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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation.

The decimal numeral system (also called base ten or occasionally denary) has ten as its base. It is the numerical base most widely used by modern civilizations.

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Decimal notation often refers to the base-10 positional notation such as the Hindu-Arabic numeral system, however it can also be used more generally to refer to non-positional systems such as Roman or Chinese numerals which are still based on powers of ten.

Decimal notation

Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals, and Chinese numerals, as well as the Hindu-Arabic numerals used by speakers of English. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals had symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese has symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 1044.

However, when people who use Hindu-Arabic numerals speak of decimal notation, they often mean not just decimal numeration, as above, but also decimal fractions, all conveyed as part of a positional system. Positional decimal systems include a zero and use symbols (called digits) for the ten values (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) to represent any number, no matter how large or how small. These digits are often used with a decimal separator which indicates the start of a fractional part, and with a symbol such as the plus sign + (for positive) or minus sign − (for negative) adjacent to the numeral to indicate its polarity.

Positional notation uses positions for each power of ten: units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier (power of ten) multiplied with that digit—each position has a value ten times that of the position to its right. There were two independent sources of positional decimal systems in ancient civilization: the Chinese counting rod system and the Hindu-Arabic numeral system, which descended from Brahmi numerals.

Ten is the number which is the count of fingers and thumbs on both hands (or toes on the feet). In many languages the word digit or its translation is also the anatomical term referring to fingers and toes. In English, decimal (decimus Lat.) means tenth, decimate means reduce by a tenth, and denary (denarius Lat.) means the unit of ten. The symbols for the digits in common use around the globe today are called Arabic numerals by Europeans and Indian numerals by Arabs, the two groups' terms both referring to the culture from which they learned the system. However, the symbols used in different areas are not identical; for instance, Western Arabic numerals (from which the European numerals are derived) differ from the forms used by other Arab cultures.

Decimal fractions

A decimal fraction is a fraction where the denominator is a power of ten.

Decimal fractions are commonly expressed without a denominator, the decimal separator being inserted into the numerator (with leading zeros added if needed), at the position from the right corresponding to the power of ten of the denominator. e.g., 8/10, 83/100, 83/1000, and 8/10000 are expressed as: 0.8, 0.83, 0.083, and 0.0008. In English-speaking and many Asian countries, a period (.) or raised period () is used as the decimal separator; in many other countries, a comma is used.

The integer part or integral part of a decimal number is the part to the left of the decimal separator (see also floor function). The part from the decimal separator to the right is the fractional part; if considered as a separate number, a zero is often written in front. Especially for negative numbers, we have to distinguish between the fractional part of the notation and the fractional part of the number itself, because the latter gets its own minus sign. It is usual for a decimal number whose absolute value is less than one to have a leading zero.

Trailing zeros after the decimal point are not necessary, although in science, engineering and statistics they can be retained to indicate a required precision or to show a level of confidence in the accuracy of the number: Whereas 0.080 and 0.08 are numerically equal, in engineering 0.080 suggests a measurement with an error of up to 1 part in two thousand (±0.0005), while 0.08 suggests a measurement with an error of up to 1 in two hundred (see Significant figures).

Other rational numbers

Any rational number which cannot be expressed as a finite decimal fraction has a unique infinite decimal expansion ending with recurring decimals.

The decimal fractions are those with denominator divisible by only 2 and or 5.

1/2 = 0.5
1/20 = 0.05
1/5 = 0.2
1/50 = 0.02

1/4 = 0.25
1/40 = 0.025
1/25 = 0.04

1/8 = 0.125
1/125= 0.008

1/10 = 0.1

1/3 = 0.333333… (with 3 repeating)
1/9 = 0.111111… (with 1 repeating)

100-1=99=9×11

1/11 = 0.090909… (with 09 or 90 repeating)

1000-1=9×111=27×37

1/27 = 0.037037037…
1/37 = 0.027027027…
1/111 = 0 .009009009…

also:
1/81= 0.012345679012… (with 012345679 repeating)

Other prime factors in the denominator will give longer recurring sequences, see for instance 7, 13.

That a rational number must have a finite or recurring decimal expansion can be seen to be a consequence of the long division algorithm, in that there are only q-1 possible nonzero remainders on division by q, so that the recurring pattern will have a period less than q. For instance to find 3/7 by long division:

0.4 2 8 5 7 1 4 ...
7 ) 3.0 0 0 0 0 0 0 0
2 8                         30/7 = 4 r 2
2 0
1 4                       20/7 = 2 r 6
6 0
5 6                     60/7 = 8 r 4
4 0
3 5                   40/7 = 5 r 5
5 0
4 9                 50/7 = 7 r 1
1 0
7               10/7 = 1 r 3
3 0
2 8             30/7 = 4 r 2
2 0
etc

The converse to this observation is that every recurring decimal represents a rational number p/q. This is a consequence of the fact the recurring part of a decimal representation is, in fact, an infinite geometric series which will sum to a rational number. For instance,
0.0123123123\cdots = \frac{123}{10000} \sum_{k=0}^\infty 0.001^k = \frac{123}{10000}\ \frac{1}{1-0.001} = \frac{123}{9990} = \frac{41}{3330}

Real numbers

Every real number has a (possibly infinite) decimal representation, i.e., it can be written as

x = \mathop{\rm sign}(x) \sum_{i\in\mathbb Z} a_i\,10^i
where
• sign() is the sign function,
• ai ∈ { 0,1,…,9 } for all iZ, are its decimal digits, equal to zero for all i greater than some number (that number being the common logarithm of |x|).

Such a sum converges as i decreases, even if there are infinitely many nonzero ai.

Rational numbers (e.g. p/q) with prime factors in the denominator other than 2 and 5 (when reduced to simplest terms) have a unique recurring decimal representation.

Non-uniqueness of decimal representation

Consider those rational numbers which have only the factors 2 and 5 in the denominator, i.e. which can be written as p/(2a5b). In this case there is a terminating decimal representation. For instance 1/1=1, 1/2=0.5, 3/5=0.6, 3/25=0.12 and 1306/1250=1.0448. Such numbers are the only real numbers which do not have a unique decimal representation, as they can also be written as a representation that has a recurring 9, for instance 1=0.99999…, 1/2=0.499999…, etc.

The number 0=0/1 is special in that it has no representation with recurring 9.

This leaves the irrational numbers. They also have unique infinite decimal representation, and can be characterised as the numbers whose decimal representations neither terminate nor recur.

So in general the decimal representation is unique, if one excludes representations that end in a recurring 9.

The same trichotomy holds for other base-n positional numeral systems:
• Terminating representation: rational where the denominator divides some nk
• Recurring representation: other rational
• Non-terminating, non-recurring representation: irrational
and a version of this even holds for irrational-base numeration systems, such as golden mean base representation.

History

History of decimal numbers

According to the Cambridge University scholars, decimal numbers originated in China . The earliest evidence of use dates back to the 14th century BC, although it was almost certainly in use long before that.

History of the Hindu-Arabic numeral system

The modern numeral system format, known as the Hindu-Arabic numeral system, originated in India by the 9th century. Its ideas were transmitted to China and the Islamic world during and after that time. It was notably introduced to the west through Muhammad ibn Mūsā al-Khwārizmī's On the Calculation with Hindu Numerals.

History of decimal fractions

While the mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggrenn notes that decimal fractions were used five centuries before him by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. Furthermore, according to the Cambridge University scholars, decimal fractions were first developed and used by the Chinese in the 1st century BC, and then spread to the Middle East and from there to Europe.

Natural languages

A straightforward decimal system, in which 11 is expressed as ten-one and 23 as two-ten-three, is found in Chinese languages, 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.

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.

Alternative bases

Some cultures do, or did, use other numeral systems, most notably

In addition, it has been suggested that many other cultures developed alternative numeral systems (although the extent is debated):

Computer hardware and software systems commonly use a binary representation, internally (although a few of the earliest computers, such as ENIAC , did use decimal representation internally). 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 the equivalent decimal values for presentation to and manipulation by humans.

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, especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic). Decimal arithmetic is used in computers so that decimal fractional results can be computed exactly, which is not possible using a binary fractional representation.This is often important for financial and other calculations.

References

1. The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
2. Fingers or Fists? (The Choice of Decimal or Binary Representation), Werner Buchholz, Communications of the ACM, Vol. 2 #12, pp3–11, ACM Press, December 1959.
3. Decimal Computation, Hermann Schmid, John Wiley & Sons 1974 (ISBN 047176180X); reprinted in 1983 by Robert E. Krieger Publishing Company (ISBN 0898743184)
4. Histoire universelle des chiffres, Georges Ifrah, Robert Laffont, 1994 (Also: The Universal History of Numbers: From prehistory to the invention of the computer, Georges Ifrah, ISBN 0471393401, John Wiley and Sons Inc., New York, 2000. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk)
5. Decimal Floating-Point: Algorism for Computers, Cowlishaw, M. F., Proceedings 16th IEEE Symposium on Computer Arithmetic, ISBN 0-7695-1894-X, pp104-111, IEEE Comp. Soc., June 2003
6. Science and Civilisation in China Vol 3 (See under heading Decimal System)
7. The Genius of China by Robert Temple (See under heading Decimal System)
8. Ifrah, page 346
9. Britannica Concise Encyclopedia (2007). algebra
10. Science and Civilisation in China Vol 3 (See under heading Decimal Fractions)
11. The Genius of China by Robert Temple
12. .
13. There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819. "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0-292-75531-7.
14. Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent."
15. Dawson, J. " Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
16. Decimal Arithmetic - FAQ