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 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 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 this article. Contents 1 Main numeral systems 2 Positional systems in detail 3 Generalized variable-length integers 4 See also 5 References 6 Sources 7 External links Main numeral systems[edit]
Main article: List of numeral systems
The most commonly used system of numerals is the Hindu–Arabic
numeral system.[1] Two
( a n a n − 1 ⋯ a 1 a 0 . c 1 c 2 c 3 ⋯ ) b = ∑ k = 0 n a k b k + ∑ k = 1 ∞ c k b − k . 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 k = log b w = log b b k displaystyle 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 displaystyle k+1=log _ b w+1 , for k ≥ 0. For example, to describe the weight 1000 then four digits are needed because log 10 1000 + 1 = 3 + 1 displaystyle 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 displaystyle log _ b k+1=log _ b log _ b w+1 (in positions 1, 10, 100,... only for simplicity in the decimal example). Position 3 2 1 0 − 1 − 2 ⋯ Weight b 3 b 2 b 1 b 0 b − 1 b − 2 ⋯ Digit 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 ⋯ 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
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 aperiodic 11.001001000011111...2. 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... = 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[edit] More general is using a mixed radix notation (here written little-endian) like a 0 a 1 a 2 displaystyle a_ 0 a_ 1 a_ 2 for a 0 + a 1 b 1 + a 2 b 1 b 2 displaystyle 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, 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 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, 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. See also[edit] List of numeral systems
Computer numbering formats
Golden ratio base
History of ancient numeral systems
History of numbers
List of numeral system topics
n-ary
References[edit] ^ David Eugene Smith; Louis Charles Karpinski (1911). The Hindu-Arabic numerals. Ginn and Company. Sources[edit] 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
External links[edit] 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 Authority control GND: 41177 |