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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 graphemes or symbols in a consistent manner.It can be seen as the context that allows the numerals "11" to be interpreted as the binary symbol for three, the decimal symbol for eleven, or as other numbers in different bases.

Ideally, a numeral system will:
• Represent a useful set of numbers (e.g. all whole numbers, integers, or rational 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 digit. However, when decimal representation is used for the rational or real numbers, the representation may not be 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 for some scientific contexts where greater precision is implied 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 Indian mathematicians are credited with 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. 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 was of this type, and the 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 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 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 x 20)) or (somewhat archaic) pedwar ugain namyn un (4 x 20 - 1)

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, 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 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 systems 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 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

In 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_{n-1}\cdots a_1a_0.c_1 c_2 c_3\cdots)_b =\sum_{k=0}^n a_kb^k + \sum_{k=1}^\infty c_kb^{-k}.

The numbers bk and bk are the weight of the corresponding digits. The position k is the logarithm of the corresponding weight w, that is k = \log_{b} w = \log_{b} 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_{b} w + 1, for k \ge 0. E.g. to describe the weight 1000 then four digits are needed since \log_{10} 1000 + 1 = 3 + 1. The number of digits required to describe the position is \log_{b} k + 1 = \log_{b} \log_{b} w + 1 (in positions 1, 10, 100... only for simplicity in the decimal example).

 Position Weight Digit Decimal example weight Decimal example digit 3 2 1 0 -1 -2 ... b^3 b^2 b^1 b^0 b^{-1} b^{-2} ... a_3 a_2 a_1 a_0 c_1 c_2 ... 1000 100 10 1 0.1 0.01 ... 4 3 2 7 0 0 ...

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.

Putting overscores, , or dots,n, above the common digits is a convention used to represent repeating rational expansions. Thus:
14/11 = 1.272727272727... = 1.   or   321.3217878787878... = 321.32178 .

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.

## 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.

Unlike a regular based 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 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.

## 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".
• 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, 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.