# Dictionary Definition

numeration

### Noun

1 naming numbers

2 the act of counting; "the counting continued
for several hours" [syn: count, counting, enumeration, reckoning, tally]

# User Contributed Dictionary

## English

### Noun

- The act of counting or numbering things; enumeration
- Any system of giving names to numbers

# Extensive Definition

A numeral system (or system of numeration) is a
mathematical
notation for representing numbers of a given set by symbols in
a consistent manner. It can be seen as the context that allows the
numeral "11" to be interpreted as the binary
numeral for three, the decimal numeral for eleven, or
other numbers in different bases.

Ideally, a numeral system will:

- Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers)
- Give every number represented a unique representation (or at least a standard representation)
- Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation of whole
numbers gives every whole number a unique representation as a
finite
sequence of digits,
with the operations of arithmetic (addition, subtraction,
multiplication and division) being present as the standard algorithms of arithmetic.
However, when decimal representation is used for the rational
or real numbers, the representation is no longer unique: many
rational numbers have two numerals, a standard one that terminates,
such as 2.31, and another that recurs,
such as 2.309999999... . Numerals which terminate have no non-zero
digits after a given position. For example, numerals like 2.31 and
2.310 are taken to be the same, except in the experimental
sciences, where greater precision is denoted by the trailing
zero.

Numeral systems are sometimes called number
systems, but that name is misleading, 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 not the topic of this
article.

## Types of numeral systems

The most commonly used system of numerals is
known as Hindu-Arabic
numerals, and two great Indian mathematicians could be given
credit for developing them. Aryabhatta of
Kusumapura who lived during the 5th century developed the place
value notation and Brahmagupta a
century later introduced the symbol zero.

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. In practice, the
unary system is normally only useful for small numbers, although it
plays an important role in
theoretical computer science. Also, 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 number 123 as + - - ///
without any need for zero. This is called sign-value
notation. The ancient Egyptian
system is of this type, and the Roman
system is 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 our alphabet for these
abbreviations, with A standing for "one occurrence", B "two
occurrences", and so on, we could then write C+ D/ for the number
304. The numeral system of English
is of this type ("three hundred [and] four"), as are those of
virtually all other spoken languages, regardless of what
written systems they have adopted.

More elegant is a positional
system, also known as place-value notation. Again working in
base 10, we use ten different digits 0, ..., 9 and use the position
of a digit to signify the power of ten that the digit is to be
multiplied with, as in 304 = 3×100 + 0×10 +
4×1. 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, borrowed from India, is a
positional base 10 system; it is used today throughout the
world.

Arithmetic is much easier in positional systems
than in the earlier additive ones; furthermore, additive systems
have a need for a potentially infinite number of different symbols
for the different powers of 10; positional systems need only 10
different symbols (assuming that it uses base 10).

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 system
use only the arithmetic numerals. The sign-value system does not
need arithmetic numerals because they are made by repetition
(except for the Ionic
system), and the 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 the 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 the same as unary.

## Bases used

### Computing

Switches, mimicked by their electronic successors built originally of vacuum tubes and in modern technology of transistors, have only two possible states: "open" and "closed". Substituting open=1 and closed=0 (or the other way around) yields the entire set of binary digits. This base-2 system (binary) is the basis for digital computers. It is used to perform integer arithmetic in almost all digital computers; some exotic base-3 (ternary) and base-10 computers have also been built, but those designs were discarded early in the history of computing hardware.Modern computers use transistors that represent
two states with either high or low voltages. The smallest unit of
memory for this binary state is called a bit. Bits are arranged in
groups to aid in processing, and to make the binary numbers shorter
and more manageable for humans. More recently these groups of bits,
such as bytes and words,
are sized in multiples of four. Thus base 16 (hexadecimal) is commonly
used as shorthand. Base 8 (octal) has also been used for this
purpose.

A computer does not treat all of its data as
numerical. For instance, some of it may be treated as program
instructions or data such as text. However, arithmetic and Boolean
logic constitute most internal operations. Whole numbers are
represented exactly, as integers.
Real
numbers, allowing fractional values, are usually approximated
as floating
point numbers. The computer uses different methods to do
arithmetic with these
two kinds of numbers.

### Five

A base-5 system (quinary) has been used in many cultures for counting. Plainly it is based on the number of fingers on a human hand. It may also be regarded as a sub-base of other bases, such as base 10 and base 60.### Eight

A base-8 system (octal) was devised by the Yuki of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight. There is also linguistic evidence which suggests that the Bronze Age Proto-Indo Europeans (from whom most European and Indic languages descend) might have replaced a base 8 system (or a system which could only count up to 8) with a base 10 system. The evidence is that the word for 9, newm, is suggested by some to derive from the word for 'new', newo-, suggesting that the number 9 had been recently invented and called the 'new number' (Mallory & Adams 1997).### Ten

The base-10 system (decimal) is the one most commonly used today. It is assumed to have originated because humans have ten fingers. These systems often use a larger superimposed base. See Decimal superbase.### Twelve

Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition and subtracting being just as easy. 12 is a useful base because it has many factors. It is the smallest multiple of one through four and of six. There is still a special word for "dozen" and just like there is a word for 102, hundred, there is also a word for 122, gross. Base-12 could have originated from the number of knuckles in the four fingers of a hand excluding the thumb, which is used as a pointer in counting.Twelve is a common British unit of measurement.
There are twelve inches to a foot. Prior to 1971, in British
currency, there were 12 pennies to a shilling.http://www.victorianweb.org/economics/currency.html.
English words for numbers are also 'base-12' in that there is a
unique word for the numbers one through twelve, with 'thirteen'
being the first word that was formed by combining numbers (three
and ten).

There are 24 hours per day, usually counted till
12 until noon (p.m.) and once again
until midnight (a.m.), often further
divided per 6 hours in counting (for instance in Thailand) or as
switches between using terms like 'night', 'morning', 'afternoon',
and 'evening', whereas other languages use such terms with
durations of 3 to 9 hours often according to switches at some of
the 3 hour interval marks.

Multiples of 12 have been in common use as
English units of resolution in the analog and digital printing
world, where 1 point
equals 1/72 of an inch and 12 points equal 1 pica,
and printer resolutions like 360, 600, 720, 1200 or 1440 dpi (dots
per inch) are common. These are combinations of base-12 and base-10
factors: (3×12)×10, (5×12)×10, (6×12)×10, (10×12)×10 and
(12×12)×10.

### Twenty

The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base-20 (vigesimal), possibly originating from the number of a person's fingers and toes. Evidence of base-20 counting systems is also found in the languages of central and western Africa.Possible remnants of a base-20 system also exist
in French, as seen in the names of the numbers from 60 through 99.
For example, sixty-five is soixante-cinq (literally, "sixty [and]
five"), while seventy-five is soixante-quinze (literally, "sixty
[and] fifteen"). Furthermore, for any number between 80 and 99, the
"tens-column" number is expressed as a multiple of twenty (somewhat
similar to the archaic English manner of speaking of "scores"). For
example, eighty-two is quatre-vingt-deux (literally, four twenty[s]
[and] two), while ninety-two is quatre-vingt-douze (literally, four
twenty[s] [and] twelve).

The Irish
language also used base-20 in the past, twenty being fichid,
forty dhá fhichid, sixty trí fhichid and eighty ceithre fhichid. A
remnant of this system may be seen in the modern word for 40,
daoichead.

Danish
numerals display a similar base-20 structure.

### Sixty

Base 60 (sexagesimal) was used by the Sumerians and their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). 60 also has a large number of factors, including the first six counting numbers. Base-60 systems are believed to have originated through the merging of base-10 and base-12 systems. The Chinese Calendar, for example, uses a base-60 Jia-Zi甲子 system to denote years, with each year within the 60-year cycle being named with two symbols, the first being base-10 (called Tian-Gan天干 or heavenly stems) and the second symbol being base 12 (called Di-Zhi地支 or earthly branches). Both symbols are incremented in successive years until the first pattern recurs 60 years later. The second symbol of this system is also related to the 12-animal Chinese zodiac system. The Jia-zi system can also be applied to counting days, with a year containing roughly six 60-day cycles.### Dual base (five and twenty)

Many ancient counting systems use 5 as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for 5 is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavigesimal. It is found in many languages of the Sudan region.### Base names

21 - unovigesimal / unobigesimal 22 -
duovigesimal 23 - triovigesimal 24 - quadrovigesimal /
quadriovigesimal 26 - hexavigesimal /
sexavigesimal 27 - heptovigesimal 28 - octovigesimal 29 -
novovigesimal 31 - unotrigesimal (...repeat naming pattern...) 36 -
hexatridecimal /
sexatrigesimal (...repeat naming pattern...) 41 - unoquadragesimal
(...repeat naming pattern...) 51 - unoquinquagesimal (...repeat
naming pattern...) 64 - quadrosexagesimal
(...repeat naming pattern...) 110 - decacentimal 111 -
unodecacentimal (...repeat naming pattern...) 210 - decabicentimal
211 - unodecabicentimal (...repeat naming pattern...) 800 -
octocentimal / octocentesimal 2000 - bimillesimal (...repeat naming
pattern...)

## Positional systems in detail

see also Positional notationIn a positional base-b numeral system (with b a
positive natural
number 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, we write 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:

(a_na_\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =
\sum_^n a_kb^k + \sum_^\infty c_kb^.

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 k = \log_ w = \log_ b^k. The highest used
position is close to the order of
magnitude of the number.

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 k + 1 = \log_ w + 1, for k \ge 0. E.g. to describe the
weight 1000 then 4 digits are needed since \log_ 1000 + 1 = 3 + 1.
The number of digits required to describe the position is \log_ k +
1 = \log_ \log_ w + 1 (in positions 1, 10, 100... only for
simplicity in the decimal example).

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 unperiodic (infinite amount of unrepeating
digits) in all integral bases. Thus, for example in base 2,
π = 3.1415926...10 can be
written down as the unperiodic 11.001001000011111...2.

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
numbers.

## Change of radix

A simple algorithm for converting integers between positive-integer radices is repeated division by the target radix; the remainders give the "digits" starting at the least significant. E.g., 1020304 base 10 into base 7: 1020304 / 7 = 145757 r 5 145757 / 7 = 20822 r 3 20822 / 7 = 2974 r 4 2974 / 7 = 424 r 6 424 / 7 = 60 r 4 60 / 7 = 8 r 4 8 / 7 = 1 r 1 1 / 7 = 0 r 1 => 11446435E.g., 10110111 base 2 into base 5: 10110111 / 101
= 100100 r 11 (3) 100100 / 101 = 111 r 1 (1) 111 / 101 = 1 r 10 (2)
1 / 101 = 0 r 1 (1) => 1213

To convert a "decimal" fraction, do repeated
multiplication, taking the protruding integer parts as the
"digits". Unfortunately a terminating fraction in one base may not
terminate in another. E.g., 0.1A4C base 16 into base 9: 0.1A4C × 9
= 0.ECAC 0.ECAC × 9 = 8.520C 0.520C × 9 = 2.E26C 0.E26C × 9 =
7.F5CC 0.F5CC × 9 = 8.A42C 0.A42C × 9 = 5.C58C =>
0.082785...

## Generalized variable-length integers

More general is using a notation (here written
little-endian) like a_0 a_1 a_2 for a_0 + a_1 b_1 + a_2 b_1
b_2, etc.

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 the 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 digit are c (2), then the
third digit has a weight 34 × 35 = 1190 and we have the
following sequence:

a (0), ba (1), ca (2), .., 9a (35), bb (36), cb
(37), .., 9b (70), bca (71), .., 99a (1260), bcb (1261), etc.

Note that unlike a regular base-35 numeral
system, we have numbers like 9b where 9 and b each represent 35;
yet the representation is unique because ac and aca are not
allowed.

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 nonzero.

### Properties of numerical systems with integer bases

Numeral systems with base A, where A is a
positive integer, possess the following properties:

- If A is even and A/2 is odd, all integral powers greater than zero of the number (A/2)+1 will contain (A/2)+1 as their last digit

- If both A and A/2 are even, then all integral powers greater than or equal to zero of the number (A/2)+1 will alternate between having (A/2)+1 and 1 as their last digit. (For odd powers it will be (A/2)+1, for even powers it will be 1)

Proof of the first property:

Define + 1 = x Then x is even, and all x^p for p
greater than 0 must be even. The property is equivalent to

- \!\ x^p \equiv\ x\ (\mbox\ A)

We first check the case for p=1

- \!\ x \equiv\ x\ (\mbox\ A)

x is less than A, so the result is trivial. We
then check for p=2:

- \!\ x^2 = xx
- \!\ x^2 = x(x-1) + x

Since x-1 = ( + 1) - 1 = , then for all even
N:

- \!\ = N(x-1) \equiv\ 0\ (\mbox\ A)\ (1)

Because x is even, then x(x-1) is congruent to
zero modulo A. Therefore:

- \!\ x^2 \equiv\ x\ (\mbox\ A)

Using induction, assuming that the property holds
for p-1:

- \!\ x^p = x = (x-1) + x^

Since the case holds for p-1, then \equiv\ x\
(\mbox\ A) . Since

- \!\ (x-1)

is a case of Equation 1, then (x-1) \equiv\ 0\
(\mbox\ A) . This leaves, for all p greater than 0,

- \!\ x^p \equiv\ x\ (\mbox\ A)

Proof of the second property:

Define + 1 = x Then x is odd, and all x^p for p
greater than or equal to 0 must be odd. The property is equivalent
to

- \!\ x^p \equiv\ 1\ (\mbox\ A);\ \mbox\ p \equiv\ 0\ (\mbox\ 2)
- \!\ x^p \equiv\ x\ (\mbox\ A);\ \mbox\ p \equiv\ 1\ (\mbox\ 2)

Since x-1 = ( + 1) - 1 = , then for all odd
E:

- \!\ = E(x-1) \equiv\ \ (\mbox\ A)\ (2)

The case is first checked for p=0:

- \!\ x^0 = 1
- \!\ 1 \equiv\ 1\ (\mbox\ A)

This result is trivial

Next, for p=1:

- \!\ x^1 = x
- \!\ x \equiv\ x\ (\mbox\ A)

This result is also trivial

Next, for p=2:

- \!\ x^2 = xx = x(x-1) + x

Because x is odd, then x(x-1) is a case of
Equation 2,

- x(x-1) + x \equiv\ \ (\mbox\ A)

- \!\ + x = + + 1 = A+1
- \!\ A+1 \equiv\ 1\ (\mbox\ A), (\mbox\ x(x-1) + x = x^2 \equiv\ 1\ (\mbox\ A)

Next, for p=3:

- \!\ x^3 = x = (x-1) + x^2

Because x^2 is odd, (x-1) + x^2 is a case of
Equation 2,

- \!\ (x-1) + x^2 \equiv\ \ (\mbox\ A)

Since x^2 \equiv\ 1\ (\mbox\ A) ,

- \!\ (x-1) + x^2 \equiv\ \ (\mbox\ A)

= x , so x^3 \equiv\ x\ (\mbox\ A) .

Using induction, assuming that the property holds
for p-1:

- \!\ x^p \equiv\ (x-1) + x^

If p is odd:

- \!\ x^ \equiv\ 1\ (\mbox\ A)

Since (x-1) is a case of Equation (2), (x-1) + x^
\equiv\ \ (\mbox\ A) , so

- x^p \equiv\ x\ (\mbox\ A)

If p is even:

- \!\ x^ \equiv\ x\ (\mbox\ A)

Since (x-1) is a case of Equation (2), (x-1) + x^
\equiv\ \ (\mbox\ A) .

+ x = + + 1 = A+1

A+1 \equiv\ 1\ (\mbox\ A) , so

- x^p \equiv\ 1\ (\mbox\ A)

## See also

- Babylonian numerals – a sexagesimal (base-60) system
- Computer numbering formats
- Golden ratio base
- List of numeral system topics
- Number names
- Quipu
- Recurring decimal
- Subtractive notation

## References

- Georges Ifrah. The Universal History of Numbers : From Prehistory to the Invention of the Computer, Wiley, 1999. ISBN 0-471-37568-3.
- D. Knuth. The Art of Computer Programming. Volume 2, 3rd Ed. Addison-Wesley. pp.194–213, "Positional Number Systems".
- J.P. Mallory and D.Q. Adams, Encyclopedia of Indo-European Culture, Fitzroy Dearborn Publishers, London and Chicago, 1997.
- Hans J. Nissen, P. Damerow, R. Englund, Archaic Bookkeeping, University of Chicago Press, 1993, ISBN 0-226-58659-6.
- Denise Schmandt-Besserat, How Writing Came About, University of Texas Press, 1992, ISBN 0-292-77704-3.
- Claudia Zaslavsky, Africa Counts: Number and Pattern in African Cultures, Lawrence Hill Books, 1999, ISBN 1-55652-350-5.

## External links

- Correspondences with numerals and letters (nine different alphabets)
- History of Counting and Numeral Systems-PlainMath.Net
- Online Converter for Different Numeral Systems (Base 2-36, JavaScript, GPL)
- Online Converter for Decimal/Roman Numerals (JavaScript, GPL)
- Number Sense & Numeration Lessons
- Counting Systems of Papua New Guinea
- Numerical Mechanisms and Children's Concept of Numbers
- Software for converting from one numeral system to another
- Numbers system conversion software

numeration in Arabic: أنظمة عد

numeration in Min Nan: Sò͘-jī

numeration in Belarusian (Tarashkevitsa):
Сыстэма зьлічэньня

numeration in Bosnian: Brojevni sistem

numeration in Bulgarian: Бройна система

numeration in Catalan: Sistema de
numeració

numeration in Czech: Číselná soustava

numeration in Chuvash: Шутлав йĕрки

numeration in Danish: Talsystem

numeration in German: Zahlensystem

numeration in Spanish: Sistema de
numeración

numeration in Esperanto: Cifereca sistemo

numeration in Basque: Zenbaki-sistema

numeration in French: Système de
numération

numeration in Galician: Sistema de
numeración

numeration in Korean: 기수법

numeration in Croatian: Brojevni sustav

numeration in Indonesian: Sistem bilangan

numeration in Italian: Sistema di
numerazione

numeration in Hebrew: שיטות ספירה

numeration in Hungarian: Számrendszerek

numeration in Malay (macrolanguage): Sistem
angka

numeration in Dutch: Talstelsel

numeration in Japanese: 位取り記数法

numeration in Norwegian: Tallsystem

numeration in Polish: System liczbowy

numeration in Portuguese: Sistema de
numeração

numeration in Romanian: Bază de numeraţie

numeration in Russian: Система счисления

numeration in Slovenian: Številski sistem

numeration in Serbo-Croatian: Brojevni
sistem

numeration in Finnish: Lukujärjestelmä

numeration in Swedish: Talsystem

numeration in Tamil: எண்ணுரு

numeration in Thai: ระบบเลข

numeration in Turkish: Sayı sistemi

numeration in Ukrainian: Система числення

numeration in Yiddish: נומערן סיסטעם

numeration in Chinese: 记数系统