Numeral systems by culture
Hindu-Arabic numerals
Indian
Eastern Arabic
Khmer
Indian family
Brahmi
Thai
East Asian numerals
Chinese
Counting rods
Japanese
Korean
Alphabetic numerals
Armenian
Cyrillic
Ge'ez
Hebrew
Greek (Ionian)
Āryabhaṭa

Other systems
Attic
Babylonian
Egyptian
Etruscan
Mayan
Roman
Urnfield
List of numeral system topics
Positional systems by base
Decimal (10)
2, 4, 8, 16, 32, 64
1, 3, 9, 12, 20, 24, 30, 36, 60, more…
v  d  e

## 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. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner In Mathematics and Computer science, a digit is a symbol (a number symbol e 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. In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional 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. Counting rods ( Japanese: 算木 sangi are small bars typically 3-14 cm long used by mathematicians for calculation in China, Japan Other numeric systems require more or fewer symbols.

The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc. Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits 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 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. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. ) 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. A globe is a three- Dimensional scale model of Earth ( terrestrial globe) or other spheroid celestial body such as a planet star or moon The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system Most of the positional Base 10 Numeral systems in the world have originated from India, which first developed the concept of positional numerology 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). The pre-Columbian era incorporates all period subdivisions in the history and prehistory of the Americas before the appearance of significant European influences Mesoamerica or Meso-America (Mesoamérica is a Region extending approximately from central Mexico to Honduras and Nicaragua, defined The Pre-Columbian Maya civilization used a Vigesimal ( base - twenty) Numeral system. The vigesimal or base - numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten Toes are the digits of the Foot of an animal Many animal species such as Cats walk on their toes and are described as being Digitigrade Nigeria, officially named the Federal Republic of Nigeria, is a federal Constitutional republic comprising thirty-six states and one Federal The duodecimal system (also known as base -12 or dozenal) is a Numeral system using twelve as its base. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital Sexagesimal ( base-sixty) is a Numeral system with sixty as the base. The Yuki are a Native American tribe from the zone of Round Valley, in what today is part of the territory of Mendocino County, Northern California 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. A computer is a Machine that manipulates data according to a list of instructions. The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. For external use by computer specialists, this binary representation is sometimes presented in the related octal or hexadecimal systems. 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 (also base -, hexa, or hex) is a Numeral system with a 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. 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 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 [1].

### Decimal fractions

A decimal fraction is a fraction where the denominator is a power of ten. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object

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. In a positional Numeral system, the decimal separator is a Symbol used to mark the boundary between the integral and the fractional A leading zero is any zero that leads a number string with a non-zero value 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). In Mathematics and Computer science, the floor and ceiling functions map Real numbers to nearby Integers The 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. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.

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. In Mathematics, trailing zeros are a sequence of 0s in the Decimal representation (or more generally in any positional representation) of a number Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. 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). The significant figures (also called significant digits and abbreviated sig figs) of a number are those digits that carry meaning contributing to its accuracy

### 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 Non-integer rational numbers (commonly called fractions A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is

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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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. In Mathematics, a sequence is an ordered list of objects (or events In mathematics Seven is the fourth Prime number. It is not only a Mersenne prime (since 23 &minus 1 = 7 but also a

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. In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left 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. A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is 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. In Mathematics, a geometric series is a series with a constant ratio between successive terms. 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

Further information: Decimal representation

Every real number has a (possibly infinite) decimal representation, i. This article gives a mathematical definition For a more accessible article see Decimal. In Mathematics, the real numbers may be described informally in several different ways 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|). The common logarithm is the Logarithm with base 10 It is also known as the decadic logarithm, named after its base

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

Rational numbers (e. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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 Number theory, the prime factors of a positive Integer are the Prime numbers that divide into that integer exactly without leaving a remainder A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is

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. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction 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

and a version of this even holds for irrational-base numeration systems, such as golden mean base representation. 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 Golden ratio base is a non-standard positional numeral system that uses the Golden ratio (an irrational number ≈1

## 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. Elam is the name of an ancient civilization located in what is now southwest Iran. For a topic outline on this subject see List of basic Iran topics. [2] [3]
• c. 2900 BC Egyptian hieroglyphs show counting in powers of 10 (1 million + 400,000 goats, etc. This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. ) – 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. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object See Ancient Indus Valley weights and measures
• c. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin 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. Chinese civilization originated in various city-states along the Yellow River ( valley in the Neolithic era 1200 BC In ancient India, the Vedic text Yajur-Veda states the powers of 10, up to 1055
• c. This article is about the history of South Asia prior to the Partition of British India in 1947 "Veda" redirects here For other uses see Veda (disambiguation. The Yajurveda ( Sanskrit यजुर्वेदः, a Tatpurusha compound of yajus "sacrificial formula' + veda 400 BC Pingala – develops the binary number system for Sanskrit prosody, with a clear mapping to the base-10 decimal system
• c. Pingala ( पिङ्गल piṅgalá) was an ancient Indian writer famous for his work the Chandas Shastra ( chandaḥ-śāstra 250 BC Archimedes writes the Sand Reckoner, which takes decimal calculation up to 1080,000,000,000,000,000
• c. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer The Sand Reckoner ( Greek: Ψαμμίτης Psammites) is a work by Archimedes in which he set out to determine an upper bound for the number 100–200 The Satkhandagama written in India – earliest use of decimal logarithms
• c. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country 476–550 Aryabhata – uses an alphabetic cipher system for numbers that used zero
• c. Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics 598–670 Brahmagupta – explains the Hindu-Arabic numerals (modern number system) which uses decimal integers, negative integers, and zero
• c. Brahmagupta ( (598–668 was an Indian mathematician and astronomer. The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 780–850 Muḥammad ibn Mūsā al-Ḵwārizmī – first to expound on algorism outside India
• c. Algorism is the technique of performing basic Arithmetic by writing numbers in Place value form and applying a set of memorized rules and facts to the digits India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country 920–980 Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi – earliest known direct mathematical treatment of decimal fractions. Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi was an Arab Mathematician, possibly from Damascus.
• 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. South India is the area encompassing India 's states of Andhra Pradesh, Karnataka, Kerala and Tamil Nadu as well as the union In Computing, floating point describes a system for numerical representation in which a string of digits (or Bits represents a Real number. Simon Stevin (1548/49 &ndash 1620 was a Flemish Mathematician and Engineer. "Pitiscus" redirects here For the crater see Pitiscus (crater.
• 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. For other people with the same name see John Napier (disambiguation. Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German 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?). In Mathematics, the Artin conjecture is a Conjecture on the set of primes p modulo which a given integer a
• 1925 Louis Charles KarpinskiThe History of Arithmetic [1]
• 1959 Werner BuchholzFingers or Fists? (The Choice of Decimal or Binary representation)[2]
• 1974 Hermann SchmidDecimal Computation[3]
• 2000 Georges IfrahThe Universal History of Numbers: From Prehistory to the Invention of the Computer[4]
• 2003 Mike CowlishawDecimal Floating-Point: Algorism for Computers[5]. Louis Charles Karpinski ( 5 August 1878 &ndash 25 January 1956) was an American mathematician born in Rochester New York A byte (pronounced "bite" baɪt is the basic unit of measurement of information storage in Computer science. Hermann Schmid is the author of the book Decimal Computation which was first published in 1974 by John Wiley & Sons (ISBN 047176180X and reprinted in 1983 by Robert E Georges Ifrah ( 1947 - was a professor of Mathematics, and a Historian of mathematics especially Numerals. Mike Cowlishaw is an IBM Fellow based at IBM UK’s Warwick location a Visiting Professor at the Department of Computer Science at the University of Warwick

## 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. Vietnamese ( tiếng Việt, or less commonly Việt ngữ) formerly known under French colonization as Annamese ( see Annam) Japanese, Korean, and Thai have imported the Chinese decimal system. is a language spoken by over 130 million people in Japan and in Japanese emigrant communities This article is mainly about the spoken Korean language See Hangul for details on the native Korean writing system Thai (th ภาษาไทย, transcription: phasa thai, transliteration:; pʰāːsǎːtʰāj is the national and 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. Quechua ( Runa Simi) is a Native American language of South America. Aymara ( Aymar aru) is an Aymaran language spoken by the Aymara people of the Andes.

Some psychologists suggest irregularities of numerals in a language may hinder children's counting ability[6].