Non-standard positional numeral systems here designates numeral systems that may loosely be described as positional systems, but that do not entirely comply with the following description of standard positional systems:
In a standard positional numeral system, the baseb is a positive integer, and b different numerals are used to represent all non-negativeintegers. The standard set of numerals contains the b values 0, 1, 2, etc., up to b−1, but the value is weighted according to the position of the digit in a number. The value of a digit string like pqrs in base b is given by the polynomial form
.
The numbers written in superscript represent the powers of the base used.
For instance, in hexadecimal (b=16), using the numerals A for 10, B for 11 etc., the digit string 7A3F means
,
which written in our normal decimal notation is 31295.
This article summarizes facts on some non-standard positional numeral systems. In most cases, the polynomial form in the description of standard systems still applies.
Some historical numeral systems may be described as non-standard positional numeral systems. E.g., the sexagesimalBabylonian notation and the Chinese rod numerals, which can be classified as standard systems of base 60 and 10, respectively, counting the space representing zero as a numeral, can also be classified as non-standard systems, more specifically, mixed-base systems with unary components, considering the primitive repeated glyphs making up the numerals.
However, most of the non-standard systems listed below have never been intended for general use, but were devised by mathematicians or engineers for special academic or technical use.
Bijective numeration systems
A bijective numeral system with base b uses b different numerals to represent all non-negative integers. However, the numerals have values 1, 2, 3, etc. up to and including b, whereas zero is represented by an empty digit string. For example, it is possible to have decimal without a zero.
Unary is the bijective numeral system with base b=1. In unary, one numeral is used to represent all positive integers. The value of the digit string pqrs given by the polynomial form can be simplified into p + q + r + s since bn=1 for all n. Non-standard features of this system include:
The value of a digit does not depend on its position. Thus, one can easily argue that unary is not a positional system at all.
Introducing a radix point in this system will not enable representation of non-integer values.
The single numeral represents the value 1, not the value 0=b−1.
The value 0 cannot be represented (or is implicitly represented by an empty digit string).
In some systems, while the base is a positive integer, negative digits are allowed. Non-adjacent form is a particular system where the base is b=2. In the balanced ternary system, the base is b=3, and the numerals have the values −1, 0 and +1 (rather than 0, 1 and 2 as in the standard ternary system, or 1, 2 and 3 as in the bijective ternary system).
The reflected binary code, also known as the Gray code, is closely related to binary numbers, but some bits are inverted, depending on the parity of the higher order bits.
Graphical and physical variants
Cistercian numerals are a decimal positional numeral system, but the positions are not aligned as in common decimal notation; instead, they are attached to the top-right, top-left, bottom-right and bottom-left of a vertical stem, respectively, and thus limited to four in number (so only integers from 0 to 9999 can be represented). The system has close similarities to standard positional numeral systems, but may also be compared to e.g. Greek numerals, where different sets of symbols (in fact, Greek letters) are used for the ones, tens, hundreds and thousands, likewise giving an upper limit on the numbers that can be represented.
Similarly, in computers, e.g. the long integer format is a standard binary system (apart from the sign bit), but it has a limited number of positions, and the physical locations for the representations of the digits may not be aligned. In an analog odometer, the decimal digits are aligned but limited in number.
Bases that are not positive integers
A few positional systems have been suggested in which the base b is not a positive integer.
Negative-base systems include negabinary, negaternary and negadecimal, with bases −2, −3, and −10 respectively; in base −b the number of different numerals used is b. Due to the properties of negative numbers raised to powers, all integers, positive and negative, can be represented without a sign.
In a purely imaginary base bi system, where b is an integer larger than 1 and i the imaginary unit, the standard set of digits consists of the b2 numbers from 0 to b2 − 1. It can be generalized to other complex bases, giving rise to the complex-base systems.
In non-integer bases, the number of different numerals used clearly cannot be b. Instead, the numerals 0 to are used. For example, golden ratio base (phinary), uses the 2 different numerals 0 and 1.
Mixed bases
A mixed base system one may already be familiar with is the measurement of time in mix of base-12 (for hours) & base-60 (for minutes & seconds), although the measurement is often reported in terms of base-10, e.g. 20:00:00 commonly represents twenty hours (past midnight).
It is sometimes convenient to consider positional numeral systems where the weights associated with the positions do not form a geometric sequence 1, b, b2, b3, etc., starting from the least significant position, as given in the polynomial form. In a mixed-radix system such as the factorial number system, the weights form a sequence where each weight is an integer multiple of the previous one, and the number of permitted digit values varies accordingly from position to position.
For calendrical use, the Mayan numeral system was a mixed-radix system, since one of its positions represents a multiplication by 18 rather than 20, in order to fit a 360-day calendar. Also, giving an angle in degrees, minutes and seconds (with decimals), or a time in days, hours, minutes and seconds, can be interpreted as mixed-radix systems.
Sequences where each weight is not an integer multiple of the previous weight may also be used, but then every integer may not have a unique representation. For example, Fibonacci coding uses the digits 0 and 1, weighted according to the Fibonacci sequence (1, 2, 3, 5, 8, ...); a unique representation of all non-negative integers may be ensured by forbidding consecutive 1s. Binary-coded decimal (BCD) are mixed base systems where bits (binary digits) are used to express decimal digits. E.g., in 1001 0011, each group of four bits may represent a decimal digit (in this example 9 and 3, so the eight bits combined represent decimal 93). The weights associated with these 8 positions are 80, 40, 20, 10, 8, 4, 2 and 1. Uniqueness is ensured by requiring that, in each group of four bits, if the first bit is 1, the next two must be 00.
Asymmetric numeral systems are systems used in computer science where each digit can have different bases, usually non-integer. In these, not only are the bases of a given digit different, they can be also nonuniform and altered in an asymmetric way to encode information more efficiently. They are optimized for chosen non-uniform probability distributions of symbols, using on average approximately Shannon entropy bits per symbol.[1]
The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.
Hexadecimal is a positional numeral system that represents numbers using a radix (base) of sixteen. Unlike the decimal system representing numbers using ten symbols, hexadecimal uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values 0 to 9 and "A"–"F" to represent values from ten to fifteen.
A numeral system 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.
Octal is a numeral system with eight as the base.
The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number N, a symbol representing 1 is repeated N times.
Babylonian cuneiform numerals, also used in Assyria and Chaldea, were written in cuneiform, using a wedge-tipped reed stylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.
Unary coding, or the unary numeral system and also sometimes called thermometer code, is an entropy encoding that represents a natural number, n, with a code of length n + 1, usually n ones followed by a zero or with n − 1 ones followed by a zero. For example 5 is represented as 111110 or 11110. Some representations use n or n − 1 zeros followed by a one. The ones and zeros are interchangeable without loss of generality. Unary coding is both a prefix-free code and a self-synchronizing code.
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A binary number may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two.
Mixed radix numeral systems are non-standard positional numeral systems in which the numerical base varies from position to position. Such numerical representation applies when a quantity is expressed using a sequence of units that are each a multiple of the next smaller one, but not by the same factor. Such units are common for instance in measuring time; a time of 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be expressed as a number of minutes in mixed-radix notation as:
A numerical digit or numeral is a single symbol used alone or in combinations, to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits.
Positional notation, also known as place-value notation, positional numeral system, or simply place value, usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.
Bijective numeration is any numeral system in which every non-negative integer can be represented in exactly one way using a finite string of digits. The name refers to the bijection that exists in this case between the set of non-negative integers and the set of finite strings using a finite set of symbols.
Finger binary is a system for counting and displaying binary numbers on the fingers of either or both hands. Each finger represents one binary digit or bit. This allows counting from zero to 31 using the fingers of one hand, or 1023 using both: that is, up to 25−1 or 210−1 respectively.
Base36 is a binary-to-text encoding scheme that represents binary data in an ASCII string format by translating it into a radix-36 representation. The choice of 36 is convenient in that the digits can be represented using the Arabic numerals 0–9 and the Latin letters A–Z.
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of
A negative base may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base b is equal to −r for some natural number r.
In computing, bit numbering is the convention used to identify the bit positions in a binary number.
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