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Encyclopedia > Decimal
Numeral systems by culture
Hindu-Arabic numerals
Western Arabic
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East Asian numerals
Chinese
Counting rods
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Armenian
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List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
3, 9, 12, 24, 30, 36, 60, more…
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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. For other uses, see Number (disambiguation). ... This article is about different methods of expressing numbers with symbols. ... In mathematics and computer science, a numerical digit is a symbol, e. ... The decimal separator is a symbol used to mark the boundary between the integral and the fractional parts of a decimal numeral. ... The counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ...

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. Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ...

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. This article is about the number 10. ... Digit may refer to: A finger or a toe Numerical digit, as used in mathematics or computer science Digit (unit), an ancient meterological unit Digit (magazine), an Indian information technology magazine This is a disambiguation page: a list of articles associated with the same title. ... For other uses, see Latin (disambiguation). ... Measurement is the determination of the size or magnitude of something. ... World globe A Baroque era celestial globe A globe is a three-dimensional scale model of a spheroid celestial body such as a planet, star or moon, in particular Earth, or, alternatively, a spherical representation of the sky with the stars (but without the Sun, Moon, or planets, because their... Numerals sans-serif Arabic numerals, known formally as Hindu-Arabic numerals, and also as Indian numerals, Hindu numerals, Western Arabic numerals, European numerals, or Western numerals, are the most common symbolic representation of numbers around the world. ... India has produced many numeral systems. ...

### 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). The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences on the Americas continent. ... This article is about the culture area. ... Mayan numerals. ... The vigesimal or base-20 numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). ... Toes on foot. ... The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. ... Babylonia was a state in southern Mesopotamia, in modern Iraq, combining the territories of Sumer and Akkad. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... [[Image:YukiTribe. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ...

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. This article is about the machine. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... The octal numeral system, or oct for short, is the base-8 number system, and uses the digits 0 to 7. ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0â€“9 and Aâ€“F, or aâ€“f. ...

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 . In computing and electronic systems, binary-coded decimal (BCD) is an encoding for decimal numbers in which each digit is represented by its own binary sequence. ... IEEE 754r is an ongoing revision to the IEEE 754 floating point standard. ...

### Decimal fractions

A decimal fraction is a fraction where the denominator is a power of ten. For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... A denominator is a name. ... â€œExponentâ€ redirects here. ...

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, 833/100, 83/1000, and 8/10000 are expressed as: 0.8, 8.33, 0.083, and 0.0008. In English-speaking countries, a dot (·) or period (.) is used as the decimal separator; in most other languages a comma is used. The decimal separator is a symbol used to mark the boundary between the integral and the fractional parts of a decimal numeral. ... A leading zero is any zero that proceeds a number string beginning with a non-null value. ...

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 which is less than one to have a leading zero. The floor and fractional part functions In mathematics, the floor function of a real number x, denoted or floor(x), is the largest integer less than or equal to x (formally, ). For example, floor(2. ...

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). In mathematics, trailing zeros are a sequence of 0s in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. ... This article is about the field of statistics. ... Rounding to n significant figures is a form of rounding. ...

### Other rational numbers

Any rational number which cannot be expressed as a decimal fraction has a unique infinite decimal expansion ending with recurring decimals. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ...

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: In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ...

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. For other senses of this word, see sequence (disambiguation). ... Seven Days of Creation - 1765 book, title page 7 (seven) is the natural number following 6 and preceding 8. ... 13 (thirteen) is the natural number following 12 and preceding 14. ...

That a rational must produce 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: In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder â€”an amount left overâ€” is also acknowledged. ...

`  .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, A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ... In mathematics, a geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence. ... $0.0123123123cdots = frac{123}{10000} sum_{k=0}^infty 0.001^k = frac{123}{10000} frac{1}{1-0.001} = frac{123}{9990} = frac{41}{3330}$

### Real numbers

Further information: Decimal representation

Every real number has a (possibly infinite) decimal representation, i.e., it can be written as It has been suggested that this article or section be merged with decimal. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... $x = mathop{rm sign}(x) sum_{iinmathbb Z} a_i,10^i$

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. Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... In mathematics, the common logarithm is the logarithm with base 10. ...

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. In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ...

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. In mathematics, an irrational number is any real number that is not a rational number â€” that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero. ...

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: A trichotomy is a splitting into three parts, and, apart from its normal literal meaning, can refer to: trichotomy (mathematics), in the mathematical field of order theory trichotomy (philosophy), for the idea that man has a threefold nature In taxonomy, a trichotomy is speciation of three groups from a common... A positional notation or place-value notation system is a numeral system in which each position is related to the next by a constant multiplier, a common ratio, called the base or radix of that numeral system. ...

• 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. The Golden mean (&#966;) can be used as a number base. ...

## 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.  
• 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
• 1925 Louis Charles KarpinskiThe History of Arithmetic 
• 1959 Werner BuchholzFingers 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 CowlishawDecimal 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. Chinese (written) language (pinyin: zh&#333;ngw n) written in Chinese characters The Chinese language (&#27721;&#35821;/&#28450;&#35486;, &#21326;&#35821;/&#33775;&#35486;, or &#20013;&#25991;; Pinyin: H ny&#468;, Hu y&#468;, or Zh&#333;ngw n) is a member of the Sino-Tibetan family of languages. ... Wu (Chinese: ; pinyin: ; Chinese: ; pinyin: ) is one of the major divisions of the Chinese language. ...

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. It has been suggested that this article or section be merged with Quechuan languages. ... Aymara is an Aymaran language spoken by the Aymara of the Andes. ...

Some psychologists suggest irregularities of numerals in a language may hinder children's counting ability. Results from FactBites:

 Encyclopedia4U - Decimal - Encyclopedia Article (479 words) Decimal, also called denary, is the base 10 numeral system, which uses symbols 0-9 (called digits). Decimal is the predominant numeral system used by humans, though some cultures do or did use other number systems. 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.
 Decimal - definition of Decimal in Encyclopedia (702 words) Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to represent numbers. Decimal fractions are usually expressed without a denominator, the decimal point being inserted into the numerator at a position corresponding to the power of ten of the denominator. That a rational must producing 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-1.
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