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Cyclic Number
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are :142857 × 1 = 142857 :142857 × 2 = 285714 :142857 × 3 = 428571 :142857 × 4 = 571428 :142857 × 5 = 714285 :142857 × 6 = 857142 Details To qualify as a cyclic number, it is required that consecutive multiples be cyclic permutations. Thus, the number 076923 would not be considered a cyclic number, because even though all cyclic permutations are multiples, they are not consecutive integer multiples: :076923 × 1 = 076923 :076923 × 3 = 230769 :076923 × 4 = 307692 :076923 × 9 = 692307 :076923 × 10 = 769230 :076923 × 12 = 923076 The following trivial cases are typically excluded: #single digits, e.g.: 5 #repeated digits, e.g.: 555 #repeated cyclic numbers, e.g.: 142857142857 If leading zeros are no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
142857 (number)
142,857 is the natural number following 142,856 and preceding 142,858. It is both a Kaprekar number and a Cyclic number. Cyclic number 142857 is the best-known cyclic number in base 10, being the six repeating digits of (0.). If 142857 is multiplied by 2, 3, 4, 5 or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of , , , or respectively: : 1 × 142,857 = 142,857 : 2 × 142,857 = 285,714 : 3 × 142,857 = 428,571 : 4 × 142,857 = 571,428 : 5 × 142,857 = 714,285 : 6 × 142,857 = 857,142 : 7 × 142,857 = 999,999 If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 142857: : 142857 × 8 = 1142856 : 1 + 142856 = 142857 : 142857 × 815 = 116428455 : 116 + 428455 = 428571 : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Senary
A senary () numeral system (also known as base-6, heximal, or seximal) has 6, six as its radix, 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 (mathematics), 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 Kleene closure of , where is the operation of string concatenation for . The senary number system for natural numbers is the quotient set equipped with a shortlex order, where the equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quinary
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 hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternary Numeral System
Quaternary is a numeral system with four as its base. It uses the digits 0, 1, 2, and 3 to represent any real number. Conversion from binary is straightforward. Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being thirty-six), making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers (the smallest better base being the primorial base six, senary). Quaternary shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties. Relation to other positional number systems Relation to binary and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ternary Numeral System
A ternary numeral system (also called base 3 or trinary) has 3 (number), three as its radix, base. Analogous to a bit, a ternary numerical digit, digit is a trit (trinary digit). One trit is equivalent to binary logarithm, log2 3 (about 1.58496) bits of Units of information, 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 numeral system, binary. For example, decimal 365 (number), 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 codi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Numeral System
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. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harrio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midy's Theorem
In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions ''a''/''p'' where ''p'' is a prime and ''a''/''p'' has a repeating decimal expansion with an even period . If the period of the decimal representation of ''a''/''p'' is 2''n'', so that \frac=0.\overline then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words, a_i+a_=9 a_1\dots a_n+a_\dots a_=10^n-1. For example, \frac=0.\overline\text076+923=999. \frac=0.\overline\text05882352+94117647=99999999. Extended Midy's theorem If ''k'' is any divisor of ''h'' (where ''h'' is the number of digits of the period of the decimal expansion of ''a''/''p'' (where ''p'' is again a prime)), then Midy's theorem can be generalised as follows. The extended Midy's theoremBassam Abdul-Baki''Extended Midy's Theorem'' 2005. states that if the repeating portion of the decima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Remainder
In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient ( integer division). In algebra of polynomials, the remainder is the polynomial "left over" after dividing one polynomial by another. The ''modulo operation'' is the operation that produces such a remainder when given a dividend and divisor. Alternatively, a remainder is also what is left after subtracting one number from another, although this is more precisely called the '' difference''. This usage can be found in some elementary textbooks; colloquially it is replaced by the expression "the rest" as in "Give me two dollars back and keep the rest." However, the term "remainder" is still used in this sense when a function is approximated by a series expansion, where the error expression ("the rest") is referred to as the remainder term. Integer division Gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Long Division
In arithmetic, long division is a standard division algorithm suitable for dividing multi-digit Hindu-Arabic numerals (positional notation) that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. History Related algorithms have existed since the 12th century. Al-Samawal al-Maghribi (1125–1174) performed calculations with decimal numbers that essentially require long division, leading to infinite decimal results, but without formalizing the algorithm. Caldrini (1491) is the earliest printed example of long division, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modulo Operation
In computing and mathematics, the modulo operation returns the remainder or signed remainder of a Division (mathematics), division, after one number is divided by another, the latter being called the ''modular arithmetic, modulus'' of the operation. Given two positive numbers and , modulo (often abbreviated as ) is the remainder of the Euclidean division of by , where is the Division (mathematics), dividend and is the divisor. For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to . mod 1 is always 0. When exactly one of or is negative, the basic definition breaks down, and programming languages differ in how these valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
