...).
In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of, and the other containing only 9s after some place, which is obtained by defining as the greatest number that is less than, having exactly digits after the decimal mark.
See main article: Repeating decimal.
Long division allows computing the infinite decimal expansion of a rational number. If the rational number is a decimal fraction, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a repeating decimal. For example,
= 0.012345679012... (with the group 012345679 indefinitely repeating).
The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
| For example, if x is | 0.4156156156... | |
| then 10,000x is | 4156.156156156... | |
| and 10x is | 4.156156156... | |
| so 10,000x − 10x, i.e. 9,990x, is | 4152.000000000... | |
| and x is |
Most modern computer hardware and software systems commonly use a binary representation internally (although many early computers, such as the ENIAC or the IBM 650, used decimal representation internally).[7] For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems.
For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.)
Both computer hardware and software also use internal representations which are effectively decimal for storing decimal values and doing arithmetic. Often this arithmetic is done on data which are encoded using some variant of binary-coded decimal,[8] [9] especially in database implementations, but there are other decimal representations in use (including decimal floating point such as in newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic).[10]
Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of
10
Many ancient cultures calculated with numerals based on ten, perhaps because two human hands have ten fingers. Standardized weights used in the Indus Valley Civilisation were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the Mohenjo-daro ruler – was divided into ten equal parts.[13] [14] [15] Egyptian hieroglyphs, in evidence since around 3000 BCE, used a purely decimal system,[16] as did the Linear A script of the Minoans[17] [18] and the Linear B script (c. 1400–1200 BCE) of the Mycenaeans. The Únětice culture in central Europe (2300-1600 BC) used standardised weights and a decimal system in trade.[19] The number system of classical Greece also used powers of ten, including an intermediate base of 5, as did Roman numerals.[20] Notably, the polymath Archimedes (c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 108.[20] [21] Hittite hieroglyphs (since 15th century BCE) were also strictly decimal.[22]
The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000.[23] The world's earliest positional decimal system was the Chinese rod calculus.
Starting from the 2nd century BCE, some Chinese units for length were based on divisions into ten; by the 3rd century CE these metrological units were used to express decimal fractions of lengths, non-positionally.[24] Calculations with decimal fractions of lengths were performed using positional counting rods, as described in the 3rd–5th century CE Sunzi Suanjing. The 5th century CE mathematician Zu Chongzhi calculated a 7-digit approximation of . Qin Jiushao's book Mathematical Treatise in Nine Sections (1247) explicitly writes a decimal fraction representing a number rather than a measurement, using counting rods.[25] The number 0.96644 is denoted
Chinese: 寸
.
Historians of Chinese science have speculated that the idea of decimal fractions may have been transmitted from China to the Middle East.[26]
Al-Khwarizmi introduced fractions to Islamic countries in the early 9th century CE, written with a numerator above and denominator below, without a horizontal bar. This form of fraction remained in use for centuries.[26] [27]
Positional decimal fractions appear for the first time in a book by the Arab mathematician Abu'l-Hasan al-Uqlidisi written in the 10th century.[28] The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350 but did not develop any notation to represent them.[29] The Persian mathematician Jamshid al-Kashi used, and claimed to have discovered, decimal fractions in the 15th century.
A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. Stevin's influential booklet De Thiende ("the art of tenths") was first published in Dutch in 1585 and translated into French as La Disme.[30]
John Napier introduced using the period (.) to separate the integer part of a decimal number from the fractional part in his book on constructing tables of logarithms, published posthumously in 1620.[31]
A method of expressing every possible natural number using a set of ten symbols emerged in India.[32] Several Indian languages show a straightforward decimal system. Dravidian languages have numbers between 10 and 20 expressed in a regular pattern of addition to 10.
The Hungarian language also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty").
A straightforward decimal rank system with a word for each order (10 Chinese: 十, 100 Chinese: 百, 1000 Chinese: 千, 10,000 Chinese: 万), and in which 11 is expressed as ten-one and 23 as two-ten-three, and 89,345 is expressed as 8 (ten thousands) Chinese: 万 9 (thousand) Chinese: 千 3 (hundred) Chinese: 百 4 (tens) Chinese: 十 5 is found in Chinese, and in Vietnamese with a few irregularities. Japanese, Korean, and Thai have imported the Chinese decimal system. Many other languages with a decimal system have special words for the numbers between 10 and 20, and decades. For example, in English 11 is "eleven" not "ten-one" or "one-teen".
Incan languages such as Quechua and Aymara have an almost straightforward decimal system, in which 11 is expressed as ten with one and 23 as two-ten with three.
Some psychologists suggest irregularities of the English names of numerals may hinder children's counting ability.[33]
See main article: Positional notation. Some cultures do, or did, use other bases of numbers.