TheInfoList

The decimal
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
(also called the base-ten
positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
, and occasionally called denary or decanary) is the standard system for denoting
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
and non-integer
number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deduct ...

s. It is the extension to non-integer numbers of the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''. A ''decimal numeral'' (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a
decimal separator A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...

(usually "." or "," as in or ). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the
decimal fractions The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the H ...
. That is,
fractions A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, ...
of the form , where is an integer, and is a
non-negative integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
. The decimal system has been extended to ''infinite decimals'' for representing any
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, by using an
infinite sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
of digits after the decimal separator (see
decimal representation A decimal representation of a non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive n ...
). In this context, the decimal numerals with a finite number of non-zero digits after the decimal separator are sometimes called ''terminating decimals''. A ''
repeating decimal A repeating decimal or recurring decimal is decimal representation of a number whose Numerical digit, digits are periodic function, periodic (repeating its values at regular intervals) and the infinity, infinitely repeated portion is not zero. It c ...
'' is an infinite decimal that, after some place, repeats indefinitely the same sequence of digits (e.g., ). An infinite decimal represents a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
, the
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of two integers, if and only if it is a repeating decimal or has a finite number of non-zero digits.

# Origin

Many
numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or other symbols in a consistent manner. The same s ...
s of ancient civilizations use ten and its powers for representing numbers, possibly because there are ten fingers on two hands and people started counting by using their fingers. Examples are firstly the
Egyptian numerals The system of ancient Egyptian numerals was used in Ancient Egypt Ancient Egypt was a civilization of Ancient history, ancient North Africa, concentrated along the lower reaches of the Nile, Nile River, situated in the place that is now th ...
, then the
Brahmi numerals The Brahmi numerals are a numeral system A numeral system (or system of numeration) is a writing system for expressing numbers; that is, a mathematical notation for representing numbers of a given set, using Numerical digit, digits or othe ...
,
Greek numerals Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, are a system of writing numbers using the letters of the Greek alphabet The Greek alphabet has been used to write the Greek language since the late ninth or ear ...
,
Hebrew numerals The system of Hebrew numerals is a quasi-decimal alphabetic numeral system An alphabetic numeral system is a type of numeral system. Developed in classical antiquity, it flourished during the early Middle Ages. In alphabetic numeral systems, number ...
,
Roman numerals Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...
, and
Chinese numerals Chinese numerals are words and characters used to denote number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has be ...
. Very large numbers were difficult to represent in these old numeral systems, and only the best mathematicians were able to multiply or divide large numbers. These difficulties were completely solved with the introduction of the
Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Arabic numeral system or Hindu numeral system) is a positional notation, positional decimal numeral system, and is t ...
for representing
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
s. This system has been extended to represent some non-integer numbers, called ''
decimal fractions The decimal numeral system (also called the base-ten positional numeral system, and occasionally called denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the H ...
'' or ''decimal numbers'', for forming the ''decimal numeral system''.

# Decimal notation

For writing numbers, the decimal system uses ten
decimal digit , in order of value. A numerical digit is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label ...

s, a
decimal mark A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the ar ...
, and, for
negative number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, a
minus sign The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an abstract concept arising in mathematics. In the us ...
"−". The decimal digits are , , 2, 3, 4, 5, 6, 7, 8, 9; the
decimal separator A decimal separator is a symbol used to separate the integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spok ...

is the dot "" in many countries (mostly English-speaking), and a comma "" in other countries. For representing a
non-negative number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, a decimal numeral consists of * either a (finite) sequence of digits (such as "2017"), where the entire sequence represents an integer, *:$a_ma_\ldots a_0$ *or a decimal mark separating two sequences of digits (such as "20.70828") ::$a_ma_\ldots a_0.b_1b_2\ldots b_n$. If , that is, if the first sequence contains at least two digits, it is generally assumed that the first digit is not zero. In some circumstances it may be useful to have one or more 0's on the left; this does not change the value represented by the decimal: for example, . Similarly, if the final digit on the right of the decimal mark is zero—that is, if —it may be removed; conversely, trailing zeros may be added after the decimal mark without changing the represented number; for example, and . For representing a
negative number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, a minus sign is placed before . The numeral $a_ma_\ldots a_0.b_1b_2\ldots b_n$ represents the number :$a_m10^m+a_10^+\cdots+a_10^0+\frac+\frac+\cdots+\frac$. The ''
integer part In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
'' or ''integral part'' of a decimal numeral is the integer written to the left of the decimal separator (see also
truncation In mathematics and computer science, truncation is limiting the number of numerical digit, digits right of the decimal point. Truncation and floor function Truncation of positive real numbers can be done using the floor function. Given a numb ...
). For a non-negative decimal numeral, it is the largest integer that is not greater than the decimal. The part from the decimal separator to the right is the ''
fractional part The fractional part or decimal part of a non‐negative real number Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R\$) * Central American Republi ...
'', which equals the difference between the numeral and its integer part. When the integral part of a numeral is zero, it may occur, typically in
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and software. It has sci ...

, that the integer part is not written (for example , instead of ). In normal writing, this is generally avoided, because of the risk of confusion between the decimal mark and other punctuation. In brief, the contribution of each digit to the value of a number depends on its position in the numeral. That is, the decimal system is a
positional numeral system Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any radix, base of the Hindu–Arabic numeral system (or decimal, decimal system). More generally, a positional system is a numeral syste ...
.

# Decimal fractions

Decimal fractions (sometimes called decimal numbers, especially in contexts involving explicit fractions) are the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s that may be expressed as a
fraction A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths ...
whose
denominator A fraction (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Rom ...
is a
power Power most often refers to: * Power (physics) In physics, power is the amount of energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, bu ...
of ten. For example, the decimals $0.8, 14.89, 0.00024, 1.618, 3.14159$ represent the fractions , , , and , and are therefore decimal numbers. More generally, a decimal with digits after the separator represents the fraction with denominator , whose numerator is the integer obtained by removing the separator. It follows that a number is a decimal fraction
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
it has a finite decimal representation. Expressed as a fully reduced fraction, the decimal numbers are those whose denominator is a product of a power of 2 and a power of 5. Thus the smallest denominators of decimal numbers are :$1=2^0\cdot 5^0, 2=2^1\cdot 5^0, 4=2^2\cdot 5^0, 5=2^0\cdot 5^1, 8=2^3\cdot 5^0, 10=2^1\cdot 5^1, 16=2^4\cdot 5^0, 25=2^0\cdot 5^2, \ldots$

# Real number approximation

Decimal numerals do not allow an exact representation for all
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, e.g. for the real number

. Nevertheless, they allow approximating every real number with any desired accuracy, e.g., the decimal 3.14159 approximates the real , being less than 10−5 off; so decimals are widely used in
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

,
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

and everyday life. More precisely, for every real number and every positive integer , there are two decimals and with at most ' digits after the decimal mark such that and . Numbers are very often obtained as the result of
measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ...

. As measurements are subject to
measurement uncertainty In metrology Metrology is the scientific study of measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to c ...
with a known
upper bound In mathematics, particularly in order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), sh ...
, the result of a measurement is well-represented by a decimal with digits after the decimal mark, as soon as the absolute measurement error is bounded from above by . In practice, measurement results are often given with a certain number of digits after the decimal point, which indicate the error bounds. For example, although 0.080 and 0.08 denote the same number, the decimal numeral 0.080 suggests a measurement with an error less than 0.001, while the numeral 0.08 indicates an absolute error bounded by 0.01. In both cases, the true value of the measured quantity could be, for example, 0.0803 or 0.0796 (see also
significant figures Significant figures (also known as the significant digits, ''precision'' or ''resolution'') of a number in positional notation are Numerical digit, digits in the number that are reliable and absolutely necessary to indicate the quantity of som ...
).

# Infinite decimal expansion

For a
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
and an integer , let denote the (finite) decimal expansion of the greatest number that is not greater than ' that has exactly digits after the decimal mark. Let denote the last digit of . It is straightforward to see that may be obtained by appending to the right of . This way one has :, and the difference of and amounts to :, which is either 0, if , or gets arbitrarily small as ' tends to infinity. According to the definition of a
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
, ' is the limit of when ' tends to
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

. This is written asor : , which is called an infinite decimal expansion of '. Conversely, for any integer and any sequence of digits$\;(d_n)_^$ the (infinite) expression is an ''infinite decimal expansion'' of a real number '. This expansion is unique if neither all are equal to 9 nor all are equal to 0 for ' large enough (for all ' greater than some natural number ). If all for equal to 9 and , the limit of the sequence is the decimal fraction obtained by replacing the last digit that is not a 9, i.e.: , by , and replacing all subsequent 9s by 0s (see 0.999...). Any such decimal fraction, i.e.: for , may be converted to its equivalent infinite decimal expansion by replacing by and replacing all subsequent 0s by 9s (see 0.999...). In summary, every real number that is not a decimal fraction has a unique infinite decimal expansion. Each decimal fraction has exactly two infinite decimal expansions, one containing only 0s after some place, which is obtained by the above definition of , and the other containing only 9s after some place, which is obtained by defining as the greatest number that is ''less'' than , having exactly ' digits after the decimal mark.

## Rational numbers

Long division In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'ar ...

allows computing the infinite decimal expansion of a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
. If the rational number is a
decimal fraction The decimal numeral system A numeral system (or system of numeration) is a writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share") is t ...
, the division stops eventually, producing a decimal numeral, which may be prolongated into an infinite expansion by adding infinitely many zeros. If the rational number is not a decimal fraction, the division may continue indefinitely. However, as all successive remainders are less than the divisor, there are only a finite number of possible remainders, and after some place, the same sequence of digits must be repeated indefinitely in the quotient. That is, one has a ''repeating decimal''. For example, : = 0.012345679012... (with the group 012345679 indefinitely repeating). The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational. or, dividing both numerator and denominator by 6, .

# Decimal computation

Most modern
computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer program, programs. These ...

hardware and software systems commonly use a binary representation internally (although many early computers, such as the
ENIAC ENIAC (; Electronic Numerical Integrator and Computer) was the first programmable, electronic Electronic may refer to: *Electronics Electronics comprises the physics, engineering, technology and applications that deal with the emission ...

or the
IBM 650 The IBM 650 Magnetic Drum Data-Processing Machine is an early digital computer A computer is a machine A machine is a man-made device that uses power to apply forces and control movement to perform an action. Machines can be dri ...
, used decimal representation internally). For external use by computer specialists, this binary representation is sometimes presented in the related
octal The octal numeral system, or oct for short, is the radix, base-8 number system, and uses the Numerical digit, digits 0 to 7, that is to say 10 represents 8 in decimal and 100 represents 64 in decimal. However, English uses a Base 10, base-10 num ...

or
hexadecimal In mathematics and computing, the hexadecimal (also base 16 or hex) numeral system is a Numeral system#Positional systems in detail, positional numeral system that represents numbers using a radix (base) of 16. Unlike the decimal system repres ...
systems. For most purposes, however, binary values are converted to or from the equivalent decimal values for presentation to or input from humans; computer programs express literals in decimal by default. (123.1, for example, is written as such in a computer program, even though many computer languages are unable to encode that number precisely.) 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 In computing and electronics, electronic systems, binary-coded decimal (BCD) is a class of Binary numeral system, binary encodings of decimal numbers where each numerical digit, digit is represented by a fixed number of bits, usually four or ei ...
, especially in database implementations, but there are other decimal representations in use (including
decimal floating point Decimal floating-point (DFP) arithmetic refers to both a representation and operations on Decimal data type, decimal floating-point numbers. Working directly with decimal (base-10) fractions can avoid the rounding errors that otherwise typically ...
such as in newer revisions of the IEEE 754 Standard for Floating-Point Arithmetic). Decimal arithmetic is used in computers so that decimal fractional results of adding (or subtracting) values with a fixed length of their fractional part always are computed to this same length of precision. This is especially important for financial calculations, e.g., requiring in their results integer multiples of the smallest currency unit for book keeping purposes. This is not possible in binary, because the negative powers of $10$ have no finite binary fractional representation; and is generally impossible for multiplication (or division). See
Arbitrary-precision arithmetic In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose numerical digit, digits of pre ...
for exact calculations.

# History

Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands typically having ten fingers/digits. Standardized weights used in the
Indus Valley Civilization The Indus Valley Civilisation (IVC), also known as the Indus Civilisation, was a Bronze Age The Bronze Age is a prehistoric that was characterized by the use of , in some areas , and other early features of urban . The Bronze Age is ...

() were based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, while their standardized ruler – the ''Mohenjo-daro ruler'' – was divided into ten equal parts.
Egyptian hieroglyphs Egyptian hieroglyphs () were the formal writing system A writing system is a method of visually representing verbal communication Communication (from Latin ''communicare'', meaning "to share" or "to be in relation with") is "an apparent ...
, in evidence since around 3000 BCE, used a purely decimal system, as did the
Cretan hieroglyphs Cretan hieroglyphs are a hieroglyphic writing system used in early Bronze Age Crete, during the Minoan civilization, Minoan era. They predate Linear A by about a century, but the two writing systems continued to be used in parallel for most of t ...
() of the Minoans whose numerals are closely based on the Egyptian model. The decimal system was handed down to the consecutive Bronze Age cultures of Greece, including
Linear A Linear A is a writing system that was used by the (Cretans) from 1800 to 1450 BC to write the hypothesized . Linear A was the primary script used in palace and religious writings of the Minoan civilization. It was discovered by arch ...
(c. 18th century BCE−1450 BCE) and
Linear B Linear B is a syllabic script In the linguistic Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them. The tradition ...
(c. 1375−1200 BCE) – the number system of
classical Greece Classical Greece was a period of around 200 years (the 5th and 4th centuries BC) in Ancient Greece Ancient Greece ( el, Ἑλλάς, Hellás) was a civilization belonging to a period of History of Greece, Greek history from the Greek Dar ...
also used powers of ten, including,
Roman numerals Roman numerals are a that originated in and remained the usual way of writing numbers throughout Europe well into the . Numbers in this system are represented by combinations of letters from the . Modern style uses seven symbols, each with a ...
, an intermediate base of 5. Notably, the polymath
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

(c. 287–212 BCE) invented a decimal positional system in his Sand Reckoner which was based on 108 and later led the German mathematician
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

to lament what heights science would have already reached in his days if Archimedes had fully realized the potential of his ingenious discovery.
Hittite Hittite may refer to: * Hittites, ancient Anatolian people ** Hittite language, the earliest-attested Indo-European language ** Hittite grammar ** Hittite phonology ** Hittite cuneiform ** Hittite inscriptions ** Hittite laws ** Hittite religion ** ...

hieroglyphs (since 15th century BCE) were also strictly decimal. Some non-mathematical ancient texts such as the
Vedas upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the '' Atharvaveda''. The Vedas (, , ) are a large body of religious texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the o ...

, dating back to 1700–900 BCE make use of decimals and mathematical decimal fractions. The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols. For instance, Egyptian numerals used different symbols for 10, 20 to 90, 100, 200 to 900, 1000, 2000, 3000, 4000, to 10,000. The world's earliest positional decimal system was the Chinese rod calculus.

## History of decimal fractions

Decimal fractions were first developed and used by the Chinese in the end of 4th century BCE, and then spread to the Middle East and from there to Europe. The written Chinese decimal fractions were non-positional. However, counting rod fractions were positional. Lam Lay Yong, "The Development of Hindu–Arabic and Traditional Chinese Arithmetic", ''Chinese Science'', 1996 p. 38, Kurt Vogel notation
Qin Jiushao Qin Jiushao (, ca. 1202–1261), courtesy name A courtesy name (), also known as a style name, is a name bestowed upon one at adulthood in addition to one's given name. This practice is a tradition in the Sinosphere, including China, Japan, Ko ...
in his book Mathematical Treatise in Nine Sections (1247) denoted 0.96644 by ::::: ::::: , meaning ::::: :::::096644 J. Lennart Berggren notes that positional decimal fractions appear for the first time in a book by the Arab mathematician
Abu'l-Hasan al-Uqlidisi Abu'l Hasan Ahmad ibn Ibrahim Al-Uqlidisi ( ar, أبو الحسن أحمد بن ابراهيم الإقليدسي) was a Muslim Arab Mathematics in medieval Islam, mathematician, who was active in Damascus and Baghdad. He wrote the earliest survivi ...
written in the 10th century. The Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
, but did not develop any notation to represent them. The Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century. Al Khwarizmi introduced fraction to Islamic countries in the early 9th century; a Chinese author has alleged that his fraction presentation was an exact copy of traditional Chinese mathematical fraction from
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the p ...
. This form of fraction with numerator on top and denominator at bottom without a horizontal bar was also used by al-Uqlidisi and by al-Kāshī in his work "Arithmetic Key".
A forerunner of modern European decimal notation was introduced by
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...
in the 16th century.

## Natural languages

A method of expressing every possible
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...
using a set of ten symbols emerged in India. Several Indian languages show a straightforward decimal system. Many
Indo-AryanIndo-Aryan refers to: * Indo-Aryan languages ** Indo-Aryan superstrate in Mitanni or Mitanni-Aryan * Indo-Aryan peoples, the various peoples speaking these languages See also

*Aryan invasion theory (disambiguation) *Indo-Aryan tribes (disambigua ...
and
Dravidian languages Dravidian languages (or sometimes Dravidic languages) are a family of languages In human society, family (from la, familia) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other r ...
have numbers between 10 and 20 expressed in a regular pattern of addition to 10. The
Hungarian language Hungarian () is a Uralic language The Uralic languages (; sometimes called Uralian languages ) form a language family A language is a structured system of communication used by humans, including speech ( spoken language), gestures (Sig ...
also uses a straightforward decimal system. All numbers between 10 and 20 are formed regularly (e.g. 11 is expressed as "tizenegy" literally "one on ten"), as with those between 20 and 100 (23 as "huszonhárom" = "three on twenty"). A straightforward decimal rank system with a word for each order (10 , 100 , 1000 , 10,000 ), and in which 11 is expressed as ''ten-one'' and 23 as ''two-ten-three'', and 89,345 is expressed as 8 (ten thousands) 9 (thousand) 3 (hundred) 4 (tens) 5 is found in
Chinese Chinese can refer to: * Something related to China China, officially the People's Republic of China (PRC), is a country in East Asia. It is the List of countries and dependencies by population, world's most populous country, with a populat ...
, and in
Vietnamese Vietnamese may refer to: * Something of, from, or related to Vietnam, a country in Southeast Asia ** A citizen of Vietnam. See Demographics of Vietnam. * Vietnamese people, or Kinh people, a Southeast Asian ethnic group native to Vietnam ** Oversea ...

with a few irregularities.
Japanese Japanese may refer to: * Something from or related to Japan Japan ( ja, 日本, or , and formally ) is an island country An island country or an island nation is a country A country is a distinct territory, territorial body or ...

,
Korean Korean may refer to: People and culture * Koreans, ethnic group originating in the Korean Peninsula * Korean cuisine * Korean culture * Korean language **Korean alphabet, known as Hangul or Chosŏn'gŭl **Korean dialects and the Jeju language **S ...
, and
Thai Thai or THAI may refer to: * Of or from Thailand, a country in Southeast Asia ** Thai people, the dominant ethnic group of Thailand ** Thai language, a Tai-Kadai language spoken mainly in and around Thailand *** Thai script *** Thai (Unicode block) ...

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. For example, in English 11 is "eleven" not "ten-one" or "one-teen". Incan languages such as
Quechua Quechua may refer to: *Quechua people, several indigenous ethnic groups in South America, especially in Peru *Quechuan languages, a Native South American language family spoken primarily in the Andes, derived from a common ancestral language **Sou ...
and
Aymara Aymara may refer to: Languages and people * Aymaran languages Aymaran (also Jaqi or Aru) is one of the two dominant language families in the central Andes alongside Quechua languages, Quechuan. The family consists of Aymara language, Aymara, wi ...
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.

## Other bases

Some cultures do, or did, use other bases of numbers. *
Pre-Columbian In the history of the Americas, the pre-Columbian era spans from the original settlement of North and South America in the Upper Paleolithic The Upper Paleolithic (or Upper Palaeolithic) also called the is the third and last subdivision o ...
Mesoamerica Mesoamerica is a historical and important region In geography, regions are areas that are broadly divided by physical characteristics (physical geography), human impact characteristics (human geography), and the interaction of humanity and the ...
n cultures such as the
Maya Maya may refer to: Civilizations * Maya peoples The Maya peoples () are an ethnolinguistic group of indigenous peoples Indigenous peoples, also referred to as First people, Aboriginal people, Native people, or autochthonous people, are cu ...

used a base-20 system (perhaps based on using all twenty fingers and
toe Toes are the digits (fingers) of the foot of a tetrapod. Animal Animals (also called Metazoa) are multicellular eukaryotic organisms that form the Kingdom (biology), biological kingdom Animalia. With few exceptions, animals Heterotrop ...

s). * The
Yuki Yuki, Yūki or Yuuki may refer to: Places * Yuki, Hiroshima (Jinseki), a town in Jinseki District, Hiroshima, Japan * Yuki, Hiroshima (Saeki), a town in Saeki District, Hiroshima, Japan * Yūki, Ibaraki, a city on Honshu island in Japan * Yuki, ...
language in
California California is a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper i ...

and the Pamean languages in
Mexico Mexico, officially the United Mexican States, is a country A country is a distinct territorial body or political entity A polity is an identifiable political entity—any group of people who have a collective identity, who are organi ...

have systems because the speakers count using the spaces between their fingers rather than the fingers themselves. * The existence of a non-decimal base in the earliest traces of the Germanic languages is attested by the presence of words and glosses meaning that the count is in decimal (cognates to "ten-count" or "tenty-wise"); such would be expected if normal counting is not decimal, and unusual if it were. Where this counting system is known, it is based on the "long hundred" = 120, and a "long thousand" of 1200. The descriptions like "long" only appear after the "small hundred" of 100 appeared with the Christians. Gordon'
Introduction to Old Norse
p. 293, gives number names that belong to this system. An expression cognate to 'one hundred and eighty' translates to 200, and the cognate to 'two hundred' translates to 240
Goodare
details the use of the long hundred in Scotland in the Middle Ages, giving examples such as calculations where the carry implies i C (i.e. one hundred) as 120, etc. That the general population were not alarmed to encounter such numbers suggests common enough use. It is also possible to avoid hundred-like numbers by using intermediate units, such as stones and pounds, rather than a long count of pounds. Goodare gives examples of numbers like vii score, where one avoids the hundred by using extended scores. There is also a paper by W.H. Stevenson, on 'Long Hundred and its uses in England'. * Many or all of the
Chumashan languages Chumashan (meaning "Santa Cruz Islander") is a family of languages that were spoken on the southern California coast by Native American Chumash people, from the Coastal plains and valleys of San Luis Obispo to Malibu, neighboring inland and ...
originally used a
base-4 A quaternary numeral system is base (exponentiation), base-. It uses the numerical digit, digits 0, 1, 2 and 3 to represent any real number. Four is the largest number within the subitizing range and one of two numbers that is both a square and ...

counting system, in which the names for numbers were structured according to multiples of 4 and 16. * Many languages use quinary (base-5) number systems, including
Gumatj The Yolngu or Yolŋu () are an aggregation of Aboriginal Australian Aboriginal Australians are the various Indigenous peoples Indigenous peoples, also referred to as First people, Aboriginal people, Native people, or autochthonous peo ...
, Nunggubuyu, Kuurn Kopan Noot and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5. * Some
Nigeria Nigeria (), officially the Federal Republic of Nigeria, is a country in West Africa West Africa or Western Africa is the westernmost region of . The defines Western Africa as the 17 countries of , , , , , , , , , , , , , , , and as we ...

ns use
duodecimal The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any of the (or ). More generally ...
systems. So did some small communities in India and Nepal, as indicated by their languages. * The
Huli language Huli may refer to: * Huli (dish), a lentil-based dish, also called Sambar, common in South India and Sri Lanka * Huli people The Huli are an indigenous people Indigenous peoples, also referred to as First people, Aboriginal people, Native ...
of
Papua New Guinea Papua New Guinea (PNG; , ; tpi, Papua Niugini; ho, Papua Niu Gini), officially the Independent State of Papua New Guinea ( tpi, Independen Stet bilong Papua Niugini; ho, Independen Stet bilong Papua Niu Gini), is a country in Oceania th ...

is reported to have base-15 numbers. ''Ngui'' means 15, ''ngui ki'' means 15 × 2 = 30, and ''ngui ngui'' means 15 × 15 = 225. * Umbu-Ungu language, Umbu-Ungu, also known as Kakoli, is reported to have base 24, base-24 numbers. ''Tokapu'' means 24, ''tokapu talu'' means 24 × 2 = 48, and ''tokapu tokapu'' means 24 × 24 = 576. * Ngiti language, Ngiti is reported to have a base 32, base-32 number system with base-4 cycles. * The Ndom language of
Papua New Guinea Papua New Guinea (PNG; , ; tpi, Papua Niugini; ho, Papua Niu Gini), officially the Independent State of Papua New Guinea ( tpi, Independen Stet bilong Papua Niugini; ho, Independen Stet bilong Papua Niu Gini), is a country in Oceania th ...

is reported to have base-6 numerals. ''Mer'' means 6, ''mer an thef'' means 6 × 2 = 12, ''nif'' means 36, and ''nif thef'' means 36×2 = 72.