# Dictionary Definition

decimal adj

1 numbered or proceeding by tens; based on ten;
"the decimal system" [syn: denary]

2 divided by tens or hundreds; "a decimal
fraction"; "decimal coinage"

### Noun

1 a proper fraction whose denominator is a power
of 10 [syn: decimal
fraction]

2 a number in the decimal system

# User Contributed Dictionary

## English

### Pronunciation

- /ˈdɛsɪml/, /"dEsIml/

### Noun

- In the context of "arithmetic|computing|uncountable": The number system that uses the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
- A number expressed in this system.
- A decimal place.

#### Synonyms

- (system): base 10, decimal system
- (number): decimal number
- (decimal place): decimal place, place of decimals

#### Translations

number system

- Dutch: decimaalstelsel , tientallig talstelsel
- Esperanto: dekuma sistemo
- Finnish: desimaalijärjestelmä
- German: Dezimalsystem, Zehnersystem
- Japanese: (じゅっしんすう, jusshinsū)
- Russian: десятичная система

number expressed in this system

- Dutch: decimaal
- Finnish: desimaaliluku, desimaali
- French: nombre décimal
- German: Dezimalzahl
- Japanese: (じゅっしんすう, jusshinsū)
- Russian: десятичное число

decimal place

See decimal
place

##### Translations to be checked

### Adjective

- In the context of "arithmetic|computing": Concerning numbers expressed in decimal or mathematical calculations performed using decimal.

#### Translations

concerning numbers expressed in decimal or
calculations using decimal

- Dutch: decimaal, decimale, tientallig, tientallige
- Finnish: desimaalinen
- French: décimal , décimale , décimaux m plural, décimales f plural
- Russian: десятичный

### See also

# Extensive Definition

The decimal (base ten or occasionally denary)
numeral
system has ten as its
base.
It is the most widely used numeral system, perhaps because humans
have ten digits over both hands.

## 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 or fewer symbols.The decimal system is a positional
numeral system; it has positions for units, tens, hundreds,
etc. The position of each digit conveys the multiplier (a power of
ten) to be used with that digit—each position has a value
ten times that of the position to its right.

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.

### Alternative notations

Some cultures do, or used to, use other numeral
systems, including pre-Columbian
Mesoamerican
cultures such as the Maya, who
use a vigesimal system
(using all twenty fingers and toes), some Nigerians who use
several duodecimal
(base 12) systems, the Babylonians, who
used sexagesimal
(base 60), and the Yuki, who
reportedly used octal
(base 8).

Computer hardware
and software systems commonly use a binary
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 binary-coded
decimal, but there are other decimal representations in use
(see IEEE
754r), especially in database implementations. 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 http://www2.hursley.ibm.com/decimal/decifaq.html.

### Decimal fractions

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
(.) is used as the decimal separator; in many other languages, 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 decimal fraction has a unique infinite 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)
- 1/4 = 0.25
- 1/5 = 0.2
- 1/6 = 0.166666… (with 6 repeating)
- 1/8 = 0.125
- 1/9 = 0.111111… (with 1 repeating)
- 1/10 = 0.1
- 1/11 = 0.090909… (with 09 repeating)
- 1/12 = 0.083333… (with 3 repeating)
- 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:

.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
(again) 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 \sum_^\infty 0.001^k = \frac\ \frac = \frac = \frac

### Real numbers

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

- x = \mathop(x) \sum_ a_i\,10^i

- sign() is the sign function,
- ai ∈ for all i ∈ Z, 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.

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
don't 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.

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.

Naturally, 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

## 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. http://www.chn.ir/english/eshownews.asp?no=1622 http://www.mpiwg-berlin.mpg.de/Preprints/P183.PDF
- 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 1080,000,000,000,000,000
- 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
- 1765 Johann Heinrich Lambert – discusses (with few if any proofs) patterns in decimal expansions of rational numbers and notes a connection with Fermat's little theorem in the case of prime denominators
- 1800 Karl Friedrich Gauss – uses number theory to systematically explain patterns in recurring decimal expansions of rational numbers (e.g., the relation between period length of the recurring part and the denominator, which fractions with the same denominator have recurring decimal parts which are shifts of each other, like 1/7 and 2/7) and also poses questions which remain open to this day (e.g., a special case of Artin's conjecture on primitive roots: is 10 a generator modulo p for infinitely many primes p?).
- 1925 Louis Charles Karpinski – The History of Arithmetic
- 1959 Werner Buchholz – Fingers or Fists? (The Choice of Decimal or Binary representation)
- 1974 Hermann Schmid – Decimal Computation
- 2000 Georges Ifrah – The Universal History of Numbers: From Prehistory to the Invention of the Computer
- 2003 Mike Cowlishaw – Decimal Floating-Point: Algorism for Computers.

## 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 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
numerals in a language may hinder children's counting
ability.

## See also

## References

## External links

decimal in Belarusian: Дзесятковая сістэма
злічэння

decimal in Belarusian (Tarashkevitsa):
Дзесятковая сыстэма зьлічэньня

decimal in Catalan: Nombre decimal

decimal in Czech: Desítková soustava

decimal in Danish: Decimal

decimal in German: Dezimalsystem

decimal in Modern Greek (1453-): Δεκαδικό
σύστημα

decimal in Esperanto: Dekuma sistemo

decimal in Spanish: Sistema decimal

decimal in Finnish: Kymmenjärjestelmä

decimal in French: Système décimal

decimal in Hebrew: השיטה העשרונית

decimal in Haitian: Sistèm desimal

decimal in Hungarian: Tízes számrendszer

decimal in Indonesian: Sistem bilangan
desimal

decimal in Icelandic: Tugakerfi

decimal in Italian: Sistema numerico
decimale

decimal in Japanese: 十進法

decimal in Korean: 십진법

decimal in Dutch: Decimaal

decimal in Norwegian Nynorsk:
Titalssystemet

decimal in Norwegian: Titallsystemet

decimal in Polish: Dziesiętny system
liczbowy

decimal in Portuguese: Sistema de numeração
decimal

decimal in Quechua: Chunkantin huchha
llika

decimal in Russian: Десятичная система
счисления

decimal in Simple English: Decimal

decimal in Slovak: Desiatková číselná
sústava

decimal in Slovenian: Desetiški številski
sistem

decimal in Serbian: Декадни систем

decimal in Swedish: Decimala talsystemet

decimal in Thai: เลขฐานสิบ

decimal in Ukrainian: Десяткова система
числення

decimal in Vietnamese: Hệ thập phân

decimal in Yiddish: דעצימאל

decimal in Chinese: 十进制

# Synonyms, Antonyms and Related Words

algorismic, algorithmic, aliquot, cardinal, decagonal, decahedral, decasyllabic, decuple, denary, differential, digital, even, exponential, figural, figurate, figurative, finite, fractional, imaginary, impair, impossible, infinite, integral, irrational, logarithmic, logometric, negative, numeral, numerary, numerative, numeric, odd, ordinal, pair, positive, possible, prime, radical, rational, real, reciprocal, submultiple, surd, tenfold, tenth, tithe, transcendental