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Perfect Digital Invariant
In number theory, a perfect digital invariant (PDI) is a number in a given number base (b) that is the sum of its own digits each raised to a given power (p). 0 F_ : \mathbb \rightarrow \mathbb is defined as: :F_(n) = \sum_^ d_i^p. where k = \lfloor \log_ \rfloor + 1 is the number of digits in the number in base b, and :d_i = \frac is the value of each digit of the number. A natural number n is a perfect digital invariant if it is a fixed point for F_, which occurs if F_(n) = n. 0 and 1 are trivial perfect digital invariants for all b and p, all other perfect digital invariants are nontrivial perfect digital invariants. For example, the number 4150 in base b = 10 is a perfect digital invariant with p = 5, because 4150 = 4^5 + 1^5 + 5^5 + 0^5. A natural number n is a sociable digital invariant if it is a periodic point for F_, where F_^k(n) = n for a positive integer k (here F_^k is the kth iterate of F_), and forms a cycle of period k. A perfect digital invariant is a sociable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer n is divisible by a nonzero integer m if there exists an integer k such that n=km. This is written as : m\mid n. This may be read as that m divides n, m is a divisor of n, m is a factor of n, or n is a multiple of m. If m does not divide n, then the notation is m\not\mid n. There are two conventions, distinguished by whether m is permitted to be zero: * With the convention without an additional constraint on m, m \mid 0 for every integer m. * With the convention that m be nonzero, m \mid 0 for every nonzero integer m. General Divisors can be negative as well as positive, although often the term is restricted to posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed-digit Representation
In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because it can eliminate chains of dependent carries. In the binary numeral system, a special case signed-digit representation is the '' non-adjacent form'', which can offer speed benefits with minimal space overhead. History Challenges in calculation stimulated early authors Colson (1726) and Cauchy (1840) to use signed-digit representation. The further step of replacing negated digits with new ones was suggested by Selling (1887) and Cajori (1928). In 1928, Florian Cajori noted the recurring theme of signed digits, starting with Colson (1726) and Cauchy (1840). In his book ''History of Mathematical Notations'', Cajori titled the section "Negative numerals". For completeness, Colson uses examples and describes addition (pp. 163–4), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-12
The duodecimal system, also known as base twelve or dozenal, is a positional numeral system using twelve as its base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 units; in the decimal system, this number is instead written as "12" meaning 1 ten and 2 units, and the string "10" means ten. In duodecimal, "100" means twelve squared (144), "1,000" means twelve cubed (1,728), and "0.1" means a twelfth (0.08333...). Various symbols have been used to stand for ten and eleven in duodecimal notation; this page uses and , as in hexadecimal, which make a duodecimal count from zero to twelve read 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , , and finally 10. The Dozenal Societies of America and Great Britain (organisations promoting the use of duodecimal) use turned digits in their published material: (a turned 2) for ten (dek, pronounced dɛk) and (a turned 3) for eleven (el, pronounced ɛl). The number twelve, a superior highly composite number, is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-9
A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information. Although ''ternary'' most often refers to a system in which the three digits are all non–negative numbers; specifically , , and , the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers. Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 or senary corresponds to binary (nine bits) and to ternary (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27). : : : As for rational numbers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-6
A senary () numeral system (also known as base-6, heximal, or seximal) has six as its base. It has been adopted independently by a small number of cultures. Like the decimal base 10, the base is a semiprime, though it is unique as the product of the only two consecutive numbers that are both prime (2 and 3). As six is a superior highly composite number, many of the arguments made in favor of the duodecimal system also apply to the senary system. Formal definition The standard set of digits in the senary system is \mathcal_6 = \lbrace 0, 1, 2, 3, 4, 5\rbrace, with the linear order 0 < 1 < 2 < 3 < 4 < 5. Let be the of , where is the operation of |
Base-3
A ternary numeral system (also called base 3 or trinary) has three as its base. Analogous to a bit, a ternary digit is a trit (trinary digit). One trit is equivalent to log2 3 (about 1.58496) bits of information. Although ''ternary'' most often refers to a system in which the three digits are all non–negative numbers; specifically , , and , the adjective also lends its name to the balanced ternary system; comprising the digits −1, 0 and +1, used in comparison logic and ternary computers. Comparison to other bases Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 or senary corresponds to binary (nine bits) and to ternary (six digits). However, they are still far less compact than the corresponding representations in bases such as decimal – see below for a compact way to codify ternary using nonary (base 9) and septemvigesimal (base 27). : : : As for ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-20
A vigesimal ( ) or base-20 (base-score) numeral system is based on twenty (in the same way in which the decimal numeral system is based on ten). ''Vigesimal'' is derived from the Latin adjective , meaning 'twentieth'. Places In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A, or , where the 20 means base , to write nineteen as , and the numbers between with the corresponding letters of the alphabet. This is similar to the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A–F". Another less common method skips over the letter "I", in order to avoid confusion between I20 as eighteen and one, so that the number eighteen is written as J20, and nineteen is written as K20. The number twenty is written as . According to this notation: : is equivalent to forty in decimal = : is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-8
Octal (base 8) is a numeral system with eight as the base. In the decimal system, each place is a power of ten. For example: : \mathbf_ = \mathbf \times 10^1 + \mathbf \times 10^0 In the octal system, each place is a power of eight. For example: : \mathbf_8 = \mathbf \times 8^2 + \mathbf \times 8^1 + \mathbf \times 8^0 By performing the calculation above in the familiar decimal system, we see why 112 in octal is equal to 64+8+2=74 in decimal. Octal numerals can be easily converted from binary representations (similar to a quaternary numeral system) by grouping consecutive binary digits into groups of three (starting from the right, for integers). For example, the binary representation for decimal 74 is 1001010. Two zeroes can be added at the left: , corresponding to the octal digits , yielding the octal representation 112. Usage In China The eight bagua or trigrams of the I Ching correspond to octal digits: * 0 = ☷, 1 = ☳, 2 = ☵, 3 = ☱, * 4 = ☶, 5 = � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-5
Quinary (base 5 or pental) is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand. In the quinary place system, five numerals, from 0 to 4, are used to represent any real number. According to this method, five is written as 10, twenty-five is written as 100, and sixty is written as 220. As five is a prime number, only the reciprocals of the powers of five terminate, although its location between two highly composite numbers ( 4 and 6) guarantees that many recurring fractions have relatively short periods. Comparison to other radices Usage Many languages use quinary number systems, including Gumatj, Nunggubuyu, Kuurn Kopan Noot, Luiseño, and Saraveca. Gumatj has been reported to be a true "5–25" language, in which 25 is the higher group of 5. The Gumatj numerals are shown below: However, Harald Hammarström reports that "one would not usually use exact numbers for counting this hig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-16
Hexadecimal (also known as base-16 or simply hex) 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. Software developers and system designers widely use hexadecimal numbers because they provide a convenient representation of binary-coded values. Each hexadecimal digit represents four bits (binary digits), also known as a nibble (or nybble). For example, an 8-bit byte is two hexadecimal digits and its value can be written as to in hexadecimal. In mathematics, a subscript is typically used to specify the base. For example, the decimal value would be expressed in hexadecimal as . In programming, several notations denote hexadecimal numbers, usually involving a prefix. The prefix 0x is used in C, which would denote this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Base-10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as ''decimal notation''. A decimal numeral (also often just ''decimal'' or, less correctly, ''decimal number''), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in or ). ''Decimal'' may also refer specifically to the digits after the decimal separator, such as in " is the approximation of to ''two decimals''". Zero-digits after a decimal separator serve the purpose of signifying the precision of a value. The numbers that may be represented in the decimal system are the decimal fractions. That is, fractions of the form , where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
