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A. W. Faber Model 366
The A. W. Faber Model 366 was an unusual model of slide rule, manufactured in Germany by the A. W. Faber Company around 1909, with scales that followed a system invented by Johannes Schumacher (1858-1930) that used discrete logarithms to calculate products of integers without approximation. The Model 366 is notable for its table of numbers, mapping the numbers 1 to 100 to a permutation of the numbers 0 to 99 in a pattern based on discrete logarithms. The markings on the table are: : The slide rule has two scales on each side of the upper edge of the slider marked with the integers 1 to 100 in a different permuted order, evenly spaced apart. The ordering of the numbers on these scales is : which corresponds to the inverse permutation to the one given by the number table. There are also two scales on each side of the lower edge of the slider, consisting of the integers 0 to 100 similarly spaced, but in ascending order, with the zero on the lower scales lining up with the 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slide Rule
The slide rule is a mechanical analog computer which is used primarily for multiplication and division, and for functions such as exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which is usually performed using other methods. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results. Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form, with slide rule scales inscribed with standardized graduated markings. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule is closely related to nomograms used for application-specific computations. Though similar in name and appearance to a standard ruler, the slide rule is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Logarithms
In mathematics, for given real numbers ''a'' and ''b'', the logarithm log''b'' ''a'' is a number ''x'' such that . Analogously, in any group ''G'', powers ''b''''k'' can be defined for all integers ''k'', and the discrete logarithm log''b'' ''a'' is an integer ''k'' such that . In number theory, the more commonly used term is index: we can write ''x'' = ind''r'' ''a'' (mod ''m'') (read "the index of ''a'' to the base ''r'' modulo ''m''") for ''r''''x'' ≡ ''a'' (mod ''m'') if ''r'' is a primitive root of ''m'' and gcd(''a'',''m'') = 1. Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution. Definition Let ''G'' be any group. Denote its group operation by mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Permutation
Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when added to the original number, yields zero * Compositional inverse, a function that "reverses" another function * Inverse element * Inverse function, a function that "reverses" another function **Generalized inverse, a matrix that has some properties of the inverse matrix but not necessarily all of them * Multiplicative inverse (reciprocal), a number which when multiplied by a given number yields the multiplicative identity, 1 ** Inverse matrix of an Invertible matrix Other uses * Invert level, the base interior level of a pipe, trench or tunnel * ''Inverse'' (website), an online magazine * An outdated term for an LGBT person; see Sexual inversion (sexology) See also * Inversion (other) Inversion or inversions may refer to: Ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Indices
In mathematics, the ''Canon arithmeticus'' is a table of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by . The tables were at one time used for arithmetical calculations modulo prime powers, though like many mathematical tables they have now been replaced by digital computers. Jacobi also reproduced Burkhardt's table of the periods of decimal fractions of 1/''p'', and Ostrogradsky's tables of primitive roots of primes less than 200, and gave tables of indices of some odd numbers modulo powers of 2 with respect to the base 3 . Although the second edition of 1956 has Jacobi's name on the title, it has little in common with the first edition apart from the topic: the tables were completely recalculated, usually with a different choice of primitive root, by Wilhelm Patz. Jacobi's original tables use 10 or –10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Percy Ludgate
Percy Edwin Ludgate (2 August 1883 – 16 October 1922) was an Irish amateur scientist who designed the second analytical engine (general-purpose Turing-complete computer) in history. Life Ludgate was born on 2 August 1883 in Skibbereen, County Cork, to Michael Ludgate and Mary McMahon. In the 1901 census, he is listed as Civil Servant National Education (Boy Copyist) in Dublin. In the 1911 census, he is also in Dublin, as a Commercial Clerk (Corn Merchant). He studied accountancy at Rathmines College of Commerce, earning a gold medal based on the results his final examinations in 1917. At some date before or after then, he joined Kevans & Son, accountants. Work on analytical engine It seems that Ludgate worked as a clerk for an unknown corn merchants, in Dublin, and pursued his interest in calculating machines at night. Charles Babbage in 1843 and Ludgate in 1909 designed the only two mechanical analytical engines before the electromechanical analytical engine of Le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Remak (mathematician)
Robert Erich Remak (14 February 1888 – 13 November 1942) was a German mathematician. He is chiefly remembered for his work in group theory ( Remak decomposition). His other interests included algebraic number theory, mathematical economics and geometry of numbers. Robert Remak was the son of the neurologist Ernst Julius Remak and the grandson of the embryologist Robert Remak. Biography Robert Remak was born in Berlin. He studied at Humboldt University of Berlin under Ferdinand Georg Frobenius and received his doctorate in 1911. His dissertation, ''Über die Zerlegung der endlichen Gruppen in indirekte unzerlegbare Faktoren'' ("On the decomposition of a finite group into indirect indecomposable factors") established that any two decompositions of a finite group into a direct product are related by a central automorphism. A weaker form of this statement, uniqueness, was first proved by Joseph Wedderburn in 1909. Later the theorem was generalized by Wolfgang Krull and Otto S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irish Logarithms
Irish logarithms were a system of number manipulation invented by Percy Ludgate for machine multiplication. The system used a combination of mechanical cams as look-up tables and mechanical addition to sum pseudo-logarithmic indices to produce partial products, which were then added to produce results. The technique is similar to Zech logarithms (also known as Jacobi logarithms), but uses a system of indices original to Ludgate. Ludgate's algorithm compresses the multiplication of two single decimal numbers into two table lookups (to convert the digits into indices), the addition of the two indices to create a new index which is input to a second lookup table that generates the output product. Because both lookup tables are one-dimensional, and the addition of linear movements is simple to implement mechanically, this allows a less complex mechanism than would be needed to implement a two-dimensional 10x10 multiplication lookup table. Pseudocode The following is an implementat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canon Arithmeticus
In mathematics, the ''Canon arithmeticus'' is a table of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by . The tables were at one time used for arithmetical calculations modulo prime powers, though like many mathematical tables they have now been replaced by digital computers. Jacobi also reproduced Burkhardt's table of the periods of decimal fractions of 1/''p'', and Ostrogradsky's tables of primitive roots of primes less than 200, and gave tables of indices of some odd numbers modulo powers of 2 with respect to the base 3 . Although the second edition of 1956 has Jacobi's name on the title, it has little in common with the first edition apart from the topic: the tables were completely recalculated, usually with a different choice of primitive root, by Wilhelm Patz. Jacobi's original tables use 10 or –10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasionally written as Carolus Gustavus Iacobus Iacobi in his Latin books, and his first name is sometimes given as Karl. Jacobi was the first Jewish mathematician to be appointed professor at a German university. Biography Jacobi was born of Ashkenazi Jewish parentage in Potsdam on 10 December 1804. He was the second of four children of banker Simon Jacobi. His elder brother Moritz von Jacobi would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oughtred Society
William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, ''A General Dictionary, Historical and Critical'', (James Bettenham, for G. Strachan and J. Clarke, London 1734/1739), Vol. VIIIpp. 77-86(Google). After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, Oughtred was the first to use two such scales sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622. He also introduced the "×" symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions. Clerical life Education The son of Benjamin Oughtred of Eton in Buckinghamshire (now part of Berkshire), William was born there on 5 March 1574/75 and was educated at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
