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 digits or other symbols in a consistent
manner. It can be seen as the context that allows the symbols "11" to
be interpreted as the binary symbol for three, the decimal symbol for
eleven, or a symbol for other numbers in different bases.
The number the numeral represents is called its value.
Ideally, a numeral system will:
Represent a useful set of numbers (e.g. all integers, or rational
Give every number represented a unique representation (or at least a
Reflect the algebraic and arithmetic structure of the numbers.
For example, the usual decimal representation of whole numbers gives
every nonzero whole number a unique representation as a finite
sequence of digits, beginning with a non-zero digit. However, when
decimal representation is used for the rational or real numbers, such
numbers in general have an infinite number of representations, for
example 2.31 can also be written as 2.310, 2.3100000, 2.309999999...,
etc., all of which have the same meaning except for some scientific
and other contexts where greater precision is implied by a larger
number of figures shown.
Numeral systems are sometimes called number systems, but that name is
ambiguous, as it could refer to different systems of numbers, such as
the system of real numbers, the system of complex numbers, the system
of p-adic numbers, etc. Such systems are, however, not the topic of
1 Main numeral systems
2 Positional systems in detail
3 Generalized variable-length integers
4 See also
7 External links
Main numeral systems
Main article: List of numeral systems
The most commonly used system of numerals is the Hindu–Arabic
numeral system. Two
Indian mathematicians are credited with
Aryabhata of Kusumapura developed the place-value
notation in the 5th century and a century later Brahmagupta
introduced the symbol for zero. The numeral system and the zero
concept, developed by the Hindus in India, slowly spread to other
surrounding countries due to their commercial and military activities
with India. The Arabs adopted and modified it. Even today, the Arabs
call the numerals which they use "Rakam Al-Hind" or the Hindu numeral
system. The Arabs translated Hindu texts on numerology and spread them
to the western world due to their trade links with them. The Western
world modified them and called them the Arabic numerals, as they
learned them from the Arabs. Hence the current western numeral system
is the modified version of the Hindu numeral system developed in
India. It also exhibits a great similarity to the
Sanskrit–Devanagari notation, which is still used in India and
The simplest numeral system is the unary numeral system, in which
every natural number is represented by a corresponding number of
symbols. If the symbol / is chosen, for example, then the number seven
would be represented by ///////.
Tally marks represent one such system
still in common use. The unary system is only useful for small
numbers, although it plays an important role in theoretical computer
science. Elias gamma coding, which is commonly used in data
compression, expresses arbitrary-sized numbers by using unary to
indicate the length of a binary numeral.
The unary notation can be abbreviated by introducing different symbols
for certain new values. Very commonly, these values are powers of 10;
so for instance, if / stands for one, − for ten and + for 100, then
the number 304 can be compactly represented as +++ //// and the number
123 as + − − /// without any need for zero. This is called
sign-value notation. The ancient
Egyptian numeral system
Egyptian numeral system was of this
type, and the
Roman numeral system
Roman numeral system was a modification of this idea.
More useful still are systems which employ special abbreviations for
repetitions of symbols; for example, using the first nine letters of
the alphabet for these abbreviations, with A standing for "one
occurrence", B "two occurrences", and so on, one could then write C+
D/ for the number 304. This system is used when writing Chinese
numerals and other East Asian numerals based on Chinese. The number
system of the
English language is of this type ("three hundred [and]
four"), as are those of other spoken languages, regardless of what
written systems they have adopted. However, many languages use
mixtures of bases, and other features, for instance 79 in French is
soixante dix-neuf (60 + 10 + 9) and in Welsh is pedwar ar bymtheg a
thrigain (4 + (5 + 10) + (3 × 20)) or (somewhat archaic) pedwar ugain
namyn un (4 × 20 − 1). In English, one could say "four score less
one", as in the famous
Gettysburg Address representing "87 years ago"
as "four score and seven years ago".
More elegant is a positional system, also known as place-value
notation. Again working in base 10, ten different digits
0, ..., 9 are used and the position of a digit is used to signify
the power of ten that the digit is to be multiplied with, as in 304 =
3×100 + 0×10 + 4×1 or more precisely 3×102 + 0×101 + 4×100. Note
that zero, which is not needed in the other systems, is of crucial
importance here, in order to be able to "skip" a power. The
Hindu–Arabic numeral system, which originated in India and is now
used throughout the world, is a positional base 10 system.
Arithmetic is much easier in positional systems than in the earlier
additive ones; furthermore, additive systems need a large number of
different symbols for the different powers of 10; a positional system
needs only ten different symbols (assuming that it uses base 10).
The positional decimal system is presently universally used in human
writing. The base 1000 is also used (albeit not universally), by
grouping the digits and considering a sequence of three decimal digits
as a single digit. This is the meaning of the common notation
1,000,234,567 used for very large numbers.
In computers, the main numeral systems are based on the positional
system in base 2 (binary numeral system), with two binary digits,
0 and 1. Positional systems obtained by grouping binary digits by
three (octal numeral system) or four (hexadecimal numeral system) are
commonly used. For very large integers, bases 232 or 264
(grouping binary digits by 32 or 64, the length of the machine word)
are used, as, for example, in GMP.
The numerals used when writing numbers with digits or symbols can be
divided into two types that might be called the arithmetic numerals
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and the geometric numerals (1, 10, 100,
1000, 10000 ...), respectively. The sign-value systems use only the
geometric numerals and the positional systems use only the arithmetic
numerals. A sign-value system does not need arithmetic numerals
because they are made by repetition (except for the Ionic system), and
a positional system does not need geometric numerals because they are
made by position. However, the spoken language uses both arithmetic
and geometric numerals.
In certain areas of computer science, a modified base k positional
system is used, called bijective numeration, with digits 1,
2, ..., k (k ≥ 1), and zero being represented by an empty
string. This establishes a bijection between the set of all such
digit-strings and the set of non-negative integers, avoiding the
non-uniqueness caused by leading zeros. Bijective base-k numeration is
also called k-adic notation, not to be confused with p-adic numbers.
Bijective base 1 is the same as unary.
Positional systems in detail
See also: Positional notation
In a positional base b numeral system (with b a natural number greater
than 1 known as the radix), b basic symbols (or digits) corresponding
to the first b natural numbers including zero are used. To generate
the rest of the numerals, the position of the symbol in the figure is
used. The symbol in the last position has its own value, and as it
moves to the left its value is multiplied by b.
For example, in the decimal system (base 10), the numeral 4327 means
(4×103) + (3×102) + (2×101) + (7×100), noting that 100 = 1.
In general, if b is the base, one writes a number in the numeral
system of base b by expressing it in the form anbn + an − 1bn − 1
+ an − 2bn − 2 + ... + a0b0 and writing the enumerated digits anan
− 1an − 2 ... a0 in descending order. The digits are natural
numbers between 0 and b − 1, inclusive.
If a text (such as this one) discusses multiple bases, and if
ambiguity exists, the base (itself represented in base 10) is
added in subscript to the right of the number, like this: numberbase.
Unless specified by context, numbers without subscript are considered
to be decimal.
By using a dot to divide the digits into two groups, one can also
write fractions in the positional system. For example, the base 2
numeral 10.11 denotes 1×21 + 0×20 + 1×2−1 + 1×2−2 = 2.75.
In general, numbers in the base b system are of the form:
displaystyle (a_ n a_ n-1 cdots a_ 1 a_ 0 .c_ 1 c_ 2 c_ 3 cdots
)_ b =sum _ k=0 ^ n a_ k b^ k +sum _ k=1 ^ infty c_ k b^ -k .
The numbers bk and b−k are the weights of the corresponding digits.
The position k is the logarithm of the corresponding weight w, that is
displaystyle k=log _ b w=log _ b b^ k
. The highest used position is close to the order of magnitude of the
The number of tally marks required in the unary numeral system for
describing the weight would have been w. In the positional system, the
number of digits required to describe it is only
displaystyle k+1=log _ b w+1
, for k ≥ 0. For example, to describe the weight 1000 then four
digits are needed because
displaystyle log _ 10 1000+1=3+1
. The number of digits required to describe the position is
displaystyle log _ b k+1=log _ b log _ b w+1
(in positions 1, 10, 100,... only for simplicity in the decimal
Decimal example weight
Decimal example digit
displaystyle begin array lrrrrrrr text Position
&3&2&1&0&-1&-2&cdots \hline text Weight
&b^ 3 &b^ 2 &b^ 1 &b^ 0 &b^ -1 &b^ -2
&cdots \ text Digit &a_ 3 &a_ 2 &a_ 1 &a_ 0
&c_ 1 &c_ 2 &cdots \hline text
Decimal example weight
&1000&100&10&1&0.1&0.01&cdots \ text
Decimal example digit &4&3&2&7&0&0&cdots
Note that a number has a terminating or repeating expansion if and
only if it is rational; this does not depend on the base. A number
that terminates in one base may repeat in another (thus 0.310 =
0.0100110011001...2). An irrational number stays aperiodic (with an
infinite number of non-repeating digits) in all integral bases. Thus,
for example in base 2, π = 3.1415926...10 can be written as the
Putting overscores, n, or dots, ṅ, above the common digits is a
convention used to represent repeating rational expansions. Thus:
14/11 = 1.272727272727... = 1.27 or 321.3217878787878...
If b = p is a prime number, one can define base-p numerals whose
expansion to the left never stops; these are called the p-adic
Generalized variable-length integers
More general is using a mixed radix notation (here written
displaystyle a_ 0 a_ 1 a_ 2
displaystyle a_ 0 +a_ 1 b_ 1 +a_ 2 b_ 1 b_ 2
This is used in punycode, one aspect of which is the representation of
a sequence of non-negative integers of arbitrary size in the form of a
sequence without delimiters, of "digits" from a collection of 36:
a–z and 0–9, representing 0–25 and 26–35 respectively. A digit
lower than a threshold value marks that it is the most-significant
digit, hence the end of the number. The threshold value depends on the
position in the number. For example, if the threshold value for the
first digit is b (i.e. 1) then a (i.e. 0) marks the end of the number
(it has just one digit), so in numbers of more than one digit, range
is only b–9 (1–35), therefore the weight b1 is 35 instead of 36.
Suppose the threshold values for the second and third digits are c
(2), then the third digit has a weight 34 × 35 = 1190 and we have the
a (0), ba (1), ca (2), .., 9a (35), bb (36), cb (37), .., 9b (70), bca
(71), .., 99a (1260), bcb (1261), etc.
Unlike a regular based numeral system, there are numbers like 9b where
9 and b each represents 35; yet the representation is unique because
ac and aca are not allowed – the a would terminate the number.
The flexibility in choosing threshold values allows optimization
depending on the frequency of occurrence of numbers of various sizes.
The case with all threshold values equal to 1 corresponds to bijective
numeration, where the zeros correspond to separators of numbers with
digits which are non-zero.
List of numeral systems
Computer numbering formats
Golden ratio base
History of ancient numeral systems
History of numbers
List of numeral system topics
Residue numeral system
Short and long scales
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numerals. Ginn and Company.
Georges Ifrah. The Universal History of Numbers : From Prehistory
to the Invention of the Computer, Wiley, 1999.
D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed.
Addison–Wesley. pp. 194–213, "Positional
A.L. Kroeber (Alfred Louis Kroeber) (1876–1960), Handbook of the
Indians of California, Bulletin 78 of the Bureau of American Ethnology
of the Smithsonian Institution (1919)
J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture,
Fitzroy Dearborn Publishers, London and Chicago, 1997.
Hans J. Nissen; Peter Damerow; Robert K. Englund (1993). Archaic
Bookkeeping: Early Writing and Techniques of Economic Administration
in the Ancient Near East. University Of Chicago Press.
Schmandt-Besserat, Denise (1996). How Writing Came About. University
of Texas Press. ISBN 978-0-292-77704-0.
Zaslavsky, Claudia (1999). Africa counts: number and pattern in
African cultures. Chicago Review Press.
Look up numeration or numeral in Wiktionary, the free dictionary.
Numerical Mechanisms and Children's Concept of Numbers
Online conversion of fractional numbers between numeral systems
Ethiopic Numeral Names, Aberra Molla