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Transposable Integer
In mathematics, the transposable integers are integers that permute or shift cyclically when they are multiplied by another integer n. Examples are: *142857 × 3 = 428571 (shifts cyclically one place left) *142857 × 5 = 714285 (shifts cyclically one place right) *128205 × 4 = 512820 (shifts cyclically one place right) *076923 × 9 = 692307 (shifts cyclically two places left) These transposable integers can be but are not always cyclic numbers. The characterization of such numbers can be done using repeating decimals (and thus the related fractions), or directly. General For any integer coprime to 10, its reciprocal is a repeating decimal without any non-recurring digits. E.g. = 0.006993... While the expression of a single series with vinculum on top is adequate, the intention of the above expression is to show that the six cyclic permutations of 006993 can be obtained from this repeating decimal if we select six consecutive digits from the repeating ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factor
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number , called trial division, tests whether is a multiple of any integer between 2 and . Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David G
David (; , "beloved one") was a king of ancient Israel and Judah and the Kings of Israel and Judah, third king of the Kingdom of Israel (united monarchy), United Monarchy, according to the Hebrew Bible and Old Testament. The Tel Dan stele, an Canaanite and Aramaic inscriptions, Aramaic-inscribed stone erected by a king of Aram-Damascus in the late 9th/early 8th centuries BCE to commemorate a victory over two enemy kings, contains the phrase (), which is translated as "Davidic line, House of David" by most scholars. The Mesha Stele, erected by King Mesha of Moab in the 9th century BCE, may also refer to the "House of David", although this is disputed. According to Jewish works such as the ''Seder Olam Rabbah'', ''Seder Olam Zutta'', and ''Sefer ha-Qabbalah'' (all written over a thousand years later), David ascended the throne as the king of Judah in 885 BCE. Apart from this, all that is known of David comes from biblical literature, Historicity of the Bible, the historicit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford A
Clifford may refer to: People * Clifford (name), an English given name and surname, includes a list of people with that name *William Kingdon Clifford * Baron Clifford *Baron Clifford of Chudleigh *Baron de Clifford * Clifford baronets * Clifford family (bankers) * Jaryd Clifford * Justice Clifford (other) * Lord Clifford (other) Arts, entertainment, and media *''Clifford the Big Red Dog'', a series of children's books ** Clifford (character), the central character of ''Clifford the Big Red Dog'' ** ''Clifford the Big Red Dog'' (2000 TV series), 2000 animated TV series **'' Clifford's Puppy Days'', 2003 animated TV series **'' Clifford's Really Big Movie'', 2004 animated movie ** ''Clifford the Big Red Dog'' (2019 TV series), 2019 animated TV series ** ''Clifford the Big Red Dog'' (film), 2021 live-action movie * ''Clifford'' (film), a 1994 film directed by Paul Flaherty * Clifford (Muppet) Mathematics *Clifford algebra, a type of associative algebra, named after ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duodecimal
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parasitic Number
In mathematics, an ''n''-parasitic number (in base 10) is a positive natural number which, when multiplied by ''n'', results in movement of the last digit of its decimal representation to its front. Here ''n'' is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift by one place. For example: :4 × 128205 = 512820, so 128205 is 4-parasitic. Most mathematicians do not allow leading zeros to be used, and that is a commonly followed convention. So even though 4 × 25641 = 102564, the number 25641 is ''not'' 4-parasitic. Derivation An ''n''-parasitic number can be derived by starting with a digit ''k'' (which should be equal to ''n'' or greater) in the rightmost (units) place, and working up one digit at a time. For example, for ''n'' = 4 and ''k'' = 7 :4 × 7 = 28 :4 × 87 = 348 :4 × 487 = 1948 :4 × 9487 = 37948 :4 × 79487 = 317948 :4& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Full Reptend Prime
In number theory, a full reptend prime, full repetend prime, proper primeDickson, Leonard E., 1952, ''History of the Theory of Numbers, Volume 1'', Chelsea Public. Co. or long prime in base ''b'' is an odd prime number ''p'' such that the Fermat quotient : q_p(b) = \frac (where ''p'' does not divide ''b'') gives a cyclic number. Therefore, the base ''b'' expansion of 1/p repeats the digits of the corresponding cyclic number infinitely, as does that of a/p with rotation of the digits for any ''a'' between 1 and ''p'' − 1. The cyclic number corresponding to prime ''p'' will possess ''p'' − 1 digits if and only if ''p'' is a full reptend prime. That is, the multiplicative order = ''p'' − 1, which is equivalent to ''b'' being a primitive root modulo ''p''. The term "long prime" was used by John Conway and Richard Guy in their ''Book of Numbers''. Base 10 < ...
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Parasitic Numbers
In mathematics, an ''n''-parasitic number (in base 10) is a positive natural number which, when multiplied by ''n'', results in movement of the last digit of its decimal representation to its front. Here ''n'' is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift by one place. For example: :4 × 128205 = 512820, so 128205 is 4-parasitic. Most mathematicians do not allow leading zeros to be used, and that is a commonly followed convention. So even though 4 × 25641 = 102564, the number 25641 is ''not'' 4-parasitic. Derivation An ''n''-parasitic number can be derived by starting with a digit ''k'' (which should be equal to ''n'' or greater) in the rightmost (units) place, and working up one digit at a time. For example, for ''n'' = 4 and ''k'' = 7 :4 × 7 = 28 :4 × 87 = 348 :4 × 487 = 1948 :4 × 9487 = 37948 :4 × 79487 = 317948 :4& ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repeating Decimal
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be ''terminating'', and is not considered as repeating. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is , whose decimal becomes periodic at the ''second'' digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... Another example of this is , which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830.... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Permutation
In mathematics, and in particular in group theory, a cyclic permutation is a permutation consisting of a single cycle. In some cases, cyclic permutations are referred to as cycles; if a cyclic permutation has ''k'' elements, it may be called a ''k''-cycle. Some authors widen this definition to include permutations with fixed points in addition to at most one non-trivial cycle. In cycle notation, cyclic permutations are denoted by the list of their elements enclosed with parentheses, in the order to which they are permuted. For example, the permutation (1 3 2 4) that sends 1 to 3, 3 to 2, 2 to 4 and 4 to 1 is a 4-cycle, and the permutation (1 3 2)(4) that sends 1 to 3, 3 to 2, 2 to 1 and 4 to 4 is considered a 3-cycle by some authors. On the other hand, the permutation (1 3)(2 4) that sends 1 to 3, 3 to 1, 2 to 4 and 4 to 2 is not a cyclic permutation because it separately permutes the pairs and . For the wider definition of a cyclic permutation, allowing fixed points, these fixe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
