# Decimal: Wikis

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# Encyclopedia

This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation.
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Decimal (10)
1, 2, 3, 4, 5, 8, 12, 16, 20, 60 more…

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.[1][2]

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.

In some contexts, especially mathematics education, the term decimal can refer specifically to decimal fractions, described below. In such cases, a single decimal fraction is called a "decimal", and non-fractional numbers, even when written in base 10, are not considered "decimals".

## 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

The basis for modern decimal notation was first introduced by Simon Stevin.[3]

#### History of the Hindu-Arabic numeral system

The modern numeral system format, known as the Hindu-Arabic numeral system, originated in Indian mathematics[4] by the 9th century. Its ideas were transmitted to Chinese mathematics and Islamic mathematics during and after that time.[5] It was notably introduced to the west through Muhammad ibn Mūsā al-Khwārizmī's On the Calculation with Hindu Numerals.

#### History of decimal numbers

Non-positional decimal numerals were used in China,[6][7] possibly as early as the 14th century BC. The earliest positional decimal numbers, however, were the Indian numerals developed in India.

#### History of decimal fractions

According to Joseph Needham, 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.[6] The Chinese decimal fractions were non-positional, however.[6][7] The incorporation of decimal fractions into a positional decimal system, namely the Arabic numerals, occurred in the Islamic world. The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, though J. Lennart Berggrenn notes that positional decimal fractions were used five centuries before him by Arab mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century.[8]

#### 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.[9]

#### 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):

• Many or all of the Chumashan languages originally used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.[10]
• Many languages[11] use quinary number systems, including Gumatj, Nunggubuyu,[12] Kuurn Kopan Noot[13] and Saraveca. Of these, Gumatj is the only true "5-25" language known, in which 25 is the higher group of 5.
• Some Nigerians use base 12 systems[citation needed]
• The Huli language of Papua New Guinea is reported to have base 15 numerals.[14] Ngui means 15, ngui ki means 15×2 = 30, and ngui ngui means 15×15 = 225.
• Umbu-Ungu, also known as Kakoli, is reported to have base-24 numerals.[15][16] Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.
• Base 27 is used in two natural languages, the Telefol language and the Oksapmin language of Papua New Guinea.[citation needed]
• Ngiti is reported to have a base 32 numeral system with base 4 cycles.[17]

## Decimal computation

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). [18] 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,[19] especially in database implementations, but there are other decimal representations in use (such as in the new IEEE 754 Standard for Floating-Point Arithmetic). [20].

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.[21]

## References

1. ^ The History of Arithmetic, Louis Charles Karpinski, 200pp, Rand McNally & Company, 1925.
2. ^ 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)
3. ^ B. L. van der Waerden (1985). A History of Algebra. From al-Khwarizmi to Emmy Noether. Berlin: Springer-Verlag.
4. ^ Ifrah, page 346
5. ^ Britannica Concise Encyclopedia (2007). algebra
6. ^ a b c Joseph Needham (1959). "Decimal System". Science and Civilisation in China, Volume III, Mathematics and the Sciences of the Heavens and the Earth. Cambridge University Press.
7. ^ a b Robert K. G. Temple (1998). "Decimal System". The Genius of China: 3,000 Years of Science, Discovery, and Invention. Prion. ISBN 9781853752926.
8. ^ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 9780691114859.
9. ^ Azar, Beth (1999). "English words may hinder math skills development". American Psychology Association Monitor 30 (4).
10. ^ 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.
11. ^ Harald Hammarström, Rarities in Numeral Systems: "Bases 5, 10, and 20 are omnipresent."
12. ^ Harris, John (1982), Hargrave, Susanne, ed., "Facts and fallacies of aboriginal number systems", Work Papers of SIL-AAB Series B 8: 153–181
13. ^ Dawson, J. "Australian Aborigines: The Languages and Customs of Several Tribes of Aborigines in the Western District of Victoria (1881), p. xcviii.
14. ^ Cheetham, Brian (1978), "Counting and Number in Huli", Papua New Guinea Journal of Education 14: 16–35
15. ^ Gordon, Raymond G., Jr., ed. (2005), "Umbu-Ungu", Ethnologue: Languages of the World (15 ed.), retrieved 2008-03-16
16. ^ Bowers, Nancy; Lepi, Pundia (1975), "Kaugel Valley systems of reckoning", Journal of the Polynesian Society 84 (3): 309–324
17. ^ Hammarström, Harald (2006), "Rarities in Numeral Systems", Proceedings of Rara & Rarissima Conference
18. ^ 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.
19. ^ Decimal Computation, Hermann Schmid, John Wiley & Sons 1974 (ISBN 047176180X); reprinted in 1983 by Robert E. Krieger Publishing Company (ISBN 0898743184)
20. ^ 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
21. ^ Decimal Arithmetic - FAQ

# Simple English

The decimal is a way of managing numbers that has ten as a starting point, or base. It is sometimes called the base ten or denary numeral system. The word "decimal" is also used instead of the word "period" to point out the dot that is sometimes used separates the positions of the numbers in this system. Almost everyone uses this nowadays and prefers the convenience of it probably because it shows up most often in calculations in nature and has "one" as another starting point for the system. The number one is usually the easiest to work with in calculations.

## Decimal notation

Decimal notation is the writing of numbers in the base-ten numeral system, which uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a decimal separator which indicates the start of a fractional part, and with one of the sign symbols + (positive) or − (negative) in front of the numerals to indicate sign.

There are only two truly positional decimal systems in ancient civilization, the Chinese counting rods system and Hindu-Arabic numeric system, both required no more than ten symbols. Other numeric systems require more symbols.

### Other rational numbers

Any rational number can be expressed as a unique decimal expansion ending with recurring decimals.

Ten is the product of the first and third prime numbers, is one greater than the square of the second prime number, and is one less than the fifth prime number. This leads to plenty of simple decimal fractions:

1/2 = 0.5
1/3 = 0.333333… (with 3 repeating forever, also called recurring)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.166666… (with 6 recurring)
1/8 = 0.125
1/9 = 0.111111… (with 1 recurring)
1/10 = 0.1
1/11 = 0.090909… (with 09 recurring)
1/12 = 0.083333… (with 3 recurring)
1/81 = 0.012345679012… (with 012345679 recurring)

## History

There follows a chronological list of recorded decimal writers.

### Decimal writers

• c. 3500 - 2500 BC Elamites of Iran possibly used early forms of decimal system. [1] [2]
• c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc.) – see Ifrah, below
• c. 2600 BC Indus Valley Civilization, earliest known physical use of decimal fractions in ancient weight system: 1/20, 1/10, 1/5, 1/2. See Ancient Indus Valley weights and measures
• c. 1400 BC Chinese writers show familiarity with the concept: for example, 547 is written 'Five hundred plus four decades plus seven of days' in some manuscripts
• c. 1200 BC In ancient India, the Vedic text Yajur-Veda states the powers of 10, up to 1055
• c. 400 BC Pingala – develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system
• c. 250 BC Archimedes writes the Sand Reckoner, which takes decimal calculation up to $10^\left\{8 \times 10^\left\{16\right\}\right\}$
• c. 100–200 The Satkhandagama written in India – earliest use of decimal logarithms
• c. 476–550 Aryabhata – uses an alphabetic cipher system for numbers that used zero
• c. 598–670 Brahmagupta – explains the Hindu-Arabic numerals (modern number system) which uses decimal integers, negative integers, and zero
• c. 780–850 Muḥammad ibn Mūsā al-Ḵwārizmī – first to expound on algorism outside India
• c. 920–980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions.
• c. 1300–1500 The Kerala School in South India – decimal floating point numbers
• 1548/49–1620 Simon Stevin – author of De Thiende ('the tenth')
• 1561–1613 Bartholemaeus Pitiscus – (possibly) decimal point notation.
• 1550–1617 John Napier – use of decimal logarithms as a computational tool
• 1925 Louis Charles Karpinski – classic book The History of Arithmetic (Rand McNally & Company)
• 1959 Werner Buchholz Fingers or Fists? (The Choice of Decimal or Binary representation) (Communications of the ACM, Vol. 2 #12, pp3-11)
• 1974 Hermann Schmid Decimal Computation (ISBN 0-471-76180-X)
• 2000 Georges Ifrah The Universal History of Numbers: From Prehistory to the Invention of the Computer (ISBN 0-471-39340-1).

## 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 except Wu, 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 teens 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 numerals in a language may hinder children's counting ability (Azar 1999).