They are not transparent when it comes to their logarithms.
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This seemingly basic problem begs for the introduction of logarithms.
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I remember learning a concept, logarithms, and when I learned it in the previous math class I took, I had no clue what was going on because of how fast the teacher taught the lesson.
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She exhibited a particular talent for science and mathematics — one of the earliest documents here is a ledger of logarithms, meant for determining syzygies of the moon and sun, that she worked out at age 14.
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The existence of e is implicit in John Napier's 1614 work on logarithms, and natural logarithms are sometimes inexactly dubbed Napierian logarithms.
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Common logarithms are sometimes also called "Briggsian logarithms" after Henry Briggs, a 17th- century British mathematician. In 1616 and 1617, Briggs visited John Napier at Edinburgh, the inventor of what are now called natural (base-e) logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first chiliad of his logarithms. Because base 10 logarithms were most useful for computations, engineers generally simply wrote "log(x)" when they meant log10(x).
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By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions. Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well. For the history of such tables, see log table.
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Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha. Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl G. J. Jacobi who used them for number theoretic investigations.
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In 1544, Michael Stifel published Arithmetica integra, which contains a table of integers and powers of 2 that has been considered an early version of a logarithmic table. The method of logarithms was publicly propounded by John Napier in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). The book contained fifty-seven pages of explanatory matter and ninety pages of tables related to natural logarithms. The English mathematician Henry Briggs visited Napier in 1615, and proposed a re-scaling of Napier's logarithms to form what is now known as the common or base-10 logarithms.
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The median of the natural logarithms of a sample is a robust estimator of \mu. The median absolute deviation of the natural logarithms of a sample is a robust estimator of \sigma.
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The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter as the base of natural logarithms. Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example).
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He later computed a new table of logarithms formatted in base 10.
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Title page of John Napier's Logarithmorum from 1620 The history of logarithms is the story of a correspondence (in modern terms, a group isomorphism) between multiplication on the positive real numbers and addition on the real number line that was formalized in seventeenth century Europe and was widely used to simplify calculation until the advent of the digital computer. The Napierian logarithms were published first in 1614. Henry Briggs introduced common (base 10) logarithms, which were easier to use. Tables of logarithms were published in many forms over four centuries.
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Napier delegated to Briggs the computation of a revised table. In 1617, they published Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a table for the first 1000 integers calculated to the 14th decimal place. The computational advance available via common logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker.
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Tables containing common logarithms (base-10) were extensively used in computations prior to the advent of electronic calculators and computers because logarithms convert problems of multiplication and division into much easier addition and subtraction problems. Base-10 logarithms have an additional property that is unique and useful: The common logarithm of numbers greater than one that differ only by a factor of a power of ten all have the same fractional part, known as the mantissa. Tables of common logarithms typically included only the mantissas; the integer part of the logarithm, known as the characteristic, could easily be determined by counting digits in the original number. A similar principle allows for the quick calculation of logarithms of positive numbers less than 1.
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Henry Briggs (February 1561 – 26 January 1630) was an English mathematician notable for changing the original logarithms invented by John Napier into common (base 10) logarithms, which are sometimes known as Briggsian logarithms in his honour. Briggs was a committed PuritanDavid C. Lindberg, Ronald L. Numbers (1986). "God and Nature", p. 201.Cedric Clive Brown (1993), "Patronage, Politics, and Literary Traditions in England, 1558-1658", Wayne State University Press. p.
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A key tool that enabled the practical use of logarithms was the table of logarithms., section 2 The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written.
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The symbol e for the base of the Napierian logarithms was introduced by Euler.
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The BKM algorithm is a shift-and-add algorithm for computing elementary functions, first published in 1994 by Jean-Claude Bajard, Sylvanus Kla, and Jean-Michel Muller. BKM is based on computing complex logarithms (L-mode) and exponentials (E-mode) using a method similar to the algorithm Henry Briggs used to compute logarithms. By using a precomputed table of logarithms of negative powers of two, the BKM algorithm computes elementary functions using only integer add, shift, and compare operations. BKM is similar to CORDIC, but uses a table of logarithms rather than a table of arctangents.
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We may ignore any powers of inside of the logarithms. The set is exactly the same as . The logarithms differ only by a constant factor (since ) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent.
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Because base–emitter voltage varies as the logarithm of the base–emitter and collector–emitter currents, a BJT can also be used to compute logarithms and anti-logarithms. A diode can also perform these nonlinear functions but the transistor provides more circuit flexibility.
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John Speidell (fl. 1600–1634) was an English mathematician. He is known for his early work on the calculation of logarithms. Speidell was a mathematics teacher in London and one of the early followers of the work John Napier had previously done on natural logarithms.
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We can then get . Similarly, factorials can be approximated by summing the logarithms of the terms.
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The dimensions of the ellipsoid axes were defined by logarithms in keeping with former calculation methods.
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The 1797 Encyclopædia Britannica explanation of logarithms By simplifying difficult calculations before calculators and computers became available, logarithms contributed to the advance of science, especially astronomy. They were critical to advances in surveying, celestial navigation, and other domains. Pierre-Simon Laplace called logarithms ::"...[a]n admirable artifice which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations.", p.
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Logarithms can be defined for any positive base other than 1, not only . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter. For instance, the base-2 logarithm (also called the binary logarithm) is equal to the natural logarithm divided by , the natural logarithm of 2. Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity.
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The natural logarithm of 10, which has the decimal expansion 2.30258509..., plays a role for example in the computation of natural logarithms of numbers represented in scientific notation, as a mantissa multiplied by a power of 10: : \ln(a\cdot 10^n) = \ln a + n \ln 10. This means that one can effectively calculate the logarithms of numbers with very large or very small magnitude using the logarithms of a relatively small set of decimals in the range [1,10).
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In 1619 Speidell published a table entitled "New Logarithmes" in which he calculated the natural logarithms of sines, tangents, and secants. He then diverged from Napier's methods in order to ensure all of the logarithms were positive. A new edition of "New Logarithmes" was published in 1622 and contained an appendix with the natural logarithms of all numbers 1-1000. Along with William Oughtred and Richard Norwood, Speidell helped push toward the abbreviations of trigonometric functions.
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Julius Zech, photography by Georg Friedrich Brandseph, before 1862 Julius August Christoph Zech (24 February 1821 Stuttgart, Germany − 13 July 1864 Berg) was a German astronomer and mathematician. In 1849, Zech published a table of logarithms; as a result, Zech logarithms for finite fields are named after him.
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CAVEAT: If the problem is not scalar one cannot take logarithms. In general explicit solutions are very rare.
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Euler established the application of binary logarithms to music theory, long before their applications in information theory and computer science became known. As part of his work in this area, Euler published a table of binary logarithms of the integers from 1 to 8, to seven decimal digits of accuracy...
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These tables listed the values of for any number in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.
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Other uses for semiconductor diodes include the sensing of temperature, and computing analog logarithms (see Operational amplifier applications#Logarithmic output).
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In 1616 Henry Briggs visited John Napier at Edinburgh in order to discuss the suggested change to Napier's logarithms. The following year he again visited for a similar purpose. During these conferences the alteration proposed by Briggs was agreed upon, and on his return from his second visit to Edinburgh, in 1617, he published the first chiliad of his logarithms. In 1624 Briggs published his Arithmetica Logarithmica, in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000).
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This gives rise to a logarithmic spiral. Benford's law on the distribution of leading digits can also be explained by scale invariance., chapter 6, section 64 Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions.
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Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.
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His achievements in logarithms and mathematics have been noted by several authors and sources. These include the Lexicon Technicum. It also includes Diderot's Encyclopédie.
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Juan Caramuel y Lobkowitz worked extensively on logarithms including logarithms with base 2. Thomas Harriot's manuscripts contained a table of binary numbers and their notation, which demonstrated that any number could be written on a base 2 system. Regardless, Leibniz simplified the binary system and articulated logical properties such as conjunction, disjunction, negation, identity, inclusion, and the empty set. He anticipated Lagrangian interpolation and algorithmic information theory.
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The four exponentials conjecture rules out a special case of non-trivial, homogeneous, quadratic relations between logarithms of algebraic numbers. But a conjectural extension of Baker's theorem implies that there should be no non-trivial algebraic relations between logarithms of algebraic numbers at all, homogeneous or not. One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture.Waldschmidt, "Variations…" (2005), consequence 1.9.
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A page from Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. A page from a table of logarithms of trigonometric functions from the 2002 American Practical Navigator. Columns of differences are included to aid interpolation.
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In 1625, De Decker entered a contract with Adriaan Vlacq for the publication of several translations of books by John Napier, Edmund Gunter and Henry Briggs. A first book was published in 1626, with several translations done by Vlacq. A second book was made of the logarithms of the first 10000 numbers from Briggs' Arithmetica logarithmica published in 1624. The logarithms were shortened to 10 places.
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Euclid Speidell (died 1702) was an English customs official and mathematics teacher known for his writing on logarithms. Speidell published revised and expanded versions of texts by his father, John Speidell. He also published a book called Logarithmotechnia, or, The making of numbers called logarithms to twenty five places from a geometrical figure in 1688. Speidell's name appears on an instrument made by his contemporary Henry Sutton.
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Common logarithms were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
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His mathematical work included works on the determinant, hyperbolic functions, and parabolic logarithms and trigonometry.This is about connecting the rectified length of line segments along a parabola, giving logarithms for appropriate coordinates, and trigonometric values for suitable angles, in a similar way as the area under a hyperbola defines the natural logarithm, and a hyperbolic angle is defined via the area of a hyperbolically truncated triangle.
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Alexander John Thompson (1885 in Plaistow, Essex - 17 June 1968 in Wallington, Surrey) is the author of the last great table of logarithms, published in 1952. This table, the Logarithmetica britannica gives the logarithms of all numbers from 1 to 100000 to 20 places and supersedes all previous tables of similar scope, in particular the tables of Henry Briggs, Adriaan Vlacq and Gaspard de Prony..
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On this basis, Michael Stifel has been credited with publishing the first known table of binary logarithms in 1544. His book Arithmetica Integra contains several tables that show the integers with their corresponding powers of two. Reversing the rows of these tables allow them to be interpreted as tables of binary logarithms. .. A copy of the same table with two more entries appears on p.
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The history of logarithms in seventeenth-century Europe is the discovery of a new function that extended the realm of analysis beyond the scope of algebraic methods. The method of logarithms was publicly propounded by John Napier in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). Prior to Napier's invention, there had been other techniques of similar scopes, such as the prosthaphaeresis or the use of tables of progressions, extensively developed by Jost Bürgi around 1600. Napier coined the term for logarithm in Middle Latin, "logarithmorum," derived from the Greek, literally meaning, "ratio-number," from logos "proportion, ratio, word" + arithmos "number".
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The binary logarithm function may be defined as the inverse function to the power of two function, which is a strictly increasing function over the positive real numbers and therefore has a unique inverse.. Alternatively, it may be defined as , where is the natural logarithm, defined in any of its standard ways. Using the complex logarithm in this definition allows the binary logarithm to be extended to the complex numbers.For instance, Microsoft Excel provides the `IMLOG2` function for complex binary logarithms: see . As with other logarithms, the binary logarithm obeys the following equations, which can be used to simplify formulas that combine binary logarithms with multiplication or exponentiation:.
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Several important formulas, sometimes called logarithmic identities or logarithmic laws, relate logarithms to one another.All statements in this section can be found in , , or , for example.
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Although this remains open, a similar statement on the asymptotic ratio of the logarithms of the numbers of matroids and sparse paving matroids has been proven.
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The log semiring also arises when working with numbers that are logarithms (measured on a logarithmic scale), such as decibels (see ), log probability, or log-likelihoods.
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Liouville's theorem states that elementary antiderivatives, if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.
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Jan Mikołaj Smogulecki (1610–1656), of the Grzymała coat of arms, was a Polish nobleman, politician, missionary, scholar and Jesuit credited with introducing logarithms to China.
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Encyclopedia of science and technology by James S. Trefil 2001 Page cxxxiii Boltzmann's equation S = k loge W (inscribed on his tombstone) first related entropy with logarithms.
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The arithmetic–geometric mean can be used to compute – among others – logarithms, complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions.
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It has been said that Speidell's logarithms were to the base , but this is not entirely true due to complications with the values being expressed as integers.
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Citation error. See inline comment how to fix. His invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters they discussed were a re-scaling of Napier's logarithms, in which the presence of the mathematical constant now known as e (more accurately, e times a large power of 10 rounded to an integer) was a practical difficulty.
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Henrion wrote a tract concerning logarithms.. Glaisher writes that Henrion, Adriaan Vlacq, and Ezechiel de Decker were rivals for being "the first foreigner who published Briggian logarithms"; he notes Henrion's Traicté des Logarithmes (Paris, 1926). He translated Euclid's Elements from Latin into French. He published Problemata nobilissima duo (Paris, 1616), a book against Marin Ghetaldi and attacking Viète and Regiomontanus. Later reorganized, the book was republished by its author.
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The regulator, a calculation of volume in 'logarithmic space' as divided by the logarithms of the units of the cyclotomic field, can be set against the quantities from the L(1) recognisable as logarithms of cyclotomic units. There result formulae stating that the class number is determined by the index of the cyclotomic units in the whole group of units. In Iwasawa theory, these ideas are further combined with Stickelberger's theorem.
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The distribution of the product of two random variables which have lognormal distributions is again lognormal. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms. Thus, in cases where a simple result can be found in the list of convolutions of probability distributions, where the distributions to be convolved are those of the logarithms of the components of the product, the result might be transformed to provide the distribution of the product. However this approach is only useful where the logarithms of the components of the product are in some standard families of distributions.
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This page shows the logarithms for numbers from 1000 to 1500 to five decimal places. The complete table covers values up to 9999. Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule.
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Urania Propitia was a simplification of the Rudolphine Tables written by Johannes Kepler in 1627. Kepler's dedication to Emperor Ferdinand II which was originally dedicated to Rudolf II was filled with complex and tedious logarithms. It was because of Kepler's ingenious yet tiresome use of logarithms that led Cunitz to simplify the Rudolphine Tables and make Kepler's work more accessible to the public. The tables are mostly astrological, but the instructions are completely astronomical.
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John M. Pollard (born 1941) is a British mathematician who has invented algorithms for the factorization of large numbers and for the calculation of discrete logarithms. His factorization algorithms include the rho, p − 1, and the first version of the special number field sieve, which has since been improved by others. His discrete logarithm algorithms include the rho algorithm for logarithms and the kangaroo algorithm. He received the RSA Award for Excellence in Mathematics.
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He wrote works about sun spots and eclipses. He was a teacher of the Chinese scholar and astronomer Xue Fengzuo, who would be the first Chinese to publish work using logarithms.
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In number theory, Selberg's identity is an approximate identity involving logarithms of primes found by . Selberg and Erdős both used this identity to give elementary proofs of the prime number theorem.
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The Wang LOCI-2 Logarithmic Computing Instrument desktop calculator (the earlier LOCI-1 in September 1964 was not a real product) was introduced in January 1965. Using factor combining it was probably the first desktop calculator capable of computing logarithms, quite an achievement for a machine without any integrated circuits. The electronics included 1,275 discrete transistors. It actually performed multiplication by adding logarithms, and roundoff in the display conversion was noticeable: 2 times 2 yielded 3.999999999.
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The first two are the easiest because they each only require two tables. Using the second formula, however, has the unique advantage that if only a cosine table is available, it can be used to estimate inverse cosines by searching for the angle with the nearest cosine value. Notice how similar the above algorithm is to the process for multiplying using logarithms, which follows these steps: scale down, take logarithms, add, take inverse logarithm, scale up.
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Jain literature covered multiple topics of mathematics around 150 AD including the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, bi- quadric equations, permutations, combinations and logarithms.
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9, p.212 identified in Johannes Werner's 16th-century manuscript on conic sections. Now recognized as one of Werner's formulas, it was essential for the development of prosthaphaeresis and logarithms decades later.
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The logarithms in this formula are usually taken (as shown in the graph) to the base 2. See binary logarithm. When p=\tfrac 1 2, the binary entropy function attains its maximum value.
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All these complex logarithms of are on a vertical line in the complex plane with real part . Since any nonzero complex number has infinitely many complex logarithms, the complex logarithm cannot be defined to be a single-valued function on the complex numbers, but only as a multivalued function. Settings for a formal treatment of this are, among others, the associated Riemann surface, branches, or partial inverses of the complex exponential function. Sometimes the instead of is used when addressing the complex logarithm.
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BEIC digital library.) William Gardiner (died 1752) was an English mathematician.Gardiner, William His logarithmic tables of sines and tangents (Tables of logarithms, 1742) had various reprints and saw use by scientists and other mathematicians.
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The formulation of Stark's conjectures led Harold Stark to define what is now called the Stark regulator, similar to the classical regulator as a determinant of logarithms of units, attached to any Artin representation.
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The common logarithm of a number is the index of that power of ten which equals the number.William Gardner (1742) Tables of Logarithms Speaking of a number as requiring so many figures is a rough allusion to common logarithm, and was referred to by Archimedes as the "order of a number".R.C. Pierce (1977) "A brief history of logarithm", Two-Year College Mathematics Journal 8(1):22–26. The first real logarithms were heuristic methods to turn multiplication into addition, thus facilitating rapid computation.
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Logarithms occur in several laws describing human perception:, pp. 355–56, p. 48 Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have., p.
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Decimal Arithmetick gives instructions for calculations involving decimals, methods of extracting roots, and an overview of the concept of logarithms. There are many worked examples, some of which involve solid geometry or the calculation of interest.
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Averaging and converting the attenuation values into logarithms are possible only if the attenuation does not vary over time. Thus, for relatively small variances, the difference is small but for large variances, the difference is significant.
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He published tables of logarithms, emphasizing their practical use in the fields of astronomy and geography. Cavalieri also constructed a hydraulic pump for a monastery he was in management of. The Duke of Mantua obtained one similar.
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Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors: : \log_b(xy) = \log_b x + \log_b y, \, provided that , and are all positive and .
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In algebraic number theory, Leopoldt's conjecture, introduced by , states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by . Leopoldt proposed a definition of a p-adic regulator Rp attached to K and a prime number p. The definition of Rp uses an appropriate determinant with entries the p-adic logarithm of a generating set of units of K (up to torsion), in the manner of the usual regulator.
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If base 2 logarithms and powers of 2 are used instead, then the unit of information is the bit, or shannon, which is the information content of an event if the probability of that event occurring is . Natural logarithms and powers of e define the nat. One ban corresponds to ln(10) nat = log2(10) bit or Sh, or approximately 2.303 nat, or 3.322 bit. A deciban is one tenth of a ban (or about 0.332 bit); the name is formed from ban by the SI prefix deci-.
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His major work was Thesaurus Logarithmorum Completus (Treasury of all Logarithms) that was first published 1794 in Leipzig (its 90th edition was published in 1924). This mathematical table was actually based on Adriaan Vlacq's tables, but corrected a number of errors and extended the logarithms of trigonometric functions for the small angles. An engineer, Franc Allmer, honourable senator of the Graz University of Technology, has found Vega's logarithmic tables with 10 decimal places in the Museum of Carl Friedrich Gauss in Göttingen. Gauss used this work frequently and he has written in it several calculations.
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An interesting tidbit is this treatise contains the earliest written reference to the decimal point (though its usage would not come into general use for another century.) The computing devices in Rabdology were overshadowed by his seminal work on logarithms as they proved more useful and more widely applicable. Nevertheless, these devices (as indeed are logarithms) are examples of Napier's ingenious attempts to discover easier ways to multiply, divide and find roots of numbers. Location arithmetic in particular foreshadowed the ease of and power of mechanizing binary arithmetic, but was never fully appreciated.
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386 The antiderivative of the natural logarithm is: : \int \ln(x) \,dx = x \ln(x) - x + C. Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.
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This model can be used for classic ad-hoc tasks. In-expC2 Inverse Expected Document Frequency model with Bernoulli after-effect and normalization 2. The logarithms are base e. This model can be used for classic ad-hoc tasks.
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Moreover, because the logarithmic function grows very slowly for large , logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.
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In its simplest form, the frequency and recency rating can be added to form a frecency rating. The ratings can be found by sorting items by most recent and most frequent respectively. A decayed calculation using logarithms can also be used.
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John Napier, 8th Laird of Merchiston, inventor of logarithms John Napier's heir, Alexander, and also his grandson were both killed in 1513 at the Battle of Flodden. Another Napier heir was killed at the Battle of Pinkie Cleugh in 1547.
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The theorem then follows from the lemma. Theorem (Alphonse Antonio de Sarasa 1649) As area measured against the asymptote increases in arithmetic progression, the projections upon the asymptote increase in geometric sequence. Thus the areas form logarithms of the asymptote index.
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Thus a single table of common logarithms can be used for the entire range of positive decimal numbers.E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913). See common logarithm for details on the use of characteristics and mantissas.
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An ivory set of Napier's Bones from around 1650 A set of Napier's calculating tables from around 1680 His work, Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms (see Napierian logarithm). The book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napier's Rules of Circular Parts. See also Pentagramma mirificum. Modern English translations of both Napier's books on logarithms and their description can be found on the web, as well as a discussion of Napier's bones and Promptuary (another early calculating device).
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Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "closed-form numbers", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "elementary numbers", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers explicitly defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as or ln 2\.
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One important application of the Cantor–Zassenhaus algorithm is in computing discrete logarithms over finite fields of prime-power order. Computing discrete logarithms is an important problem in public key cryptography. For a field of prime-power order, the fastest known method is the index calculus method, which involves the factorisation of field elements. If we represent the prime-power order field in the usual way – that is, as polynomials over the prime order base field, reduced modulo an irreducible polynomial of appropriate degree – then this is simply polynomial factorisation, as provided by the Cantor–Zassenhaus algorithm.
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Binary logarithms allow for a convenient comparison of expression rates: a doubled expression rate can be described by a log ratio of , a halved expression rate can be described by a log ratio of , and an unchanged expression rate can be described by a log ratio of zero, for instance.. Data points obtained in this way are often visualized as a scatterplot in which one or both of the coordinate axes are binary logarithms of intensity ratios, or in visualizations such as the MA plot and RA plot that rotate and scale these log ratio scatterplots..
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Cover page of Rabdologiæ In 1617 a treatise in Latin titled Rabdologiæ and written by John Napier was published in Edinburgh. Printed three years after his treatise on the discovery of logarithms and in the same year as his death, it describes three devices to aid arithmetic calculations. The devices themselves don't use logarithms, rather they are tools to reduce multiplication and division of natural numbers to simple addition and subtraction operations. The first device, which by then was already popularly used and known as Napier's bones, was a set of rods inscribed with the multiplication table.
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The identities of logarithms can be used to approximate large numbers. Note that , where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, . To get the base-10 logarithm, we would multiply 32,582,657 by , getting .
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The fact that there are two logarithms (log of a log) in the limit for the Meissel–Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler–Mascheroni constant.
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Gel'fond (1934) and Schneider (1934). Alan Baker also used the method in the 1960s for his work on linear forms in logarithms and ultimately Baker's theorem.Baker and Wüstholz (2007). Another example of the use of this method from the 1960s is outlined below.
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This can be used as an alternative definition of the real-number power and agrees with the definition given above using rational exponents and continuity. The definition of exponentiation using logarithms is more common in the context of complex numbers, as discussed below.
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For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used in finance to solve problems involving compound interest.
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Ahmad completed his BA in mathematics (with distinction) from Muhammadan Anglo-Oriental College in 1895. He was the first Muslim to obtain a D.Sc. (Mathematics), from Allahabad University. His field was complex logarithms applications. He published in differential geometry and algebraic geometry.
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BB2 Bernoulli-Einstein model with Bernoulli after-effect and normalization 2. IFB2 Inverse Term Frequency model with Bernoulli after-effect and normalization 2. In-expB2 Inverse Expected Document Frequency model with Bernoulli after-effect and normalization 2. The logarithms are base 2.
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This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).J J O'Connor and E F Robertson: Alan Baker. The MacTutor History of Mathematics archive 1998.
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The decisional Diffie–Hellman (DDH) assumption is a computational hardness assumption about a certain problem involving discrete logarithms in cyclic groups. It is used as the basis to prove the security of many cryptographic protocols, most notably the ElGamal and Cramer–Shoup cryptosystems.
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Analysis of algorithms is a branch of computer science that studies the performance of algorithms (computer programs solving a certain problem)., pp. 1–2 Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems., p.
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Invented and described by S. Dunn, octavo (London), 1793. # The Longitude Logarithms, in their regular and shortest order, made easy for use in taking the Latitude and Longitude at Sea and Land, octavo, London, 1793 (British Museum Cat.; Watt, '"Bibl. Brit"'. i. 324 f.).
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The National Cyclopaedia of Useful Knowledge, Vol III, (1847), London, Charles Knight, p.808 At this time, Briggs obtained a copy of Mirifici Logarithmorum Canonis Descriptio, in which Napier introduced the idea of logarithms. It has also been suggested that he knew of the method outlined in Fundamentum Astronomiae published by the Swiss clockmaker Jost Bürgi, through John Dee. Napier's formulation was awkward to work with, but the book fired Briggs' imagination – in his lectures at Gresham College he proposed the idea of base 10 logarithms in which the logarithm of 10 would be 1; and soon afterwards he wrote to the inventor on the subject.
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A page from Henry Briggs' 1617 Logarithmorum Chilias Prima showing the base-10 (common) logarithm of the integers 0 to 67 to fourteen decimal places. In 1616 Briggs visited Napier at Edinburgh in order to discuss the suggested change to Napier's logarithms. The following year he again visited for a similar purpose. During these conferences the alteration proposed by Briggs was agreed upon; and on his return from his second visit to Edinburgh, in 1617, he published the first chiliad of his logarithms. In 1619 he was appointed Savilian Professor of Geometry at the University of Oxford, and resigned his professorship of Gresham College in July 1620.
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A 12-digit arithmometer sold for 300 francs in 1853, which was 30 times the price of a table of logarithms book and 1,500 times the cost of a first-class stamp (20 French cents), but, unlike a table of logarithms book, it was simple enough to be used for hours by an operator without any special qualifications.(fr) Annales de la Société d'émulation du département des Vosges, 1853 Gallica web site An advertisement taken from a magazine published in 1855 shows that a 10-digit machine sold for 250 francs and a 16-digit machine sold for 500 francs.(fr) Cosmos July 1855 www.arithmometre.org. Retrieved 2010-09-22.
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A necessary condition for a three-pass algorithm to be secure is that an attacker cannot determine any information about the message m from the three transmitted messages E(s,m), E(r,E(s,m)) and E(r,m). For the encryption functions used in the Shamir algorithm and the Massey–Omura algorithm described above, the security relies on the difficulty of computing discrete logarithms in a finite field. If an attacker could compute discrete logarithms in GF(p) for the Shamir method or GF(2n) for the Massey–Omura method then the protocol could be broken. The key s could be computed from the messages mr and mrs.
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In its logarithmic form it is the following conjecture. Let λ1, λ2, and λ3 be any three logarithms of algebraic numbers and γ be a non-zero algebraic number, and suppose that λ1λ2 = γλ3. Then λ1λ2 = γλ3 = 0\. The exponential form of this conjecture is the following.
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All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.
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Factor bases are used in, for example, Dixon's factorization, the quadratic sieve, and the number field sieve. The difference between these algorithms is essentially the methods used to generate (x, y) candidates. Factor bases are also used in the Index calculus algorithm for computing discrete logarithms.
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Shorey has done significant work on transcendental number theory, in particular best estimates for linear forms in logarithms of algebraic numbers. He has obtained some new applications of Baker’s method to Diophantine equations and Ramanujan’s T-function. Shorey's contribution to irreducibility of Laguerre polynomials is extensive.
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The authors needed several thousand CPU cores for a week to precompute data for a single 512-bit prime. Once that was done, however, individual logarithms could be solved in about a minute using two 18-core Intel Xeon CPUs. Originally published in Proc. 22nd Conf.
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During 1858 he solved the equation of the fifth degree by elliptic functions; and during 1873 he proved e, the base of the natural system of logarithms, to be transcendental. This last was used by Ferdinand von Lindemann to prove during 1882 the same for π.
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In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms. Gisbert Wüstholz proved it in the 1980s. It marked a breakthrough in the theory of transcendental numbers.
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One cent compared to a semitone on a truncated monochord. The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to express small intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is too small to be perceived between successive notes. Cents, as described by Alexander J. Ellis, follow a tradition of measuring intervals by logarithms that began with Juan Caramuel y Lobkowitz in the 17th century.Caramuel mentioned the possible use of binary logarithms for music in a letter to Athanasius Kircher in 1647; this usage often is attributed to Leonhard Euler in 1739 (see Binary logarithm). Isaac Newton described musical logarithms using the semitone () as base in 1665; Gaspard de Prony did the same in 1832. Joseph Sauveur in 1701, and Felix Savart in the first half of the 19th century, divided the octave in 301 or 301,03 units.
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Mathematical tables containing common logarithms (base-10) were extensively used in computations prior to the advent of computers and calculators, not only because logarithms convert problems of multiplication and division into much easier addition and subtraction problems, but for an additional property that is unique to base-10 and proves useful: Any positive number can be expressed as the product of a number from the interval and an integer power of This can be envisioned as shifting the decimal separator of the given number to the left yielding a positive, and to the right yielding a negative exponent of Only the logarithms of these normalized numbers (approximated by a certain number of digits), which are called mantissas, need to be tabulated in lists to a similar precision (a similar number of digits). These mantissas are all positive and enclosed in the interval . The common logarithm of any given positive number is then obtained by adding its mantissa to the common logarithm of the second factor. This logarithm is called the characteristic of the given number.
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In a seminal work, he showed that the Diffie-Hellman problem is (under certain conditions) equivalent to solving the discrete log problem.Ueli Maurer: Towards the equivalence of breaking the Diffie-Hellman protocol and computing discrete logarithms. In: Advances in Cryptology - Crypto '94. Springer-Verlag, 1994, S. 271−281.
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There are no official variations or sequels to the game. More mathematically skilled players may wish to include operations such as exponentiation, logarithms, and modulo, or incorporate sequences using any number of adjacent tiles in a line (such as 4 × 2 + 2 = 10, where only the 10 is played).
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Assuming the axiom of choice and, given an infinite cardinal κ and a finite cardinal μ greater than 1, there may or may not be a cardinal λ satisfying \mu^\lambda = \kappa. However, if such a cardinal exists, it is infinite and less than κ, and any finite cardinality ν greater than 1 will also satisfy u^\lambda = \kappa. The logarithm of an infinite cardinal number κ is defined as the least cardinal number μ such that κ ≤ 2μ. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of cardinal invariants of topological spaces, though they lack some of the properties that logarithms of positive real numbers possess.
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The natural unit of information (symbol: nat), sometimes also nit or nepit, is a unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the bit. This unit is also known by its unit symbol, the nat. The nat is the coherent unit for information entropy. The International System of Units, by assigning the same units (joule per kelvin) both to heat capacity and to thermodynamic entropy implicitly treats information entropy as a quantity of dimension one, with . Physical systems of natural units that normalize the Boltzmann constant to 1 are effectively measuring thermodynamic entropy in nats.
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Filipowski afterwards was employed as an actuary for the Colonial and Standard Life offices in Edinburgh, a position he kept for about eight years. He later worked as an actuary of the Mercantile, Professional, and General Life and Guarantee Insurance Company, and as an assistant computer at the Royal and Briton companies. In this capacity he published A Table of Anti- Logarithms (1849), which included a testimonial by mathematician Augustus de Morgan and established Filipowski's name among mathematicians. He later published Napier's Canon of Logarithms (1857), a translation of John Napier's Logarithmorum Canonis Descriptio from the Latin into English, and in 1864–66 he edited Francis Baily's Doctrine of Life Annuities and Assurance.
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The binary logarithm function is the inverse function of the power of two function. As well as , alternative notations for the binary logarithm include , , (the notation preferred by ISO 31-11 and ISO 80000-2), and (with a prior statement that the default base is 2) . Historically, the first application of binary logarithms was in music theory, by Leonhard Euler: the binary logarithm of a frequency ratio of two musical tones gives the number of octaves by which the tones differ. Binary logarithms can be used to calculate the length of the representation of a number in the binary numeral system, or the number of bits needed to encode a message in information theory.
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The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed.
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A alt=A photograph of a nautilus' shell. Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor.
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Bonaventura Francesco Cavalieri (; 1598 - 30 November 1647) was an Italian mathematician and a Jesuate. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of infinitesimal calculus, and the introduction of logarithms to Italy. Cavalieri's principle in geometry partially anticipated integral calculus.
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It is a large collection of small tables, with some seven-figure logarithms. This he dedicated to William Jones. The same year he started the publication of The Mathematical Miscellany, containing analytical and algebraic solutions of a large number of problems in various branches of mathematics. His preface to vol.
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Universidade de São Paulo, Departamento de História, Sociedade de Estudos Históricos (Brazil), Revista de História (1965), ed. 61-64, p. 350 Jean Picard performed the first modern meridian arc measurement in 1669–1670. He measured a baseline using wooden rods, a telescope (for his angular measurements), and logarithms (for computation).
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In the following, the Nernst slope (or thermal voltage) is used, which has a value of 0.02569... V at STP. When base-10 logarithms are used, VT λ = 0.05916... V at STP where λ=ln[10]. There are three types of line boundaries in a Pourbaix diagram: Vertical, horizontal, and sloped.
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Let be a root of the primitive polynomial . The traditional representation of elements of this field is as polynomials in α of degree 2 or less. A table of Zech logarithms for this field are , , , , , , , and . The multiplicative order of α is 7, so the exponential representation works with integers modulo 7.
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His Latinized name was Ioannes Neper. John Napier is best known as the discoverer of logarithms. He also invented the so-called "Napier's bones" and made common the use of the decimal point in arithmetic and mathematics. Napier's birthplace, Merchiston Tower in Edinburgh, is now part of the facilities of Edinburgh Napier University.
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An elementary number is one formalization of the concept of a closed-form number. The elementary numbers form an algebraically closed field containing the roots of arbitrary equations using field operations, exponentiation, and logarithms. The set of the elementary numbers is subdivided into the explicit elementary numbers and the implicit elementary numbers.
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Conway's surreal numbers fall into category 2. They are a system designed to be as rich as possible in different sizes of numbers, but not necessarily for convenience in doing analysis. Certain transcendental functions can be carried over to the surreals, including logarithms and exponentials, but most, e.g., the sine function, cannot.
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This development resulted in the first scientific handheld calculator, the HP-35 in 1972. Based on hyperbolic CORDIC, Yuanyong Luo et al. further proposed a Generalized Hyperbolic CORDIC (GH CORDIC) to directly compute logarithms and exponentials with an arbitrary fixed base in 2019. Theoretically, Hyperbolic CORDIC is a special case of GH CORDIC.
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Canon logarithmorum As the common log of ten is one, of a hundred is two, and a thousand is three, the concept of common logarithms is very close to the decimal-positional number system. The common log is said to have base 10, but base 10,000 is ancient and still common in East Asia. In his book The Sand Reckoner, Archimedes used the myriad as the base of a number system designed to count the grains of sand in the universe. As was noted in 2000:Ian Bruce (2000) "Napier’s Logarithms", American Journal of Physics 68(2):148, doi: 10.1119/1.19387 :In antiquity Archimedes gave a recipe for reducing multiplication to addition by making use of geometric progression of numbers and relating them to an arithmetic progression.
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Cotes's major original work was in mathematics, especially in the fields of integral calculus, logarithms, and numerical analysis. He published only one scientific paper in his lifetime, titled Logometria, in which he successfully constructs the logarithmic spiral.O'Connor & Robertson (2005)In Logometria, Cotes evaluated e, the base of natural logarithms, to 12 decimal places. See: Roger Cotes (1714) "Logometria," Philosophical Transactions of the Royal Society of London, 29 (338) : 5-45; see especially the bottom of page 10. From page 10: "Porro eadem ratio est inter 2,718281828459 &c; et 1, … " (Furthermore, the same ratio is between 2.718281828459… and 1, … ) After his death, many of Cotes's mathematical papers were hastily edited by his cousin Robert Smith and published in a book, Harmonia mensurarum.
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Using Gibbs free energy: ΔG = −RT ln(Keq), where R is the universal gas constant, T is the temperature in kelvins, and Keq is the equilibrium constant of a reaction in equilibrium. The deprotonation of His31 is an acid equilibrium reaction with a special Keq known as the acid dissociation constant, Ka: His31-H+ His31 + H+. The pKa is then related to Ka by the following: pKa = −log(Ka). Calculation of the free energy difference of the mutant and wild-type can now be done using the free energy equation, the definition of pKa, the observed pKa values, and the relationship between natural logarithms and logarithms. In the T4 lysozyme example, this approach yielded a calculated contribution of about 3 kcal/mol to the overall free energy.
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In these equations e is the base of natural logarithms, h is the Planck constant, kB is the Boltzmann constant and T the absolute temperature. R' is the ideal gas constant in units of (bar·L)/(mol·K). The factor is needed because of the pressure dependence of the reaction rate. R' = 8.3145 × 10−2 (bar·L)/(mol·K).
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The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface.
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The idea of logarithms was also used to construct the slide rule, which became ubiquitous in science and engineering until the 1970s. A breakthrough generating the natural logarithm was the result of a search for an expression of area against a rectangular hyperbola, and required the assimilation of a new function into standard mathematics.
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Among Napier's early followers were the instrument makers Edmund Gunter and John Speidell. The development of logarithms is given credit as the largest single factor in the general adoption of decimal arithmetic. The Trissotetras (1645) of Thomas Urquhart builds on Napier's work, in trigonometry. Henry Briggs (mathematician) was an early adopter of the Napierian logarithm.
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In 1627, De Decker's "Tweede deel" was published and it contained the logarithms of all numbers from 1 to 100000, to 10 places. Only very few copies of this book are known and its publication was apparently stopped or delayed. In 1628, Vlacq's Arithmetica logarithmica was published and contained exactly the tables published in 1627.
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The logarithm of the odds ratio, the difference of the logits of the probabilities, tempers this effect, and also makes the measure symmetric with respect to the ordering of groups. For example, using natural logarithms, an odds ratio of 27/1 maps to 3.296, and an odds ratio of 1/27 maps to −3.296.
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Dials were laid out using straight edges and compasses. In the late nineteenth century sundials became objects of academic interest. The use of logarithms allowed algebraic methods of laying out dials to be employed and studied. No longer utilitarian, sundials remained as popular ornaments, and several popular books promoted that interest- and gave constructional details.
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The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature scheme, which should not be confused with ElGamal encryption. ElGamal encryption can be defined over any cyclic group G, like multiplicative group of integers modulo n. Its security depends upon the difficulty of a certain problem in G related to computing discrete logarithms.
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A large part of the volume consists of mathematical and astronomical tables, since Harris intended his work to serve as a small mathematical library. He provided tables of logarithms, sines, tangents, and secants, a two-page list of books, and an index of the articles in both volumes under 26 heads, filling 50 pages.
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S. F. Hotchkin, The First Six Bishops of Pennsylvania (Diocese of Pennsylvania Church House, 1911), 22-28. In 1823, Potter published his first book. It was A Tractate on Logarithms, by which he came known "as a proficient mathematician." In 1825, Geneva College, now Hobart College, offered the twenty-five-year-old Potter its presidency.
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This gives: : b=\sqrt[y] It is less easy to make the subject of the expression. Logarithms allow us to do this: : y= This expression means that is equal to the power that you would raise to, to get . This operation undoes exponentiation because the logarithm of tells you the exponent that the base has been raised to.
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Exponentiation has two inverse operations; roots and logarithms. Analogously, the inverses of tetration are often called the super-root, and the super-logarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function {^3}y=x, the two inverses are the cube super-root of and the super logarithm base of .
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In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation. Divided differences is a recursive division process. The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.
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Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions. The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero.
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A unique but somewhat arbitrary solution called the principal value can be chosen using a general rule which also applies for nonrational powers. Complex powers and logarithms are more naturally handled as single valued functions on a Riemann surface. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a branch cut.
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In this process, the intrinsic charges of polypeptides becomes negligible when compared to the negative charges contributed by SDS. Thus polypeptides after treatment become rod-like structures possessing a uniform charge density, that is same net negative charge per unit length. The electrophoretic mobilities of these proteins will be a linear function of the logarithms of their molecular weights.
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Mercator's comma is a name often used for a closely related interval because of its association with Nicholas Mercator.W. Holder, A Treatise..., ibid., writes that Mersenne had calculated 58¼ commas in the octave; Mercator "working by the Logarithms, finds out but 55, and a little more." One of these intervals was first described by Ching-Fang in 45 BCE.
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Part of a 20th-century table of common logarithms in the reference book Abramowitz and Stegun. Before the advent of computers, lookup tables of values were used to speed up hand calculations of complex functions, such as in trigonometry, logarithms, and statistical density functions. In ancient (499 AD) India, Aryabhata created one of the first sine tables, which he encoded in a Sanskrit-letter-based number system. In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144"Maher, David.
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Since the 1980s the security of cryptographic key exchanges and digital signatures over the Internet has been primarily based on a small number of public key algorithms. The security of these algorithms is based on a similarly small number of computationally hard problems in classical computing. These problems are the difficulty of factoring the product of two carefully chosen prime numbers, the difficulty to compute discrete logarithms in a carefully chosen finite field, and the difficulty of computing discrete logarithms in a carefully chosen elliptic curve group. These problems are very difficult to solve on a classical computer (the type of computer the world has known since the 1940s through today) but are rather easily solved by a relatively small quantum computer using only 5 to 10 thousand of bits of memory.
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Nowadays, the word mantissa is generally used to describe the fractional part of a floating-point number on computers, though the recommended term is significand. Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g., 1000 to 9999.
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Hume was no mathematical reductionist, like Hobbes. The only 17th- century Scottish philosopher, other than James I, that Hume applauds is John Napier of Merchiston, the inventor of logarithms. However Napier, Newton and James I are criticised for producing eschatological literature predicting the final days. Writings of this sort were a potent factor in the politico- religious ferment of the time.
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He attached his pocket watch to his wrist with a piece of string. In 1577, the minute hand was added by a Swiss clock maker, Jost Burgi (who also is a contender for the invention of logarithms), and was incorporated into a clock Burgi made for astronomer Tycho Brahe, who had a need for more accuracy as he charted the heavens.
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Base-e (natural) logarithms and exponentiation can be used, but not base-10. However, workarounds exist for many of those limitations. Complex numbers can be entered in either rectangular form (using the key) or polar form (using the key), and displayed in either form regardless of how they were entered. They can be decomposed using the (radius r) and (angle Θ) functions.
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The graphs of the functions are shown for (dotted), (blue), and (dashed). They all pass through the point , but the red line (which has slope ) is tangent to only there. The value of the natural log function for argument , i.e. , equals The principal motivation for introducing the number , particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.
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In 1878 he published an expanded version of Mathematical Problems, and in 1888 Examples for Practice in the Use of Seven- figure Logarithms. He was a close friend of Leslie Stephen from his undergraduate studies at Cambridge. Virginia Woolf used his personality for the character Augustus Carmichael in her novel To the Lighthouse. His sister was the feminist Elizabeth Clarke Wolstenholme Elmy.
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H. B. Goodwin, The haversine in nautical astronomy, Naval Institute Proceedings, vol. 36, no. 3 (1910), pp. 735–746: Evidently if a Table of Haversines is employed we shall be saved in the first instance the trouble of dividing the sum of the logarithms by two, and in the second place of multiplying the angle taken from the tables by the same number.
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Merchiston Tower, also known as Merchiston Castle, was probably built by Alexander Napier, the 2nd Laird of Merchiston around 1454. It serves as the seat for Clan Napier. It was the home of John Napier, the 8th Laird of Merchiston and the inventor of logarithms, who was born there in 1550. The tower stands at the centre of Edinburgh Napier University's Merchiston campus.
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One way to make some time series stationary is to compute the differences between consecutive observations. This is known as differencing. Transformations such as logarithms can help to stabilize the variance of a time series. Differencing can help stabilize the mean of a time series by removing changes in the level of a time series, and so eliminating trend and seasonality.
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But, as advanced as they were, they attributed no refraction whatever above 45° altitude for solar refraction, and none for starlight above 20° altitude. To perform the huge number of multiplications needed to produce much of his astronomical data, Tycho relied heavily on the then-new technique of prosthaphaeresis, an algorithm for approximating products based on trigonometric identities that predated logarithms.
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He was in Frankfurt-on-Oder in 1573, teaching mathematics and logic. He returned to Scotland in 1584. Craig may have been the person who gave John Napier of Merchiston the hint which led to his discovery of logarithms. Anthony à Wood wrote that Napier himself informed Tycho Brahe, via Craig, of his discovery, some twenty years before it was made public.
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Electronic calculators began to be owned at school from the early 1980s, becoming widespread from the mid-1980s. Parents and teachers believed that calculators would diminish abilities of mental arithmetic. Scientific calculators came to the aid for those working out logarithms and trigonometric functions. Since 1988, exams in Mathematics at age sixteen, except Scotland, have been provided by the GCSE.
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This table was later extended by Adriaan Vlacq, but to 10 places, and by Alexander John Thompson to 20 places in 1952. Briggs was one of the first to use finite-difference methods to compute tables of functions. He also completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places and the tangents and secants for the same to ten places, all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica; this work was probably a successor to his 1617 Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th decimal place.
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Stark later pointed out that Baker's proof, involving linear forms in 3 logarithms, could be reduced to only 2 logarithms, when the result was already known from 1949 by Gelfond and Linnik. Stark's 1969 paper also cited the 1895 text by Heinrich Martin Weber and noted that if Weber had "only made the observation that the reducibility of [a certain equation] would lead to a Diophantine equation, the class-number one problem would have been solved 60 years ago". Bryan Birch notes that Weber's book, and essentially the whole field of modular functions, dropped out of interest for half a century: "Unhappily, in 1952 there was no one left who was sufficiently expert in Weber's Algebra to appreciate Heegner's achievement." Deuring, Siegel, and Chowla all gave slightly variant proofs by modular functions in the immediate years after Stark.
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The logarithm base (that is ) is called the common logarithm and is commonly used in science and engineering. The natural logarithm has the number (that is ) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base (that is ) and is commonly used in computer science. Logarithms are examples of concave functions.
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Some of these methods used tables derived from trigonometric identities.Enrique Gonzales-Velasco (2011) Journey through Mathematics – Creative Episodes in its History, §2.4 Hyperbolic logarithms, p. 117, Springer Such methods are called prosthaphaeresis. Invention of the function now known as the natural logarithm began as an attempt to perform a quadrature of a rectangular hyperbola by Grégoire de Saint-Vincent, a Belgian Jesuit residing in Prague.
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Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation. The non-sliding logarithmic scale, Gunter's rule, was invented shortly after Napier's invention. William Oughtred enhanced it to create the slide rule—a pair of logarithmic scales movable with respect to each other. Numbers are placed on sliding scales at distances proportional to the differences between their logarithms.
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In the 16th and early 17th centuries an algorithm called prosthaphaeresis was used to approximate multiplication and division. This used the trigonometric identity :\cos\alpha\cos\beta = \frac12[\cos(\alpha+\beta) + \cos(\alpha-\beta)] or similar to convert the multiplications to additions and table lookups. However, logarithms are more straightforward and require less work. It can be shown using Euler's formula that the two techniques are related.
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In GWAS Manhattan plots, genomic coordinates are displayed along the X-axis, with the negative logarithm of the association p-value for each single nucleotide polymorphism (SNP) displayed on the Y-axis, meaning that each dot on the Manhattan plot signifies a SNP. Because the strongest associations have the smallest p-values (e.g., 10−15), their negative logarithms will be the greatest (e.g., 15).
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Sarason, Section IV.9. This construction is analogous to the real logarithm function , which is the inverse of the real exponential function , satisfying for positive real numbers . If a non-zero complex number is given in polar form as ( and real numbers, with ), then is one logarithm of . Since exactly for all integer , adding integer multiples to the argument gives all the numbers that are logarithms of : :.
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In 1882, Lindemann published the result for which he is best known, the transcendence of . His methods were similar to those used nine years earlier by Charles Hermite to show that e, the base of natural logarithms, is transcendental. Before the publication of Lindemann's proof, it was known that if was transcendental, then it would be impossible to square the circle by compass and straightedge.
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Arithmometre, designed by Charles Xavier Thomas, c. 1820, for the four rules of arithmetic, manufactured 1866-1870 AD. Exhibit in the Tekniska museet, Stockholm, Sweden. Charles Babbage designed machines to tabulate logarithms and other functions in 1837. His Difference engine can be considered an advanced mechanical calculator and his Analytical Engine a forerunner of the modern computer, though none were built in Babbage's lifetime.
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Microsoft Windows NT Calculator Version 3.1 A simple arithmetic calculator was first included with Windows 1.0.Windows 1.01 - Graphical User Interface Gallery In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
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A Gauss sum is a type of exponential sum. The best known classical algorithm for estimating these sums takes exponential time. Since the discrete logarithm problem reduces to Gauss sum estimation, an efficient classical algorithm for estimating Gauss sums would imply an efficient classical algorithm for computing discrete logarithms, which is considered unlikely. However, quantum computers can estimate Gauss sums to polynomial precision in polynomial time.
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After Simson's death, restorations of Apollonius's treatise De section determinata and of Euclid's treatise De Porismatibus were printed for private circulation in 1776, at the expense of Earl Stanhope, in a volume with the title Roberti Simson opera quaedam reliqua. The volume contains also dissertations on Logarithms and on the Limits of Quantities and Ratios, and a few problems illustrating the ancient geometrical analysis.
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Following the death of spanish king Charles II in 1700 Kresa went back to Prague. He obtained a doctorate in theology at Charles University and also started to teach theology there. At the same time he was privately teaching mathematics and was acquiring mathematical apparatus for the Department of Mathematics. He was engaged in arithmetic, fractions and logarithms, trigonometry, astronomy, algebra, as well as military architecture.
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Steps of the Pohlig–Hellman algorithm. In group theory, the Pohlig–Hellman algorithm, sometimes credited as the Silver–Pohlig–Hellman algorithm,Mollin 2006, pg. 344 is a special-purpose algorithm for computing discrete logarithms in a finite abelian group whose order is a smooth integer. The algorithm was introduced by Roland Silver, but first published by Stephen Pohlig and Martin Hellman (independent of Silver).
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Briggs was one of the first to use finite-difference methods to compute tables of functions. He also completed a table of logarithmic sines and tangents for the hundredth part of every degree to fourteen decimal places, with a table of natural sines to fifteen places, and the tangents and secants for the same to ten places; all of which were printed at Gouda in 1631 and published in 1633 under the title of Trigonometria Britannica; this work was probably a successor to his 1617 Logarithmorum Chilias Prima ("The First Thousand Logarithms"), which gave a brief account of logarithms and a long table of the first 1000 integers calculated to the 14th decimal place. Briggs discovered, in a somewhat concealed form and without proof, the binomial theorem. English translations of Briggs's Arithmetica and the first part of his Trigonometria Britannica are available on the web.
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In 1786 Zaborowski published the textbook Jeometria praktyczna (Polish for Practical geometry), where he described methods of geodesic measurement. The book was awarded the Merentibus by Stanisław August Poniatowski. Zaborowski was an author of Logarytmy dlá szkół narodowych (Polish for Logarithms for national schools) written by order of Commission of National Education and published in 1787. He also wrote Tablice matematyczne (Polish for Mathematical tables) which was published in 1797.
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Advanced use of the rods can extract square roots. Napier's bones are not the same as logarithms, with which Napier's name is also associated, but are based on dissected multiplication tables. The complete device usually includes a base board with a rim; the user places Napier's rods inside the rim to conduct multiplication or division. The board's left edge is divided into nine squares, holding the numbers 1 to 9.
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Addition, multiplication, and exponentiation are three of the most fundamental arithmetic operations. Addition, the simplest of these, is undone by subtraction: when you add to to get , to reverse this operation you need to subtract from . Multiplication, the next-simplest operation, is undone by division: if you multiply by to get , you then can divide by to return to the original expression . Logarithms also undo a fundamental arithmetic operation, exponentiation.
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The Sierpinski triangle (at the right) is constructed by repeatedly replacing alt=Parts of a triangle are removed in an iterated way. Logarithms occur in definitions of the dimension of fractals. Fractals are geometric objects that are self- similar: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length.
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An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa.This use of the word mantissa stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text.
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Neither Napier nor Briggs actually discovered the constant e; that discovery was made decades later by Jacob Bernoulli. Napier delegated to Briggs the computation of a revised table. The computational advance available via logarithms, the inverse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics, and other areas of physics.
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Therefore, VSH can be useful in embedded environments where code space is limited. Two major variants of VSH were proposed. For one, finding a collision is as difficult as finding a nontrivial modular square root of a very smooth number modulo n. The other one uses a prime modulus p (with no trapdoor), and its security proof relies on the hardness of finding discrete logarithms of very smooth numbers modulo p.
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An introduction to some of Buergi's methods survives in a copy by Kepler; it discusses the basics of Algebra (or Coss as it was known at the time), and of decimal fractions. Some authors consider Bürgi as one of the inventors of logarithms. His legacy also includes the engineering achievement contained in his innovative mechanical astronomical models.Jost Bürgi; by Ludwig Oechslin; Publisher: Verlag Ineichen, Luzern, 2001, 108 p.
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For some mathematical functions, a gold "f−1" prefix key would access the inverse of the gold-printed functions, e.g. "f−1" followed by "4" would calculate the inverse sine (sin^{-1}). Functions included square root, inverse, trigonometric (sine, cosine, tangent and their inverses), exponentiation, logarithms and factorial. The HP-65 was one of the first calculators to include a base conversion function, although it only supported octal (base 8) conversion.
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G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 4th Ed., Oxford 1975, footnote to paragraph 1.7: "log x is, of course, the 'Naperian' logarithm of x, to base e. 'Common' logarithms have no mathematical interest". Extract of page 9 Parentheses are sometimes added for clarity, giving , , or . This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
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Around 1620, Harriot's unpublished papers includes the early basis of continuous compounding. Harriot uses modern mathematical concepts to explain the process behind continuous compounding. The concept of compounded interest occurs when the more times interest is added within the year assuming the rate stays the same then the final interest will be larger. Based on this observation, Harriot created mathematical equations that included logarithms and series calculations to illustrate his concepts.
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Its most well-known proponent was Tycho Brahe, who used it extensively for astronomical calculations such as those described above. It was also used by John Napier, who is credited with inventing the logarithms that would supplant it. Nicholas Copernicus mentions 'prosthaphaeresis' several times in his 1543 work De Revolutionibus Orbium Coelestium, meaning the "great parallax" caused by the displacement of the observer due to the Earth's annual motion.
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The entropy will then be obtained from . In this expression the error on the second term is negligible compared to the error on the first term. The magnifying factor is then ÷ 298 K, so for an error of 0.05 in the logarithms the error on will be of the order of . When equilibrium constants are measured at three or more temperatures, values of will be obtained by straight-line fitting.
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He also taught astronomy and mathematics, introducing logarithms to China, and was much respected by Chinese scholars. His fame as a scholar and teacher spread, and in 1653 he was invited by the Shunzhi Emperor to his court in Beijing. Smogulecki requested permission to leave the court to continue his missionary travels. He went to Manchuria, then to Yunnan, where another civil war made him travel to Guangzhou.
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Bernice Weldon Sargent, (24 September 1906 – 17 December 1993) was a Canadian physicist who worked at the Manhattan Project's Montreal Laboratory during the Second World War as head of its nuclear physics division. In his 1932 doctoral thesis, he discovered the relationship between the radioactive disintegration constants of beta particle-emitting radioisotopes and corresponding logarithms of their maximum beta particle energies. These plots are known as "Sargent curves".
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By his position as Brahe's assistant, Johannes Kepler was first exposed to and seriously interacted with the topic of planetary motion. Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinates.
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Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouville numbers (not to be confused with Liouville numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouville numbers, denoted L, form the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in . L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in , denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations.
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Each independent sample's maximum likelihood estimate is a separate estimate of the "true" parameter set describing the population sampled. Successive estimates from many independent samples will cluster together with the population’s "true" set of parameter values hidden somewhere in their midst. The difference in the logarithms of the maximum likelihood and adjacent parameter sets’ likelihoods may be used to draw a confidence region on a plot whose co- ordinates are the parameters θ1 ... θp.
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The Mathematical Tables Project was one of the largest and most sophisticated computing organizations that operated prior to the invention of the digital electronic computer. Begun in the United States in 1938 as a project of the Works Progress Administration (WPA), it employed 450 unemployed clerks to tabulate higher mathematical functions, such as exponential functions, logarithms, and trigonometric functions. These tables were eventually published in a 28 volume set by Columbia University Press.
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The pattern of spacing of nodes in horsetails, wherein those toward the apex of the shoot are increasingly close together, inspired John Napier to invent logarithms. A superficially similar but entirely unrelated flowering plant genus, mare's tail (Hippuris), is occasionally referred to as "horsetail", and adding to confusion, the name "mare's tail" is sometimes applied to Equisetum. Despite centuries of use in traditional medicine, there is no evidence that Equisetum has any medicinal properties.
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To justify the definition of logarithms, it is necessary to show that the equation :b^x = y \, has a solution and that this solution is unique, provided that is positive and that is positive and unequal to 1. A proof of that fact requires the intermediate value theorem from elementary calculus., section III.3 This theorem states that a continuous function that produces two values ' and ' also produces any value that lies between ' and '.
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Arithmetic expressions involving operations such as additions, subtractions, multiplications, divisions, minima, maxima, powers, exponentials, logarithms, square roots, absolute values, etc., are commonly used in risk analyses and uncertainty modeling. Convolution is the operation of finding the probability distribution of a sum of independent random variables specified by probability distributions. We can extend the term to finding distributions of other mathematical functions (products, differences, quotients, and more complex functions) and other assumptions about the intervariable dependencies.
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Stephen Pohlig (deceased April 14, 2017) was an electrical engineer who worked in the MIT Lincoln Laboratory. As a graduate student of Martin Hellman's at Stanford University in the mid-1970s, he helped develop the underlying concepts of Diffie-Hellman key exchange, including the Pohlig–Hellman exponentiation cipher and the Pohlig–Hellman algorithmOral history interview with Martin Hellman, 2004, Palo Alto, California. Charles Babbage Institute, University of Minnesota, Minneapolis. for computing discrete logarithms.
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The subroutine concept led to the availability of a substantial subroutine library. By 1951, 87 subroutines in the following categories were available for general use: floating point arithmetic; arithmetic operations on complex numbers; checking; division; exponentiation; routines relating to functions; differential equations; special functions; power series; logarithms; miscellaneous; print and layout; quadrature; read (input); nth root; trigonometric functions; counting operations (simulating repeat until loops, while loops and for loops); vectors; and matrices.
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Meteorological registers kept by him over fourteen years were communicated to the same Royal Society in 1740.ibid. xl. 686. His observation of twenty-one sunspots on 21 July 1736 obtained no public record.Stukeley, Memoirs, i. 432. He became in 1719 a member of the Spalding Society, to which he presented an extensive table of logarithms compiled by himself, and he joined William Stukeley in founding The Brazen-Nose Society at Stamford, Lincolnshire in 1736.
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Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Hermite polynomials, Hermite interpolation, Hermite normal form, Hermitian operators, and cubic Hermite splines are named in his honor. One of his students was Henri Poincaré. He was the first to prove that e, the base of natural logarithms, is a transcendental number.
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The earth atmosphere's scale height is about 8.5km, as can be confirmed from this diagram of air pressure p by altitude h: At an altitude of 0, 8.5, and 17 km, the pressure is about 1000, 370, and 140 hPa, respectively. In various scientific contexts, a scale height, usually denoted by the capital letter H, is a distance over which a quantity decreases by a factor of e (the base of natural logarithms, approximately 2.718).
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For sufficiently small finite fields, a table of Zech logarithms allows an especially efficient implementation of all finite field arithmetic in terms of a small number of integer addition/subtractions and table look- ups. The utility of this method diminishes for large fields where one cannot efficiently store the table. This method is also inefficient when doing very few operations in the finite field, because one spends more time computing the table than one does in actual calculation.
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Mathematicians, on the other hand, wrote "log(x)" when they meant loge(x) for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes "ln(x)" when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.
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W.F. Raphael Weldon, the first scientist supported by the committee worked with his wife, Florence Tebb Weldon, who was his computer. Weldon used logarithms and mathematical tables created by August Leopold Crelle and had no calculating machine. Karl Pearson, who had a lab at the University of London, felt that the work Weldon did was "hampered by the committee". However, Pearson did create a mathematical formula that the committee was able to use for data correlation.
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Joseph Morrill Wells was born at Roxbury, Massachusetts on March 1, 1853,New York City Municipal Death Records. the son of Thomas Foster Wells (1822–1903) and his wife, Sarah Morrill Wells (1828–1897). Samuel Adams, the Boston brewer and patriot, was a great-great-grandfather. Joseph Wells's brother, Webster Wells, (1851–1916) was a professor of mathematics at the Massachusetts Institute of Technology, and the author of numerous textbooks such as the Elementary Treatise on Logarithms (1878).
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Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itself came later); there is a story from Anthony à Wood, perhaps not well substantiated, that Napier had a hint from Craig that Longomontanus, a follower of Brahe, was working in a similar direction. It has been shown that Craig had notes on a method of Paul Wittich that used trigonometric identities to reduce a multiplication formula for the sine function to additions.
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In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge alone. Many well-known curves are W-curves, among them conics, logarithmic spirals, powers (like y = x3), logarithms and the helix, but not e.g.
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In a three player game, the indifferent player may choose who he sides with in the case of a challenge. A player who correctly challenges another player wins the game. The loser of a game gains two points, The winner six, and the sider (if he sided with the winner) gains four or two (if he sided with the loser). Equations games become more intricate with the use of factorials, vulgar fractions, and even logarithms, in the Senior division.
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1660)', Oxford Dictionary of National Biography (2004). 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.
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Murphy contributed other mathematical papers to the Cambridge Philosophical Transactions (1831–1836), Philosophical Magazine (1833–1842), and the Philosophical Transactions (1837). Encouraged by Augustus De Morgan, Murphy wrote articles for the Society for the Diffusion of Useful Knowledge and for the Penny Cyclopaedia. His final works were Remark on Primitive Radices (1841), Calculations of Logarithms by Means of Algebraic Fractions (1841), and On Atmospheric Refraction (1842). De Morgan claimed "He had a true genius for mathematical invention".
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Don Coppersmith (born 1950) is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysis. He also improved the quantum Fourier transform discovered by Peter Shor in the same year (1994). He has also worked on algorithms for computing discrete logarithms, the cryptanalysis of RSA, methods for rapid matrix multiplication (see Coppersmith–Winograd algorithm) and IBM's MARS cipher.
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A pentagram can be drawn as a star polygon on a sphere, composed of five great circle arcs, whose all internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici logarithmorum canonis descriptio (Description of the wonderful rule of logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). It was studied later by Carl Friedrich Gauss.
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Each trigonometric and hyperbolic function has its own name and abbreviation both for the reciprocal (for example, ), and its inverse (for example ). A similar convention exists for logarithms, where today usually means , not . To avoid ambiguity, some mathematicians choose to use to denote the compositional meaning, writing for the -th iterate of the function , as in, for example, meaning . For the same purpose, was used by Benjamin Peirce whereas Alfred Pringsheim and Jules Molk suggested instead.
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Mathematical tables are lists of numbers showing the results of a calculation with varying arguments. Tables of trigonometric functions were used in ancient Greece and India for applications to astronomy and celestial navigation. They continued to be widely used until electronic calculators became cheap and plentiful, in order to simplify and drastically speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks, and specialized tables were published for numerous applications.
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The hartley (symbol Hart), also called a ban, or a dit (short for decimal digit), is a logarithmic unit which measures information or entropy, based on base 10 logarithms and powers of 10. One hartley is the information content of an event if the probability of that event occurring is . It is therefore equal to the information contained in one decimal digit (or dit), assuming a priori equiprobability of each possible value. It is named after Ralph Hartley.
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An ivory set of Napier's Bones, an early calculating device invented by John Napier John Napier introduced logarithms as a powerful mathematical tool. With the help of the prominent mathematician Henry Briggs their logarithmic tables embodied a computational advance that made calculations by hand much quicker. His Napier's bones used a set of numbered rods as a multiplication tool using the system of lattice multiplication. The way was opened to later scientific advances, particularly in astronomy and dynamics.
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His research led to an internal technical report proposing the CORDIC algorithm to solve sine and cosine functions and a prototypical computer implementing it. The report also discussed the possibility to compute hyperbolic coordinate rotation, logarithms and exponential functions with modified CORDIC algorithms. Utilizing CORDIC for multiplication and division was also conceived at this time. Based on the CORDIC principle, Dan H. Daggett, a colleague of Volder at Convair, developed conversion algorithms between binary and binary-coded decimal (BCD).
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Leonhard Euler was the first to apply binary logarithms to music theory, in 1739. The powers of two have been known since antiquity; for instance, they appear in Euclid's Elements, Props. IX.32 (on the factorization of powers of two) and IX.36 (half of the Euclid–Euler theorem, on the structure of even perfect numbers). And the binary logarithm of a power of two is just its position in the ordered sequence of powers of two.
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A microarray for approximately 8700 genes. The expression rates of these genes are compared using binary logarithms. In bioinformatics, microarrays are used to measure how strongly different genes are expressed in a sample of biological material. Different rates of expression of a gene are often compared by using the binary logarithm of the ratio of expression rates: the log ratio of two expression rates is defined as the binary logarithm of the ratio of the two rates.
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The simple formula for the factorial, x! = 1 \times 2 \times \cdots \times x, cannot be used directly for fractional values of x since it is only valid when is a natural number (or positive integer). There are, relatively speaking, no such simple solutions for factorials; no finite combination of sums, products, powers, exponential functions, or logarithms will suffice to express x!; but it is possible to find a general formula for factorials using tools such as integrals and limits from calculus.
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Starting with known special cases, the calculation of logarithms and trigonometric functions can be performed by looking up numbers in a mathematical table, and interpolating between known cases. For small enough differences, this linear operation was accurate enough for use in navigation and astronomy in the Age of Exploration. The uses of interpolation have thrived in the past 500 years: by the twentieth century Leslie Comrie and W.J. Eckert systematized the use of interpolation in tables of numbers for punch card calculation.
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In mathematics, the function field sieve was introduced in 1994 by Leonard Adleman as an efficient technique for extracting discrete logarithms over finite fields of small characteristic, and elaborated by Adleman and Huang in 1999. Sieving for points at which a polynomial-valued function is divisible by a given polynomial is not much more difficult than sieving over the integers – the underlying structure is fairly similar, and Gray code provides a convenient way to step through multiples of a given polynomial very efficiently.
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Their logarithms did not lead to irrational numbers, however Theon tackled this discussion head on. He acknowledged that “one can prove that” the tone of value 9/8 cannot be divided into equal parts and so it is a number in itself. Many Pythagoreans believed in the existence of irrational numbers, but did not believe in using them because they were unnatural and not positive integers. Theon also does an amazing job of relating quotients of integers and musical intervals.
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Tabulae, 1670 BEIC) Adriaan Vlacq (1600-1667) was a Dutch book publisher and author of mathematical tables. Born in Gouda, Vlacq published a table of logarithms from 1 to 100,000 to 10 decimal places in 1628 in his Arithmetica logarithmica. This table extended Henry Briggs' original tables which only covered the values 1-20,000 and 90,001 to 100,000. The new table was computed by Ezechiel de Decker and Vlacq who calculated and added 70,000 further values to complete the tables.
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In computer science, they count the number of steps needed for binary search and related algorithms. Other areas in which the binary logarithm is frequently used include combinatorics, bioinformatics, the design of sports tournaments, and photography. Binary logarithms are included in the standard C mathematical functions and other mathematical software packages. The integer part of a binary logarithm can be found using the find first set operation on an integer value, or by looking up the exponent of a floating point value.
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However this system has been broken by several attacks : one from Shamir, one by Adleman, and the low density attack. However, there exist modern knapsack cryptosystems that are considered secure so far: among them is Nasako-Murakami 2006. Knapsack cryptosystems, when not subject to classical cryptoanalysis, are believed to be difficult even for quantum computers. That is not the case for systems that rely on factoring large integers, like RSA, or computing discrete logarithms, like ECDSA, problems solved in polynomial time with Shor's algorithm.
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In probability theory and computer science, a log probability is simply a logarithm of a probability. The use of log probabilities means representing probabilities on a logarithmic scale, instead of the standard [0, 1] unit interval. Since the probability of independent events multiply, and logarithms convert multiplication to addition, log probabilities of independent events add. Log probabilities are thus practical for computations, and have an intuitive interpretation in terms of information theory: the negative of the log probability is the information content of an event.
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In it Briggs stated he had seen a map that had been brought from Holland that showed the Island of California. The tract was republished three years later (1625) in Pvrchas His Pilgrimes (vol 3, p848). In 1624 his Arithmetica Logarithmica was published, in folio, a work containing the logarithms of thirty thousand natural numbers to fourteen decimal places (1-20,000 and 90,001 to 100,000). This table was later extended by Adriaan Vlacq to 10 places, and by Alexander John Thompson to 20 places in 1952.
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If an improved algorithm can be found to solve the problem, then the system is weakened. For example, the security of the Diffie–Hellman key exchange scheme depends on the difficulty of calculating the discrete logarithm. In 1983, Don Coppersmith found a faster way to find discrete logarithms (in certain groups), and thereby requiring cryptographers to use larger groups (or different types of groups). RSA's security depends (in part) upon the difficulty of integer factorization — a breakthrough in factoring would impact the security of RSA.
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Physically, this arises from the fact that \epsilon is related to the induced dipole interactions between two particles. Given two particles with instantaneous dipole \mu_i, \mu_j respectively, their interactions correspond to the products of \mu_i, \mu_j. An arithmetic average of \epsilon_i and \epsilon_j will not however, result in the average of the two dipole products, but the average of their logarithms would be. These rules are the most widely used and are the default in many molecular simulation packages, but are not without failings.
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A set of John Napier's calculating tables from around 1680 Scottish mathematician and physicist John Napier discovered that the multiplication and division of numbers could be performed by the addition and subtraction, respectively, of the logarithms of those numbers. While producing the first logarithmic tables, Napier needed to perform many tedious multiplications. It was at this point that he designed his 'Napier's bones', an abacus-like device that greatly simplified calculations that involved multiplication and division.A Spanish implementation of Napier's bones (1617), is documented in .
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Margaret Brisbane (née Napier) was a member of the Napier family of Merchiston, Scotland, and was the great-granddaughter of John Napier, the inventor of logarithms. She was the daughter of Archibald Napier, 2nd Lord Napier and Lady Elizabeth Erskine, daughter of John Erskine, 19th Earl of Mar. Upon the death of her brother, Archibald Napier, 3rd Lord Napier, the title passed through her sister Jean to her nephew Thomas Nicolson, 4th Lord Napier. When he, too, died unmarried and without heir, the title passed to her.
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Adrian Rice, Oxford Dictionary of National Biography, Oxford University Press, 2004; online edn, May 2008, accessed 13 December 2009 He became a schoolteacher and through hard work and patronage became assistant to Nevil Maskelyne, the Astronomer Royal in 1773.. An early scientific paper was: "Theorems for Computing Logarithms." By the Rev. John Hellins; Communicated by the Rev. Nevil Maskelyne, D. D., F. R. S. and Astronomer Royal,in Philosophical Transactions Series I,(1780), volume 70, pages 307–317 [Referred to by Smithsonian/NASA ADS Astronomy Abstract Service.
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B-smooth and B-powersmooth numbers have applications in number theory, such as in Pollard's p − 1 algorithm and ECM. Such applications are often said to work with "smooth numbers," with no B specified; this means the numbers involved must be B-powersmooth, for some unspecified small number B. As B increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O(B1/2)—for groups of B-smooth order.
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Burroughs Corporation in about 1912 built a machine for the Nautical Almanac Office which was used as a difference engine of second-order. It was later replaced in 1929 by a Burroughs Class 11 (13-digit numbers and second-order differences, or 11-digit numbers and [at least up to] fifth-order differences). Alexander John Thompson about 1927 built integrating and differencing machine (13-digit numbers and fifth-order differences) for his table of logarithms "Logarithmetica britannica". This machine was composed of four modified Triumphator calculators.
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Logarithms are about 40 instructions, with anti-log taking about 20 more. The code to normalize and display the computed values are roughly the same in both the TI and Sinclair programs. The design of the algorithms meant that some calculations, such as arccos0.2, could take up to 15 seconds, whereas the HP-35 was designed to complete calculations in under a second. Accuracy in scientific functions was also limited to around three digits at most, and there were a number of bugs and limitations.
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He published Anthony Munday's translation of Amadis de Gaul (1618–19), one of the chivalric romances that were enormously popular in the era. Okes published A Short Treatise on Magnetical Bodies and Motions (1613) by Mark Ridley, a follower of William Gilbert, and John Napier's A Description of the Admirable Table of Logarithms (1616). Printers who published usually needed a retail outlet for their wares. The title page of Okes's edition of The Silver Age states that the book would be sold by Benjamin Lightfoote.
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The FP is an infectious disease with horizontal transmission. An alphaherpesvirus initially called fibropapilloma-associated turtle herpesvirus (FPTHV), and now called Chelonid alphaherpesvirus 5, is believed to be the causative agent of the disease. The reason for this belief is because nearly all tissue samples tested from turtles with lesions carry genetic material of this herpesvirus, varying between 95 and 100% depending on different studies and locations. The DNA loads of the herpesvirus in tumour tissue are 2.5–4.5 logarithms higher than in uninfected tissue.
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Von Neumann describes a detailed design of a "very high speed automatic digital computing system." He divides it into six major subdivisions: a central arithmetic part, CA, a central control part, CC, memory, M, input, I, output, O, and (slow) external memory, R, such as punched cards, Teletype tape, or magnetic wire or steel tape. The CA will perform addition, subtraction, multiplication, division and square root. Other mathematical operations, such as logarithms and trigonometric functions are to be done with table look up and interpolation, possibly biquadratic.
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By the end of the century, Europeans and Indians were aware of logarithms, electricity, the telescope and microscope, calculus, universal gravitation, Newton's Laws of Motion, air pressure and calculating machines due to the work of the first scientists of the Scientific Revolution, including Galileo Galilei, Johannes Kepler, René Descartes, Pierre Fermat, Blaise Pascal, Robert Boyle, Christiaan Huygens, Antonie van Leeuwenhoek, Robert Hooke, Isaac Newton, and Gottfried Wilhelm Leibniz. It was also a period of development of culture in general (especially theater, music, visual arts and philosophy).
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It incorporated the first multiplier-accumulator (MAC), and was the first to exploit a MAC to perform division (using multiplication seeded by reciprocal, via the convergent series ). Ludgate's engine used a mechanism similar to slide rules, but employed his unique discrete Logarithmic Indexes (now known as Irish logarithms (Boys, 1909)), and provided a very novel memory using concentric cylinders, storing numbers as displacements of rods in shuttles. His design featured several other novel features, including for program control (e.g. preemption and subroutines – or microcode, depending on viewpoint).
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The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different random factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called Gibrat's law.
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To make this correction the navigator would measure the altitudes of the moon and sun (or star) at about the same time as the lunar distance angle. Only rough values for the altitudes were required. Then a calculation with logarithms or graphical tables requiring ten to fifteen minutes' work would convert the observed angle to a geocentric lunar distance. The navigator would compare the corrected angle against those listed in the almanac for every three hours of Greenwich time, and interpolate between those values to get the actual Greenwich time aboard ship.
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Didot invented the word "stereotype", which in printing refers to the metal printing plate created for the actual printing of pages (as opposed to printing pages directly with movable type), and used the process extensively, revolutionizing the book trade by his cheap editions. His manufactory was a place of pilgrimage for the printers of the world. He first used the process in his edition of Callet’s Tables of Logarithms (1795), in which he secured an accuracy till then unattainable. He published stereotyped editions of French, English and Italian classics at a very low price.
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In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function, whether applied to real numbers or complex numbers.
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More complicated equations can sometimes be expressed as the sum of functions of the three variables. For example, the nomogram at the top of this article could be constructed as a parallel-scale nomogram because it can be expressed as such a sum after taking logarithms of both sides of the equation. The scale for the unknown variable can lie between the other two scales or outside of them. The known values of the calculation are marked on the scales for those variables, and a line is drawn between these marks.
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In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. The result, proved by , subsumed many earlier results in transcendental number theory and solved a problem posed by Alexander Gelfond nearly fifteen years earlier.See the final paragraph of Gelfond (1952). Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.
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The Indian mathematician Virasena worked with the concept of ardhaccheda: the number of times a number of the form 2n could be halved. For exact powers of 2, this equals the binary logarithm, but it differs from the logarithm for other numbers. He described a product formula for this concept and also introduced analogous concepts for base 3 (trakacheda) and base 4 (caturthacheda). Michael Stifel published Arithmetica integra in Nuremberg in 1544, which contains a table of integers and powers of 2 that has been considered an early version of a table of binary logarithms.
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It is indicated by log(x), log10(x), or sometimes Log(x) with a capital L (however, this notation is ambiguous, since it can also mean the complex natural logarithmic multi-valued function). On calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when they write "log". To mitigate this ambiguity, the ISO 80000 specification recommends that log10(x) should be written lg(x), and loge(x) should be ln(x). Page from a table of common logarithms.
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The Carlitz derivative is an operation similar to usual differentiation have been devised with the usual context of real or complex numbers changed to local fields of positive characteristic in the form of formal Laurent series with coefficients in some finite field Fq (it is known that any local field of positive characteristic is isomorphic to a Laurent series field). Along with suitably defined analogs to the exponential function, logarithms and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.
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Human computers were used to compile 18th and 19th century Western European mathematical tables, for example those for trigonometry and logarithms. Although these tables were most often known by the names of the principal mathematician involved in the project, such tables were often in fact the work of an army of unknown and unsung computers. Ever more accurate tables to a high degree of precision were needed for navigation and engineering. Approaches differed, but one was to break up the project into a form of long distance work from home piece work.
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Huygens preferred meantone temperament; he innovated in 31 equal temperament, which was not itself a new idea but known to Francisco de Salinas, using logarithms to investigate it further and show its close relation to the meantone system. In 1654, Huygens returned to his father's house in The Hague, and was able to devote himself entirely to research. The family had another house, not far away at Hofwijck, and he spent time there during the summer. His scholarly life did not allow him to escape bouts of depression.
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A Törnqvist index is a discrete approximation to a continuous Divisia index. A Divisia index is a theoretical construct, a continuous-time weighted sum of the growth rates of the various components, where the weights are the component's shares in total value. For a Törnqvist index, the growth rates are defined to be the difference in natural logarithms of successive observations of the components (i.e. their log-change) and the weights are equal to the mean of the factor shares of the components in the corresponding pair of periods (usually years).
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The pH-metric set of techniques determine lipophilicity pH profiles directly from a single acid- base titration in a two-phase water–organic-solvent system. Hence, a single experiment can be used to measure the logarithms of the partition coefficient (log P) giving the distribution of molecules that are primarily neutral in charge, as well as the distribution coefficient (log D) of all forms of the molecule over a pH range, e.g., between 2 and 12. The method does, however, require the separate determination of the pKa value(s) of the substance.
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View towards Holy Corner from Colinton Road Merchiston spans the catchment areas for Boroughmuir and Tynecastle secondary schools. Primary education is provided by Craiglockhart Primary School and Bruntsfield Primary School. Also in the area are a number of independent schools including George Watson's College, Merchiston Castle School and a Steiner School. Merchiston Tower A campus forming a major part of Edinburgh Napier University is in the area; it includes Merchiston Tower (or Castle), once the home of John Napier, 8th Laird of Merchiston and the inventor of logarithms.
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His memory was so great, that in resolving a question he could leave off and resume the operation again at the same point after the lapse of several months. His perpetual application to figures prevented the acquisition of other knowledge. Among the examples of Buxton's arithmetical feats which are given are his calculation of the product of a farthing doubled 139 times. The result, expressed in pounds, extends to thirty-nine figures, and is correct so far as it can be readily verified by the use of logarithms.
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As a mathematician, Vince wrote on many aspects of his expertise, including logarithms and imaginary numbers. His Observations on the Theory of the Motion and Resistance of Fluids and Experiments upon the Resistance of Bodies Moving in Fluids had later importance to aviation history. He was also author of the influential A Complete System of Astronomy (3 vols. 1797-1808). Vince also published the pamphlet The Credibility of Christianity Vindicated, In Answer to Mr. Hume's Objections; In Two Discourses Preached Before the University of Cambridge by the Rev.
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By 1800, Routledge had become Manager at the Round Foundry. Somehow, along the way, he learned the value of logarithms and thereby had the means of developing a method of measuring "all kinds of metals and other bodies" British Library, Call 717.a.18 needed for engineering purposes. Using the principles of Edmund Gunter's (1581–1626) logarithmic scales and William Oughtred's (1574–1660) sliding rule, Routledge combined a 12-inch brass slide containing the logarithmic scales with an ordinary 2-foot ruler to which he added a table of commonly used references called gauge points.
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He invented logarithms as a working tool for himself for his astronomical calculations, but as a "craftsman/scholar" rather than a "book scholar" he failed to publish his invention for a long time. In 1592 Rudolf II, Holy Roman Emperor in Prague received from his uncle, the Landgrave of Hesse-Kassel, a Bürgi globe and insisted that Bürgi deliver it personally. From then on Bürgi commuted between Kassel and Prague, and finally entered the service of the emperor in 1604 to work for the imperial astronomer Johannes Kepler.Ralf Kern.
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Atanasoff's father later became an electrical engineer, whereas his mother, Iva Lucena Purdy (of mixed French and Irish ancestry), was a teacher of mathematics. Young Atanasoff's ambitions and intellectual pursuits were in part influenced by his parents, whose interests in the natural and applied sciences cultivated in him a sense of critical curiosity and confidence. Atanasoff was raised in Brewster, Florida. At the age of nine he learned to use a slide rule, followed shortly by the study of logarithms, and subsequently completed high school at Mulberry High School in two years.
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Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables (see below; in the classical case, what would be the analog of free multiplicative convolution can be reduced to additive convolution by passing to logarithms of random variables). These operations have some interpretations in terms of empirical spectral measures of random matrices.Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010).
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The calculator can be set to display values in binary, octal, or hexadecimal form, as well as the default decimal. When a non-decimal base is selected, calculation results are truncated to integers. Regardless of which display base is set, non- decimal numbers must be entered with a suffix indicating their base, which involves three or more extra keystrokes. When hexadecimal is selected, the row of six keys normally used for floating-point functions (trigonometry, logarithms, exponentiation, etc.) are instead allocated to the hex digits A to F (although they are physically labelled to ).
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Methods suitable for arbitrary large numbers that do not depend on the size of its factors include the quadratic sieve and general number field sieve. As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve. the largest number known to have been factored by a general-purpose algorithm is RSA-240, which has 240 decimal digits (795 bits) and is the product of two large primes.Emmanuel Thomé, “795-bit factoring and discrete logarithms,” December 2, 2019.
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Graphical projection was once commonly taught, though this has been superseded by trigonometry, logarithms, sliderules and computers which made arithmetical calculations increasingly trivial/ Graphical projection was once the mainstream method for laying out a sundial but has been sidelined and is now only of academic interest. The first known document in English describing a schema for graphical projection was published in Scotland in 1440, leading to a series of distinct schema for horizontal dials each with characteristics that suited the target latitude and construction method of the time.
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By default, Calculator runs in standard mode, which resembles a four-function calculator. More advanced functions are available in scientific mode, including logarithms, numerical base conversions, some logical operators, operator precedence, radian, degree and gradians support as well as simple single-variable statistical functions. It does not provide support for user-defined functions, complex numbers, storage variables for intermediate results (other than the classic accumulator memory of pocket calculators), automated polar-cartesian coordinates conversion, or support for two-variables statistics. Calculator supports keyboard shortcuts; all Calculator features have an associated keyboard shortcut.
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John Napier, inventor of logarithms, was born in Merchiston Tower and lived and died in the city. His house now forms part of the original campus of Napier University which was named in his honour. He lies buried under St. Cuthbert's Church. James Clerk Maxwell, founder of the modern theory of electromagnetism, was born at 14 India Street (now the home of the James Clerk Maxwell Foundation) and educated at the Edinburgh Academy and the University of Edinburgh, as was the engineer and telephone pioneer Alexander Graham Bell.
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The concept of the natural logarithm was worked out by Gregoire de Saint-Vincent and Alphonse Antonio de Sarasa before 1649. Their work involved quadrature of the hyperbola with equation , by determination of the area of hyperbolic sectors. Their solution generated the requisite "hyperbolic logarithm" function, which had the properties now associated with the natural logarithm. An early mention of the natural logarithm was by Nicholas Mercator in his work Logarithmotechnia, published in 1668, although the mathematics teacher John Speidell had already compiled a table of what in fact were effectively natural logarithms in 1619.
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Note that, while the equal tempered pitches increase exponentially, the pitches found lower on the Non- Pythagorean scale have frequencies that are farther apart while pitches found higher on the scale are closer together. The Non-Pythagorean scale is a musical scale and tuning, based on natural logarithms, conceived and developed by Schneider. The term "Non-Pythagorean" is a reference to the Pythagorean tuning approximated by the chromatic scale. The scale was introduced in 2007 with the release of New Magnetic Wonder, the sixth studio album by The Apples in Stereo.
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In this process, the intrinsic charges of polypeptides become negligible when compared to the negative charges contributed by SDS. Thus polypeptides after treatment become rod-like structures possessing a uniform charge density, that is same net negative charge per unit weight. The electrophoretic mobilities of these proteins is a linear function of the logarithms of their molecular weights. Without SDS, different proteins with similar molecular weights would migrate differently due to differences in mass-charge ratio, as each protein has an isoelectric point and molecular weight particular to its primary structure.
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This is the case of the above, with Gal(K/Q) an abelian group, in which all the ρ can be replaced by Dirichlet characters (via class field theory) for some modulus f called the conductor. Therefore all the L(1) values occur for Dirichlet L-functions, for which there is a classical formula, involving logarithms. By the Kronecker–Weber theorem, all the values required for an analytic class number formula occur already when the cyclotomic fields are considered. In that case there is a further formulation possible, as shown by Kummer.
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Three accomplishments Durand thought noteworthy during his years at Cornell were the development of logarithmic paper, developing theoretically and mechanically an averaging radial planimeter, and research on marine propellers. In 1893 Durand developed and introduced Logarithmic graph paper, where the logarithmic scale is marked off in distances proportional to the logarithms of the values being represented. Keuffel and Esser Company listed logarithmic paper in their general catalog as "Durand's Logarithmic Paper" as late as 1936. In 1893 Durand published a paper mathematically describing a radial planimeter for averaging values plotted in polar coordinates.
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The term significand was introduced by George Forsythe and Cleve Moler in 1967 and is the word used in the IEEE standard. However, in 1946 Arthur Burks used the terms mantissa and characteristic to describe the two parts of a floating-point number (Burks et al.) and that usage remains common among computer scientists today. Mantissa and characteristic have long described the two parts of the logarithm found on tables of common logarithms. While the two meanings of exponent are analogous, the two meanings of mantissa are not equivalent.
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For this reason, the use of mantissa for significand is discouraged by some including the creator of the standard, William Kahan and prominent computer programmer and author of The Art of Computer Programming, Donald E. Knuth The confusion is because scientific notation and floating- point representation are log-linear, not logarithmic. To multiply two numbers, given their logarithms, one just adds the characteristic (integer part) and the mantissa (fractional part). By contrast, to multiply two floating-point numbers, one adds the exponent (which is logarithmic) and multiplies the significand (which is linear).
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In May 1833, Charles Chalmers took a lease of Merchiston Castle (the former home of John Napier, the inventor of logarithms) — which at that time stood in rural surroundings — and opened his academy, starting with thirty boys. Over time, the number of pupils grew and the Merchiston Castle became too small to accommodate the school. The governors decided to purchase 90 acres of ground at the Colinton House estate, four miles south-west of Edinburgh. Building began in 1928 including the Chalmers and Rogerson boarding houses, designed by Sir Robert Lorimer.
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This computation appears independent of the kind of black hole, since the given Immirzi parameter is always the same. However, Krzysztof Meissner and Marcin Domagala with Jerzy Lewandowski have corrected the assumption that only the minimal values of the spin contribute. Their result involves the logarithm of a transcendental number instead of the logarithms of integers mentioned above. The Immirzi parameter appears in the denominator because the entropy counts the number of edges puncturing the event horizon and the Immirzi parameter is proportional to the area contributed by each puncture.
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MathomaticFSF Free Software Directory entry is a free, portable, general- purpose computer algebra system (CAS) that can symbolically solve, simplify, combine, and compare algebraic equations, and can perform complex number, modular, and polynomial arithmetic, along with standard arithmetic. It does some symbolic calculus (derivative, extrema, Taylor series, and polynomial integration and Laplace transforms), numerical integration, and handles all elementary algebra except logarithms. Trigonometric functions can be entered and manipulated using complex exponentials, with the GNU m4 preprocessor. Not currently implemented are general functions like f(x), arbitrary-precision and interval arithmetic, and matrices.
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Mikhail Lomonosov was himself taught by this book,. which he called the "gates to his own erudition". This book was more an encyclopedia of mathematics than a textbook, and the first secular book to be printed in Russia.. In 1703, Magnitsky also produced a Russian edition of Adriaan Vlacq's log tables called Таблицы логарифмов и синусов, тангенсов и секансов (Tables of logarithms, sines, tangents, and secants). Legend has it that Leonty Magnitsky was nicknamed Magnitsky by Peter the Great, who considered him a "people's magnet" (магнит, or "magnit" in Russian).
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The original version of 24 is played with an ordinary deck of playing cards with all the face cards removed. The aces are taken to have the value 1 and the basic game proceeds by having 4 cards dealt and the first player that can achieve the number 24 exactly using only allowed operations (addition, subtraction, multiplication, division, and parentheses) wins the hand. Some advanced players allow exponentiation, roots, logarithms, and other operations. For short games of 24, once a hand is won, the cards go to the player that won.
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When Wang Laboratories found that the hp 9100A used an approach similar to the factor combining method in their earlier LOCI-1 (September 1964) and LOCI-2 (January 1965) Logarithmic Computing Instrument desktop calculators, they unsuccessfully accused Hewlett-Packard of infringement of one of An Wang's patents in 1968. John Stephen Walther at Hewlett-Packard generalized the algorithm into the Unified CORDIC algorithm in 1971, allowing it to calculate hyperbolic functions, natural exponentials, natural logarithms, multiplications, divisions, and square roots. The CORDIC subroutines for trigonometric and hyperbolic functions could share most of their code.
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Feldman's mentor Gelfond obtained his most famous result in 1948 in his eponymous theorem, also known as the 7th Hilbert's problem: : If α and β are algebraic numbers (with α≠0 and α≠1), and if β is not a real rational number, then any value of αβ is a transcendental number. In 1949, Feldman further improved Gelfond's method to estimate of the measure of transcendence for logarithms of algebraic numbers and periods of elliptic curves. Of special importance is his result from 1960 on the measure of the transcendence of the number \pi.
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One of the spectacular consequences of the analytic subgroup theorem was the Isogeny Theorem published by Masser and Wüstholz. A direct consequence is the Tate conjecture for abelian varieties which Gerd Faltings had proved with totally different methods which has many applications in modern arithmetic geometry. Using the multiplicity estimates for group varieties Wüstholz succeeded to get the final expected form for lower bound for linear forms in logarithms. This was put into an effective form in a joint work of him with Alan Baker which marks the current state of art.
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In order for this generator to be secure, the prime number p needs to be large enough so that computing discrete logarithms modulo p is infeasible. To be more precise, any method that predicts the numbers generated will lead to an algorithm that solves the discrete logarithm problem for that prime. There is a paper discussing possible examples of the quantum permanent compromise attack to the Blum–Micali construction. This attacks illustrate how a previous attack to the Blum–Micali generator can be extended to the whole Blum–Micali construction, including the Blum Blum Shub and Kaliski generators.
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All measurements in MEGS are done using a logarithmic scale. The units on this scale are called "Attribute Points" or "APs" in the superhero games and simply "Units" in Underground, with each unit on the scale represents exponentially increasing values for length, weight, time, etc. Because of the nature of logarithms and exponents, 0 APs/Units is a meaningful, positive value. Indeed, even negative APs/Units still represent positive values, though exponentially smaller, down to -100 APs, which is defined as absolute zero for all units. In the superhero games, 1 AP corresponds to 8 seconds, , , , $50, or a typed page of information.
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Quantum computers may become a technological reality; it is therefore important to study cryptographic schemes used against adversaries with access to a quantum computer. The study of such schemes is often referred to as post-quantum cryptography. The need for post-quantum cryptography arises from the fact that many popular encryption and signature schemes (schemes based on ECC and RSA) can be broken using Shor's algorithm for factoring and computing discrete logarithms on a quantum computer. Examples for schemes that are, as of today's knowledge, secure against quantum adversaries are McEliece and lattice-based schemes, as well as most symmetric-key algorithms.
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The basic A-Level course consists of six modules, four pure modules (C1, C2, C3, and C4) and two applied modules in Statistics, Mechanics and/or Decision Mathematics. The C1 through C4 modules are referred to by A-level textbooks as "Core" modules, encompassing the major topics of mathematics such as logarithms, differentiation/integration and geometric/arithmetic progressions. The two chosen modules for the final two parts of the A-Level are determined either by a student's personal choices, or the course choice of their school/college, though it commonly takes the form of S1 (Statistics) and M1 (Mechanics).
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Page 10 of Raymond Davis, Jr. and F. M. Walters, Jr., Scientific Papers of the Bureau of Standards, No. 439 (Part of Vol. 18) "Sensitometry of Photographic Emulsions and a Survey of the Characteristics of Plates and Films of American Manufacture," 1922. The next page starts with the H & D quote: "In a theoretically perfect negative, the amounts of silver deposited in the various parts are proportional to the logarithms of the intensities of light proceeding from the corresponding parts of the object." The assumption here, based on empirical observations, is that the "amount of silver" is proportional to the optical density.
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The definition of contrast ratio is therefore re-stated as follows : 'The ratio between the opacities of the darkest and lightest points in the film image', thus: contrast ratio = Omax. / Omin. As we have already seen, opacity is not easily measured with standard photographic equipment—but the logarithm of opacity is continually measured since, in fact, it is the unit of image saturation known as density. Since density is a logarithm we must take the ratio of the anti-logarithms of the maximum and minimum densities in the image in order to arrive at the contrast ratio.
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Part of a 20th-century precomputed mathematical table of common logarithms. In algorithms, precomputation is the act of performing an initial computation before run time to generate a lookup table that can be used by an algorithm to avoid repeated computation each time it is executed. Precomputation is often used in algorithms that depend on the results of expensive computations that don't depend on the input of the algorithm. A trivial example of precomputation is the use of hardcoded mathematical constants, such as π and e, rather than computing their approximations to the necessary precision at run time.
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The first algorithm for finding a CEEI partition in this case was developed by Reijnierse and Potters. A more computationally-efficient algorithm was developed by Aziz and Ye. In fact, every CEEI cake-partition maximizes the product of utilities, and vice versa - every partition that maximizes the product of utilities is a CEEI. Therefore, a CEEI can be found by solving a convex program maximizing the sum of the logarithms of utilities. For two agents, the adjusted winner procedure can be used to find a PEEF allocation that is also equitable (but not necessarily a CEEI).
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Bürgi constructed a table of progressions what is now understood as antilogarithmsJost Bürgi, Arithmetische und Geometrische Progress Tabulen … [Arithmetic and Geometric Progression Tables … ], (Prague, (Czech Republic): University [of Prague] Press, 1620). Available on-line at: Bavarian State Library, Germany Unfortunately, Bürgi did not include, with his table, instructions for using the table. That was published separately. The contents of that publication were reproduced in: Hermann Robert Gieswald, Justus Byrg als Mathematiker, und dessen Einleitung zu seinen Logarithmen [Justus Byrg as a mathematician, and an introduction to his logarithms] (Danzig, Prussia: St. Johannisschule, 1856), pages 26 ff.
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Two types of punched cards used to program the machine. Foreground: 'operational cards', for inputting instructions; background: 'variable cards', for inputting data Babbage's first attempt at a mechanical computing device, the Difference Engine, was a special-purpose machine designed to tabulate logarithms and trigonometric functions by evaluating finite differences to create approximating polynomials. Construction of this machine was never completed; Babbage had conflicts with his chief engineer, Joseph Clement, and ultimately the British government withdrew its funding for the project. During this project, Babbage realised that a much more general design, the Analytical Engine, was possible.
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That is, scaling by a constant c simply multiplies the original power-law relation by the constant c^{-k}. Thus, it follows that all power laws with a particular scaling exponent are equivalent up to constant factors, since each is simply a scaled version of the others. This behavior is what produces the linear relationship when logarithms are taken of both f(x) and x, and the straight-line on the log–log plot is often called the signature of a power law. With real data, such straightness is a necessary, but not sufficient, condition for the data following a power-law relation.
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His findings led to the natural logarithm function, once called the hyperbolic logarithm since it is obtained by integrating, or finding the area, under the hyperbola.Martin Flashman The History of Logarithms from Humboldt State University Before 1748 and the publication of Introduction to the Analysis of the Infinite, the natural logarithm was known in terms of the area of a hyperbolic sector. Leonhard Euler changed that when he introduced transcendental functions such as 10x. Euler identified e as the value of b producing a unit of area (under the hyperbola or in a hyperbolic sector in standard position).
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Traditionally, steel detailing was accomplished via manual drafting methods, using pencils, paper, and drafting tools such as a parallel bar or drafting machine, triangles, templates of circles and other useful shapes, and mathematical tables, such as tables of logarithms and other useful calculational aids. Eventually, hand-held calculators were incorporated into the traditional practice. Today, manual drafting has been largely replaced by computer-aided drafting (CAD). A steel detailer using computer- aided methods creates drawings on a computer, using software specifically designed for the purpose, and printing out drawings on paper only when they are complete.
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Logarithmic number systems have been independently invented and published at least three times as an alternative to fixed-point and floating- point number systems. Nicholas Kingsbury and Peter Rayner introduced "logarithmic arithmetic" for digital signal processing (DSP) in 1971. A similar LNS named "signed logarithmic number system" (SLNS) was described in 1975 by Earl Swartzlander and Aristides Alexopoulos; rather than use two's complement notation for the logarithms, they offset them (scale the numbers being represented) to avoid negative logs. Samuel Lee and Albert Edgar described a similar system, which they called the "Focus" number system, in 1977.
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Thus public key systems require longer key lengths than symmetric systems for an equivalent level of security. 3072 bits is the suggested key length for systems based on factoring and integer discrete logarithms which aim to have security equivalent to a 128 bit symmetric cipher. Elliptic curve cryptography may allow smaller-size keys for equivalent security, but these algorithms have only been known for a relatively short time and current estimates of the difficulty of searching for their keys may not survive. As early as 2004, a message encrypted using a 109-bit key elliptic curve algorithm had been broken by brute force.
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He found that none of the different theories of color were quite related to nature, so he began putting color into what he felt was their natural sequence. He began by mixing his own basic, primary or "parent" colors and from that eventually mixing colors according to his experience and ideas about how one color can become another color can become another color, etc. until it rounded the whole spectrum. Eventually it became impossible to continue by "eyeballing" colors into place so he began using logarithms and mathematical curves to graph progressions, giving the colors numerical values.
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A third (posthumous) edition, with corrections and a supplemental treatise on logarithms, appeared 1729. Nicolas de Malézieu also translated Euripides’ Iphigenia in Tauris as well as poems, songs and sketches, which were published in 1712 in Les Divertissements de Sceaux and in 1725 in the Suite des Divertissements. Among these pieces are Philémon et Baucis, Le Prince de Cathay, Les Importuns de Chatenay, La Grande Nuit de l'éclipse, L'Hôte de Lemnos, La Tarentole and L'Heautontimorumenos. Often written in a single day, these pieces were set to music and staged for the amusement of the duchess, to whom Malézieu also gave courses in astronomy.
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Parallel line assay An antibiotic standard (shown in red) and test preparation (shown in blue) are applied at three dose levels to sensitive microorganisms on a layer of agar in petri dishes. The stronger the dose the larger the zone of inhibition of growth of the microorganisms. The biological response u is in this case the zone of inhibition and the diameter of this zone f(u) can be used as the measurable response. The doses z are transformed to logarithms x=\log (z) and the method of least squares is used to fit two parallel lines to the data.
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The S-unit equation is a Diophantine equation :u + v = 1 with u, v restricted to being S-units of K. The number of solutions of this equation is finite and the solutions are effectively determined using estimates for linear forms in logarithms as developed in transcendental number theory. A variety of Diophantine equations are reducible in principle to some form of the S-unit equation: a notable example is Siegel's theorem on integral points on elliptic curves, and more generally superelliptic curves of the form yn=f(x). A computational solver for S-unit equation is available in the software SageMath.
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Attending that city's German- language school, León, who had previously learned English, Spanish, and Guaraní, learned German as well. He was 18 when he began working as a clerk for the cold-storage facilities of Swift & Co.'s slaughterhouse in the Zevallos Cué barrio of Asunción. Through a friendship struck up with the Frenchman Emile Lelieur, he learned French and gained the opportunity to read classic authors, to learn elementary mathematics and the use of logarithms. In 1919 he moved to Buenos Aires, and two years later his restless spirit led him to the jungles of Caaguazú, where he worked harvesting yerba maté.
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In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
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In Book 6, part 4, page 586, Proposition CIX, he proves that if the abscissas of points are in geometric proportion, then the areas between a hyperbola and the abscissas are in arithmetic proportion. This finding allowed Saint-Vincent's former student, Alphonse Antonio de Sarasa, to prove that the area between a hyperbola and the abscissa of a point is proportional to the abscissa's logarithm, thus uniting the algebra of logarithms with the geometry of hyperbolas. See also: Enrique A. González-Velasco, Journey through Mathematics: Creative Episodes in Its History (New York, New York: Springer, 2011), page 118.
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Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem. The goal is to compute \gamma such that \alpha ^ \gamma = \beta, where \beta belongs to a cyclic group G generated by \alpha. The algorithm computes integers a, b, A, and B such that \alpha^a \beta^b = \alpha^A \beta^B. If the underlying group is cyclic of order n, \gamma is one of the solutions of the equation (B-b) \gamma = (a-A) \pmod n.
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Optical illusion caused by lateral inhibition: the Hermann grid illusion The concept of neural inhibition (in motor systems) was well known to Descartes and his contemporaries. Sensory inhibition in vision was inferred by Ernst Mach in 1865 as depicted in his mach band. Inhibition in single sensory neurons was discovered and investigated starting in 1949 by Haldan K. Hartline when he used logarithms to express the effect of Ganglion receptive fields. His algorithms also help explain the experiment conducted by David H. Hubel and Torsten Wiesel that expressed a variation of sensory processing, including lateral inhibition, within different species.
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Rodríguez wrote many works, some of them truly revolutionary contributions to mathematics (like his treatise on logarithms), astronomy and engineering. He also wrote treatises on technology, such as the one dealing with the construction of precise clocks. Many of these works were developed for his own courses in the university; others were written to support his own investigations. In the latter category is the report on the prediction and exact measurement of eclipses, which is fundamental for calculation of exact geographic positions (longitude), because the eclipse permits synchronization of the time with that in other geographic localities.
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Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula) and was the first to use power series with confidence and to revert power series. Newton's work on infinite series was inspired by Simon Stevin's decimals.
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The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As is the case for the decibel and bel, the neper is a unit defined in the international standard ISO 80000. It is not part of the International System of Units (SI), but is accepted for use alongside the SI.Bureau International des Poids et Mesures (2006), The International System of Units (SI) Brochure, 8th edition, pp. 127–128.
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Let A1, A2,..., Ak be a sequence of events such that each event occurs with probability at most p and such that each event is independent of all the other events except for at most d of them. > Lemma I (Lovász and Erdős 1973; published 1975) If :4 p d \le 1 then there > is a nonzero probability that none of the events occurs. > Lemma II (Lovász 1977; published by Joel Spencer) If :e p (d+1) \le 1, where > e = 2.718... is the base of natural logarithms, then there is a nonzero > probability that none of the events occurs. Lemma II today is usually referred to as "Lovász local lemma".
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Both are written as exponentiation modulo a composite number, and both are related to the problem of prime factorization. Functions related to the hardness of the discrete logarithm problem (either modulo a prime or in a group defined over an elliptic curve) are not known to be trapdoor functions, because there is no known "trapdoor" information about the group that enables the efficient computation of discrete logarithms. A trapdoor in cryptography has the very specific aforementioned meaning and is not to be confused with a backdoor (these are frequently used interchangeably, which is incorrect). A backdoor is a deliberate mechanism that is added to a cryptographic algorithm (e.g.
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On each iteration, a choice of coefficient is made from a set of nine complex numbers, 1, 0, −1, i, −i, 1+i, 1−i, −1+i, −1−i, rather than only −1 or +1 as used by CORDIC. BKM provides a simpler method of computing some elementary functions, and unlike CORDIC, BKM needs no result scaling factor. The convergence rate of BKM is approximately one bit per iteration, like CORDIC, but BKM requires more precomputed table elements for the same precision because the table stores logarithms of complex operands. As with other algorithms in the shift-and-add class, BKM is particularly well-suited to hardware implementation.
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The quantitative insulin sensitivity check index (QUICKI) is derived using the inverse of the sum of the logarithms of the fasting insulin and fasting glucose: : This index correlates well with glucose clamp studies (r = 0.78), and is useful for measuring insulin sensitivity (IS), which is the inverse of insulin resistance (IR). It has the advantage of that it can be obtained from a fasting blood sample, and is the preferred method for certain types of clinical research. Values typically associated with the QUICKI calculation for insulin resistance in humans fall broadly within a range between 0.45 for unusually healthy individuals and 0.30 in diabetics. So lower numbers reflect greater insulin resistance.
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Sliding the upper scale appropriately amounts to mechanically adding logarithms, as illustrated here: alt=A slide rule: two rectangles with logarithmically ticked axes, arrangement to add the distance from 1 to 2 to the distance from 1 to 3, indicating the product 6. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. The slide rule was an essential calculating tool for engineers and scientists until the 1970s, because it allows, at the expense of precision, much faster computation than techniques based on tables.
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The Babylonians sometime in 2000–1600 BC may have invented the quarter square multiplication algorithm to multiply two numbers using only addition, subtraction and a table of quarter squares. Thus, such a table served a similar purpose to tables of logarithms, which also allow multiplication to be calculated using addition and table lookups. However, the quarter-square method could not be used for division without an additional table of reciprocals (or the knowledge of a sufficiently simple algorithm to generate reciprocals). Large tables of quarter squares were used to simplify the accurate multiplication of large numbers from 1817 onwards until this was superseded by the use of computers.
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Gerbert of Aurillac marked triples of columns with an arc (called a "Pythagorean arc"), when using his Hindu–Arabic numeral-based abacus in the 10th century. Fibonacci followed this convention when writing numbers, such as in his influential work Liber Abaci in the 13th century. Tables of logarithms prepared by John Napier in 1614 and 1619 used the period (full stop) as the decimal separator, which was then adopted by Henry Briggs in his influential 17th century work. In France, the full stop was already in use in printing to make Roman numerals more readable, so the comma was chosen.Enciclopedia Universal Santillana, 1996 by SANTILLANA S.A., Barcelona, Spain. .
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Moreover, they lacked a versatile architecture, each machine serving only very concrete purposes. In spite of this, Llull's work had a strong influence on Gottfried Leibniz (early 18th century), who developed his ideas further, and built several calculating tools using them. Indeed, when John Napier discovered logarithms for computational purposes in the early 17th century, there followed a period of considerable progress by inventors and scientists in making calculating tools. The apex of this early era of formal computing can be seen in the difference engine and its successor the analytical engine (which was never completely constructed but was designed in detail), both by Charles Babbage.
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It is known from Richardson's theorem that there may not exist an algorithm that decides if two expressions representing numbers are semantically equal, if exponentials and logarithms are allowed in the expressions. Therefore, (semantical) equality may be tested only on some classes of expressions such as the polynomials and rational fractions. To test the equality of two expressions, instead of designing specific algorithms, it is usual to put expressions in some canonical form or to put their difference in a normal form, and to test the syntactic equality of the result. Unlike in usual mathematics, "canonical form" and "normal form" are not synonymous in computer algebra.
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Dr. Haji Abdul Karim Amrullah (left), Sheikh Tahir Jalaluddin (center), and Sheikh Daud Rasyidi (right) Sheikh Tahir Jalaluddin (8 December 1869, Ampek Angkek, Agam, West Sumatra – 1956) was a famous Muslim ulama in Southeast Asia. He became the editor for Al-Imam publication which was published in Singapore between 1906 and 1908. Sheikh Muhammad Tahir published two major treatises on astronomy: Natijatul'Umur (published 1357 H./ 1936 M.) and Kitab Jadual Pati Kiraan (published in 1362 H./ 1941 M.) - logarithms for determining the direction of kiblat and times for prayer. His son Tun Hamdan Sheikh Tahir was a Penang state governor (Yang di-Pertua Negeri) from 1989–2002.
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Sample configurations of pentagramma mirificum Napier’s circles contain circular shifts of parts (a, \pi/2-B, \pi/2-c, \pi/2-A, b) Pentagramma mirificum (Latin for miraculous pentagram) is a star polygon on a sphere, composed of five great circle arcs, all of whose internal angles are right angles. This shape was described by John Napier in his 1614 book Mirifici Logarithmorum Canonis Descriptio (Description of the Admirable Table of Logarithms) along with rules that link the values of trigonometric functions of five parts of a right spherical triangle (two angles and three sides). The properties of pentagramma mirificum were studied, among others, by Carl Friedrich Gauss.
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The map of the world from the Rudolphine TablesThe book, written in Latin, contains tables for the positions of the 1,005 stars measured by Tycho Brahe, and more than 400 stars from Ptolemy and Johann Bayer, with directions and tables for locating the moon and the planets of the Solar System. Included are function tables of logarithms (a useful computational tool that had been described in 1614 by John Napier) and antilogarithms, and instructive examples for computing planetary positions. For most stars these tables were accurate to within one arc minute, and included corrective factors for atmospheric refraction.The New Encyclopædia Britannica, 1988, Volume 10, pg.
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Diffie–Hellman key exchange depends for its security on the presumed difficulty of solving the discrete logarithm problem. The authors took advantage of the fact that the number field sieve algorithm, which is generally the most effective method for finding discrete logarithms, consists of four large computational steps, of which the first three depend only on the order of the group G, not on the specific number whose finite log is desired. If the results of the first three steps are precomputed and saved, they can be used to solve any discrete log problem for that prime group in relatively short time. This vulnerability was known as early as 1992.
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Shor's algorithm can be used to break elliptic curve cryptography by computing discrete logarithms on a hypothetical quantum computer. The latest quantum resource estimates for breaking a curve with a 256-bit modulus (128-bit security level) are 2330 qubits and 126 billion Toffoli gates. In comparison, using Shor's algorithm to break the RSA algorithm requires 4098 qubits and 5.2 trillion Toffoli gates for a 2048-bit RSA key, suggesting that ECC is an easier target for quantum computers than RSA. All of these figures vastly exceed any quantum computer that has ever been built, and estimates place the creation of such computers as a decade or more away.
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It comprises a core set of ideas from Pure Mathematics. These ideas reflect those that would be met early on in a typical A Level Mathematics course: algebra, basic functions, sequences and series, coordinate geometry, trigonometry, exponentials and logarithms, differentiation, integration, graphs of functions. In addition, knowledge of the GCSE curriculum is assumed Test of Mathematics for University Admission - Specification for October 2018 retrieved 22 April 2019 . Paper 2: Mathematical Reasoning Paper 2 has 20 multiple-choice questions, with 75 minutes allowed to complete the paper. The second paper assesses a candidate’s ability to justify and interpret mathematical arguments and conjectures, and deal with elementary concepts from logic.
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The College Board's recommended preparation is a one-year college preparatory course in chemistry, a one-year course in algebra, and experience in the laboratory. However, some second-year algebra concepts (including logarithms) are tested on this subject test. Given the timed nature of the test, one of the keys of the mathematics that appears on the SAT II in Chemistry is not the difficulty, but rather the speed at which it must be completed. Furthermore, the oft- quoted prerequisite of lab-experience is sometimes unnecessary for the SAT Subject Test in Chemistry due to the nature of the questions concerning experiments; most laboratory concepts can simply be memorized beforehand.
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The development of calculus was at the forefront of 18th century mathematical research, and the Bernoullis--family friends of Euler--were responsible for much of the early progress in the field. Understanding the infinite was the major focus of Euler's research. While some of Euler's proofs may not have been acceptable under modern standards of rigor, his ideas were responsible for many great advances. First of all, Euler introduced the concept of a function, and introduced the use of the exponential function and logarithms in analytic proofs Euler frequently used the logarithmic functions as a tool in analysis problems, and discovered new ways by which they could be used.
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In 1736, Bliss became rector of St Ebbe's Church in Oxford. Supported by, among others; the 2nd Earl of Macclesfield (George Parker), Savilian Professor of Astronomy James Bradley and by William Jones, Bliss succeeded Edmond Halley as Savilian Professor of Geometry at Oxford University in February 1742 – being elected a Fellow of the Royal Society in May the same year. As Savilian Professor he lectured courses in arithmetic, algebra, plane and spherical geometry, the use of logarithms and surveying instruments. In 1762 he succeeded James Bradley to become the fourth Astronomer Royal, but held the post for only two years before his unexpected death.
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Wherever in the scene the two phases substantially cancel everywhere in the array, the difference vectors being added are in phase, yielding, for that scene point, a maximum value for the sum. The equivalence of these two methods can be seen by recognizing that multiplication of sinusoids can be done by summing phases which are complex-number exponents of e, the base of natural logarithms. However it is done, the image-deriving process amounts to "backtracking" the process by which nature previously spread the scene information over the array. In each direction, the process may be viewed as a Fourier transform, which is a type of correlation process.
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He invented a new method of computing logarithms that he later used on the Connection Machine. Other work at Los Alamos included calculating neutron equations for the Los Alamos "Water Boiler", a small nuclear reactor, to measure how close an assembly of fissile material was to criticality. On completing this work, Feynman was sent to the Clinton Engineer Works in Oak Ridge, Tennessee, where the Manhattan Project had its uranium enrichment facilities. He aided the engineers there in devising safety procedures for material storage so that criticality accidents could be avoided, especially when enriched uranium came into contact with water, which acted as a neutron moderator.
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As slide rule development progressed, added scales provided reciprocals, squares and square roots, cubes and cube roots, as well as transcendental functions such as logarithms and exponentials, circular and hyperbolic trigonometry and other functions. Slide rules with special scales are still used for quick performance of routine calculations, such as the E6B circular slide rule used for time and distance calculations on light aircraft. In the 1770s, Pierre Jaquet-Droz, a Swiss watchmaker, built a mechanical doll (automaton) that could write holding a quill pen. By switching the number and order of its internal wheels different letters, and hence different messages, could be produced.
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Around 1933 Eckert proposed interconnecting punched card tabulating machines from IBM located in Columbia's Rutherford Laboratory to perform more than simple statistical calculations. Eckert arranged with IBM president Thomas J. Watson for a donation of newly developed IBM 601 calculating punch, which could multiply instead of just adding and subtracting. In 1937 the facility was named the Thomas J. Watson Astronomical Computing Bureau. IBM support included customer service and hardware circuit modifications needed to tabulate numbers, create mathematical tables, add, subtract, multiply, reproduce, verify, create tables of differences, create tables of logarithms and perform Lagrangian interpolation, all to solve differential equations for astronomical applications.
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Arriving in India he first met the daughter of Lord Francis Napier, Hester (d. 1819). Hester was married to Samuel Johnston who worked as a civil servant at Madurai (their son Alexander Johnston later became a judge in Sri Lanka, founded the Royal Asiatic Society of Great Britain and Ireland and wrote a memoir on the life of Colin Mackenzie). Hester introduced Mackenzie to some Brahmins to obtain information on Hindu mathematical traditions as part of the biographical memoir on John Napier and the history of logarithms. The biography project appears to have been subsequently dropped but Colin continued to take an interest in antiquities.
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Mount Gunter () is a conspicuous mountain, high, with precipitous black rock cliffs on its west side, rising at the south side of Hariot Glacier, east of Briggs Peak, on the west side of the Antarctic Peninsula. It was first roughly surveyed by the British Graham Land Expedition in 1936–37, and was photographed by the Ronne Antarctic Research Expedition in November 1947 (trimetrogon air photography). It was surveyed by the Falkland Islands Dependencies Survey in 1958, and was named by the UK Antarctic Place-Names Committee after Edmund Gunter, an English mathematician whose "line of numbers" (1617) was the first step toward a slide rule; in 1620 he published tables of logarithms, sines and tangents, which revolutionized navigation.
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Since the polynomial Q can have only finitely many zeros by the fundamental theorem of algebra, such a rational function will be defined for all sufficiently large x, specifically for all x larger than the largest real root of Q. Adding and multiplying rational functions gives more rational functions, and the quotient rule shows that the derivative of rational function is again a rational function, so R(x) forms a Hardy field. Another example is the field of functions that can be expressed using the standard arithmetic operations, exponents, and logarithms, and are well-defined on some interval of the form (x,\infty). G. H. Hardy, Properties of Logarithmico- Exponential Functions, Proc. London Math. Soc.
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The Swiss mathematician Jost Bürgi constructed a table of progressions which can be considered a table of antilogarithmsJost Bürgi, Arithmetische und Geometrische Progress Tabulen … [Arithmetic and Geometric Progression Tables … ], (Prague, (Czech Republic): University [of Prague] Press, 1620). Available on-line at: Bavarian State Library, Germany Unfortunately, Bürgi did not include, with his table, instructions for using the table. Neither the table nor the instructions were published, apparently only proof sheets of the table were printed. The contents of the instructions were reproduced in: Hermann Robert Gieswald, Justus Byrg als Mathematiker, und dessen Einleitung zu seinen Logarithmen [Justus Byrg as a mathematician, and an introduction to his logarithms] (Danzig, Prussia: St. Johannisschule, 1856), pages 26 ff.
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At the request of the Sadrazam or Grand Vizier Halil Hamit Pasha ("Paşa" in modern Turkish) (1782–1785), and of the Fleet Admiral Cezayirli Hasan Pasha, he was appointed to a professorship in mathematics at the new Naval College in Kasımpaşa, on the Golden Horn, in Istanbul where he worked with other Ottoman reformers such as the Franco- Hungarian military engineer and aristocrat François Baron de Tott. Gelenbevi received an award from the Emperor Sultan Selim III for his very accurate ballistic computations. Gelenbevi Ismail published some thirty five scientific articles including a treatise on the game of chess, written in Turkish and Arabic. He is credited with the introduction of logarithms in Turkey.
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In 1629 he was appointed Chair of Mathematics at the University of Bologna, which is attributed to Galileo's support of him to the Bolognese senate. He published most of his work while at Bologna, though some of it had been written previously; his Geometria Indivisibilius, where he outlined what would later become the method of indivisibles, was written in 1627 while in Parma and presented as part of his application to Bologna, but was not published until 1635. Contemporary critical reception was mixed, and Exercitationes geometricae sex (Six Exercises in Geometry) was published in 1647, partly as a response to criticism. Also at Bologna he published tables of logarithms and information on their use, promoting their use in Italy.
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Dr. Taher Elgamal (Arabic: طاهر الجمل) (born 18 August 1955) is an Egyptian cryptographer and entrepreneur. He is recognized as the "father of SSL" for the work he did in computer security while working at Netscape, which helped in establishing a private and secure communications on the Internet. Elgamal is also known for his 1985 paper entitled "A Public key Cryptosystem and A Signature Scheme based on discrete Logarithms" in which he proposed the design of the ElGamal discrete log cryptosystem and of the ElGamal signature scheme., The latter scheme became the basis for Digital Signature Algorithm (DSA) adopted by National Institute of Standards and Technology (NIST) as the Digital Signature Standard (DSS).
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In the 1960s, Hewlett-Packard was becoming a diversified electronics company with product lines in electronic test equipment, scientific instrumentation, and medical electronics, and was just beginning its entry into computers. The corporation recognized two opportunities: it might be possible to automate the instrumentation that HP was producing, and HP's customer base were likely to buy a product that could replace the slide rules and adding machines that they were now using for computation. With this in mind, HP built the HP 9100 desktop scientific calculator. This was a full- featured calculator that included not only standard "adding machine" functions but also powerful capabilities to handle floating-point numbers, trigonometric functions, logarithms, exponentiation, and square roots.
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TWINKLE, on the other hand, works one candidate smooth number (call it X) at a time. There is one LED corresponding to each prime smaller than B. At the time instant corresponding to X, the set of LEDs glowing corresponds to the set of primes that divide X. This can be accomplished by having the LED associated with the prime p glow once every p time instants. Further, the intensity of each LED is proportional to the logarithm of the corresponding prime. Thus, the total intensity equals the sum of the logarithms of all the prime factors of X smaller than B. This intensity is equal to the logarithm of X if and only if X is B-smooth.
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A dial plate can be laid out, by a pragmatic approach, observing and marking a shadow at regular intervals throughout the day on each day of the year. If the latitude is known the dial plate can be laid out using geometrical construction techniques which rely on projection geometry, or by calculation using the known formulas and trigonometric tables usually using logarithms, or slide rules or more recently computers or mobile phones. Linear algebra has provided a useful language to describe the transformations. A sundial schema uses a compass and a straight edge to firstly to derive the essential angles for that latitude, then to use this to draw the hourlines on the dial plate.
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An early result of Vanstone (joint with Ian Blake, R. Fuji-Hara, and Ron Mullin) was an improved algorithm for computing discrete logarithms in binary fields, which inspired Don Coppersmith to develop his famous exp(n^{1/3+ε}) algorithm (where n is the degree of the field). Vanstone was one of the first to see the commercial potential of Elliptic Curve Cryptography (ECC), and much of his subsequent work was devoted to developing ECC algorithms, protocols, and standards. In 1985 he co-founded Certicom, which later became the chief developer and promoter of ECC. Vanstone authored or coauthored five widely used books and almost two hundred research articles, and he held several patents.
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These mathematical tables from 1925 were distributed by the College Entrance Examination Board to students taking the mathematics portions of the tests Tables of common logarithms were used until the invention of computers and electronic calculators to do rapid multiplications, divisions, and exponentiations, including the extraction of nth roots. Mechanical special-purpose computers known as difference engines were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions - that is, to compute large logarithmic tables. This was motivated mainly by errors in logarithmic tables made by the human computers of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery.
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An analytic expression (or expression in analytic form) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the th root), logarithms, and trigonometric functions. However, the class of expressions considered to be analytic expressions tends to be wider than that for closed- form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions.
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If the dividend has a fractional part (expressed as a decimal fraction), one can continue the algorithm past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction. A person can calculate division with an abacus by repeatedly placing the dividend on the abacus, and then subtracting the divisor the offset of each digit in the result, counting the number of divisions possible at each offset. A person can use logarithm tables to divide two numbers, by subtracting the two numbers' logarithms, then looking up the antilogarithm of the result.
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The lack of the notion of prime elements in the group of points on elliptic curves makes it impossible to find an efficient factor base to run index calculus method as presented here in these groups. Therefore this algorithm is incapable of solving discrete logarithms efficiently in elliptic curve groups. However: For special kinds of curves (so called supersingular elliptic curves) there are specialized algorithms for solving the problem faster than with generic methods. While the use of these special curves can easily be avoided, in 2009 it has been proven that for certain fields the discrete logarithm problem in the group of points on general elliptic curves over these fields can be solved faster than with generic methods.
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A log–log plot of y = x (blue), y = x2 (green), and y = x3 (red). Note the logarithmic scale markings on each of the axes, and that the log x and log y axes (where the logarithms are 0) are where x and y themselves are 1. In science and engineering, a log–log graph or log–log plot is a two- dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Monomials – relationships of the form y=ax^k – appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line.
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The Frobenius number exists as long as the set of coin denominations has no common divisor greater than 1. There is an explicit formula for the Frobenius number when there are only two different coin denominations, x and y: xy − x − y. If the number of coin denominations is three or more, no explicit formula is known; but, for any fixed number of coin denominations, there is an algorithm computing the Frobenius number in polynomial time (in the logarithms of the coin denominations forming an input). No known algorithm is polynomial time in the number of coin denominations, and the general problem, where the number of coin denominations may be as large as desired, is NP-hard.
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Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems,Quantum Algorithm Zoo – Stephen Jordan's Homepage including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely. However, quantum computers offer polynomial speedup for some problems. The most well- known example of this is quantum database search, which can be solved by Grover's algorithm using quadratically fewer queries to the database than that are required by classical algorithms.
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See antiderivative and nonelementary integral for more details. A procedure called the Risch algorithm exists which is capable of determining whether the integral of an elementary function (function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations) is elementary and returning it if it is. In its original form, Risch algorithm was not suitable for a direct implementation, and its complete implementation took a long time. It was first implemented in Reduce in the case of purely transcendental functions; the case of purely algebraic functions was solved and implemented in Reduce by James H. Davenport; the general case was solved and implemented in Axiom by Manuel Bronstein.
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149–165, Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving many definite integrals encountered by practical engineering applications. Earlier systems such as Macsyma had a few definite integrals related to special functions within a look-up table. However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, pattern matching and other manipulations, was pioneered by developers of the MapleK.O. Geddes and T.C. Scott, Recipes for Classes of Definite Integrals Involving Exponentials and Logarithms, Proceedings of the 1989 Computers and Mathematics conference, (held at MIT June 12, 1989), edited by E. Kaltofen and S.M. Watt, Springer- Verlag, New York, (1989), pp. 192–201.
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He published Essay on the Education of Youth, in which he wrote that he did not "study the interest of the boy but the embryo Man". To a non- specialist, he would have seemed deeply knowledgeable in science and mathematics, but a close inspection of his essay and curriculum revealed that the extent of his mathematical teachings was limited to algebra, trigonometry and logarithms. Thus, Green's later mathematical contributions, which exhibited knowledge of very modern developments in mathematics, could not have resulted from his tenure at the Robert Goodacre Academy. He stayed for only four terms (one school year), and it was speculated by his contemporaries that he had exhausted all they had to teach him.
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Napier Island is an island, long, in the southeastern part of Marguerite Bay, west-northwest of Mount Balfour on the Fallières Coast. Following survey and mapping as an ice rise in the Wordie Ice Shelf by the Falklands Islands Dependencies Survey in 1958, this feature was named Napier Ice Rise by the UK Antarctic Place-names Committee (UK-APC). The name was amended to Napier Island by the UK-APC after a general eastward recession of the Wordie Ice Front (around 1999) revealed it was an island. In association with the names of pioneers of navigation grouped in this area, it was named after John Napier (1550-1617), the Scottish mathematician who invented logarithms and published his first tables in 1614.
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According to a historiographical tradition widespread in the Arab world, his work would have led to the discovery of the logarithm function around 1591; 23 years before the Scottish John Napier, notoriously known to be the inventor of the function of the natural logarithm. This hypothesis is based initially on the interpretation of Sâlih Zekî of the handwritten copy of the work of Ibn Hamza, interpreted a posteriori in the Arab and Ottoman world as laying the foundations of the logarithmic function. Zekî published in 1913, a two-volume work on the history of mathematical sciences, written in Ottoman Turkish: Âsâr-ı Bâkiye (literally in Turkish: The memories that remain). where his observations on Ibn Hamza's role in the invention of logarithms appear.
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Archimedes had written The Quadrature of the Parabola in the third century BC, but a quadrature for the hyperbola eluded all efforts until Saint-Vincent published his results in 1647. The relation that the logarithm provides between a geometric progression in its argument and an arithmetic progression of values, prompted A. A. de Sarasa to make the connection of Saint-Vincent's quadrature and the tradition of logarithms in prosthaphaeresis, leading to the term "hyperbolic logarithm", a synonym for natural logarithm. Soon the new function was appreciated by Christiaan Huygens, and James Gregory. The notation Log y was adopted by Leibniz in 1675,Florian Cajori (1913) "History of the exponential and logarithm concepts", American Mathematical Monthly 20: 5, 35, 75, 107, 148, 173, 205.
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Another interpretation of this is that the "inverse" of the complex exponential function is a multivalued function taking each nonzero complex number z to the set of all logarithms of z. There are two solutions to this problem. One is to restrict the domain of the exponential function to a region that does not contain any two numbers differing by an integer multiple of 2πi: this leads naturally to the definition of branches of , which are certain functions that single out one logarithm of each number in their domains. This is analogous to the definition of on as the inverse of the restriction of to the interval : there are infinitely many real numbers θ with , but one arbitrarily chooses the one in .
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He had Euclid's "Elements of Geometry" translated into Sanskrit as also several works on trigonometry, and Napier's work on the construction and use of logarithms. Relying primarily on Indian astronomy, these buildings were used to accurately predict eclipses and other astronomical events. The observational techniques and instruments used in his observatories were also superior to those used by the European Jesuit astronomers he invited to his observatories. Termed as the Jantar Mantar they consisted of the Ram Yantra (a cylindrical building with an open top and a pillar in its center), the Jai Prakash (a concave hemisphere), the Samrat Yantra (a huge equinoctial dial), the Digamsha Yantra (a pillar surrounded by two circular walls), and the Narivalaya Yantra (a cylindrical dial).
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When considering the entire size distribution, not just the largest cities, then the city size distribution is log-normal.Eeckhout J. (2004), Gibrat's law for (All) Cities. American Economic Review 94(5), 1429–1451. The log-normality of the distribution reconciles Gibrat's law also for cities: The law of proportionate effect will therefore imply that the logarithms of the variable will be distributed following the log-normal distribution.Gibrat R. (1931) "Les Inégalités économiques", Paris, France, 1931. In isolation, the upper tail (less than 1,000 out of 24,000 cities) fits both the log-normal and the Pareto distribution: the uniformly most powerful unbiased test comparing the lognormal to the power law shows that the largest 1000 cities are distinctly in the power law regime.
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Even in PC-based implementations, it's a common optimization to speed up sieving by adding approximate logarithms of small primes together. Similarly, TWINKLE has much room for error in its light measurements; as long as the intensity is at about the right level, the number is very likely to be smooth enough for the purposes of known factoring algorithms. The existence of even one large factor would imply that the logarithm of a large number is missing, resulting in a very low intensity; because most numbers have this property, the device's output would tend to consist of stretches of low intensity output with brief bursts of high intensity output. In the above it is assumed that X is square-free, i.e.
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The first edition was published in 1754, and The Elements of Navigation went through seven editions in fifty years. The first volume included sections on logarithms, Euclidean geometry, plane trigonometry, spherics, geography, plane sailing, oblique sailing, current sailing, globular sailing, parallel sailing, middle latitudes and Mercator's sailing, great circular sailing, astronomy, use of globes, as well as estimating distances and fortification. Pupils were prepared for the Royal Navy and the standards of mathematics at the Royal Mathematical School was high. A similar curriculum was followed at the Royal Navy Academy, of which Robertson became a mathematics master in 1755. His edition of The Elements of Navigation was used by Royal Mathematical School pupils on a daily basis between 1755 and 1775.
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The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms (prosthaphaeresis). Arithmetic expressions must be evaluated according to the intended sequence of operations. There are several methods to specify this, either—most common, together with infix notation—explicitly using parentheses and relying on precedence rules, or using a prefix or postfix notation, which uniquely fix the order of execution by themselves. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws (including distributivity), is called a field.
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The Logjam attack used this vulnerability to compromise a variety of Internet services that allowed the use of groups whose order was a 512-bit prime number, so called export grade. The authors needed several thousand CPU cores for a week to precompute data for a single 512-bit prime. Once that was done, individual logarithms could be solved in about a minute using two 18-core Intel Xeon CPUs. As estimated by the authors behind the Logjam attack, the much more difficult precomputation needed to solve the discrete log problem for a 1024-bit prime would cost on the order of $100 million, well within the budget of a large national intelligence agency such as the U.S. National Security Agency (NSA).
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As slide rule development progressed, added scales provided reciprocals, squares and square roots, cubes and cube roots, as well as transcendental functions such as logarithms and exponentials, circular and hyperbolic trigonometry and other functions. Aviation is one of the few fields where slide rules are still in widespread use, particularly for solving time–distance problems in light aircraft. In 1831–1835, mathematician and engineer Giovanni Plana devised a perpetual-calendar machine, which, through a system of pulleys and cylinders could predict the perpetual calendar for every year from AD 0 (that is, 1 BC) to AD 4000, keeping track of leap years and varying day length. The tide- predicting machine invented by Sir William Thomson in 1872 was of great utility to navigation in shallow waters.
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By targeting the NTPase gene, one dose of siRNA 4 hours pre- infection was shown to control Tulane virus replication for 48 hours post- infection, reducing the viral titer by up to 2.6 logarithms. Although the Tulane virus is species-specific and does not affect humans, it has been shown to be closely related to the human norovirus, which is the most common cause of acute gastroenteritis and food-borne disease outbreaks in the United States. Human noroviruses are notorious for being difficult to study in the laboratory, but the Tulane virus offers a model through which to study this family of viruses for the clinical goal of developing therapies that can be used to treat illnesses caused by human norovirus.
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He often found that they left the lock combinations on the factory settings, wrote the combinations down, or used easily guessable combinations like dates. He found one cabinet's combination by trying numbers he thought a physicist might use (it proved to be 27–18–28 after the base of natural logarithms, e = 2.71828 ...), and found that the three filing cabinets where a colleague kept research notes all had the same combination. He left notes in the cabinets as a prank, spooking his colleague, Frederic de Hoffmann, into thinking a spy had gained access to them. Feynman's $380 monthly salary was about half the amount needed for his modest living expenses and Arline's medical bills, and they were forced to dip into her $3,300 in savings.
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CORDIC (for COordinate Rotation DIgital Computer), also known as Volder's algorithm, including Circular CORDIC (Jack E. Volder), Linear CORDIC, Hyperbolic CORDIC (John Stephen Walther), and Generalized Hyperbolic CORDIC (GH CORDIC) (Yuanyong Luo et al.), is a simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, and exponentials and logarithms with arbitrary base, typically converging with one digit (or bit) per iteration. CORDIC is therefore also an example of digit-by-digit algorithms. CORDIC and closely related methods known as pseudo-multiplication and pseudo-division or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and FPGAs), as the only operations it requires are additions, subtractions, bitshift and lookup tables.
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Crop from Thomas Hickey's painting Colin Mackenzie was born in Stornoway on Lewis, Outer Hebrides, Scotland, the second son of merchant Murdoch Mackenzie (who was the first postmaster of the town) and Barbara around 1753 or 1754. Little is known of his early life but he is thought to have started his work as a Comptroller of the Customs at Stornoway from 1778 to 1783, possibly through the influence of his father's association with the Mackenzie Earls of Seaforth. In his youth he had an interest in mathematics possibly fostered by his schoolmaster, a freemason, Alexander Anderson. Lord Kenneth Mackenzie (last Earl of Seaforth) and Francis (fifth Lord Napier) sought his help in preparing a biography of John Napier and his work on logarithms.
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Arms of Napier (same arms as Earl of Lennox): Argent, a saltire engrailed between four roses gules The earliest recorded mention of the name Napier occurred in 1290, in a charter of Maol Choluim I, Earl of Lennox, granting lands at Kilmahew to the Napiers. They are said to have taken their name from a saying by King Alexander II of Scotland to one of the Earls of Lennox, after a battle, that Lennox had na peer (no equal). Archibald Napier, son of John Napier, the inventor of logarithms, served as a Gentleman of the Bedchamber to King James VI of Scotland (I of England) and as a Lord of Session. On 2 March 1627 he was created a baronet, "of Merchistoun in the County of Midlothian", in the Baronetage of Nova Scotia.
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Anton Formann provided an alternative explanation by directing attention to the interrelation between the distribution of the significant digits and the distribution of the observed variable. He showed in a simulation study that long right-tailed distributions of a random variable are compatible with the Newcomb–Benford law, and that for distributions of the ratio of two random variables the fit generally improves. For numbers drawn from certain distributions (IQ scores, human heights) the Benford's law fails to hold because these variates obey a normal distribution which is known not to satisfy Benford's law, since normal distributions can't span several orders of magnitude and the mantissae of their logarithms will not be (even approximately) uniformly distributed. However, if one "mixes" numbers from those distributions, for example by taking numbers from newspaper articles, Benford's law reappears.
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Riddle's most valuable work was A Treatise on Navigation and Nautical Astronomy (1824; 4th edition 1842; 8th edition 1864), forming a complete course of mathematics for sailors, and combining practice and theory in just proportion, which was not usually done at that time in books of this class; the tables of logarithms were issued separately in 1841 and 1851. He re-edited Hutton's Mathematical Recreations (1840, 1854). He also published some sixteen papers on astronomical subjects, of which eight are in the Philosophical Magazine, 1818–22, 1826, 1828, five in Memoirs of the Royal Astronomical Society, 1829, 1830, 1833, 1840, 1842, and three in Monthly Notices of the Royal Astronomical Society, 1833–9, 1845–7. The most important are those on chronometers (in which the author shows how to find the rates without the help of a transit instrument) (cf.
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Scientific calculators are used widely in situations that require quick access to certain mathematical functions, especially those that were once looked up in mathematical tables, such as trigonometric functions or logarithms. They are also used for calculations of very large or very small numbers, as in some aspects of astronomy, physics, and chemistry. They are very often required for math classes from the junior high school level through college, and are generally either permitted or required on many standardized tests covering math and science subjects; as a result, many are sold into educational markets to cover this demand, and some high-end models include features making it easier to translate a problem on a textbook page into calculator input, e.g. by providing a method to enter an entire problem in as it is written on the page using simple formatting tools.
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Similar to torque and energy in physics; information-theoretic information and data storage size have the same dimensionality of units of measurement, but there is in general no meaning to adding, subtracting or otherwise combining the units mathematically. Other units of information, sometimes used in information theory, include the natural digit also called a nat or nit and defined as log2 e (≈ 1.443) bits, where e is the base of the natural logarithms; and the dit, ban, or hartley, defined as log2 10 (≈ 3.322) bits. This value, slightly less than 10/3, may be understood because 103 = 1000 ≈ 1024 = 210: three decimal digits are slightly less information than ten binary digits, so one decimal digit is slightly less than 10/3 binary digits. Conversely, one bit of information corresponds to about ln 2 (≈ 0.693) nats, or log10 2 (≈ 0.301) hartleys.
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The fundamental difference between a calculator and computer is that a computer can be programmed in a way that allows the program to take different branches according to intermediate results, while calculators are pre-designed with specific functions (such as addition, multiplication, and logarithms) built in. The distinction is not clear-cut: some devices classed as programmable calculators have programming functions, sometimes with support for programming languages (such as RPL or TI-BASIC). For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in read-only memory (ROM), and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require much multiplication. Bit serial logic designs are more common in calculators whereas bit parallel designs dominate general-purpose computers, because a bit serial design minimizes chip complexity, but takes many more clock cycles.
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Applied Welfare Economics, edited by Richard E. Just et al, Edward Elgar Publishing More generally, Vartia's expertise is axiomatic index numbers, where he is known for his "consistency in aggregation" test and his discovery, along with Kazuo Sato of the "ideal log-change index", which utilised logarithms and logarithmic mean to define the Sato-Vartia or "Vartia II" index (both 1976). He proposed also in his dissertation "Relative Changes and Index Numbers" in 1976 another index known as Montgomery-Vartia (or "Vartia I") index, which satisfies Time and Factor Reversal Tests and is Consistent in Aggregation. But it satisfies only a weaker form of Proportionality Test WPT along with, say, the factor antithesis FA(Törnqvist) of the Törnqvist index. According to WPT, if both prices and quantities change proportionally, the price index must equal the common factor of proportionality.
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The amount of memory that the algorithm needs is the space for one element and one counter. In the random access model of computing usually used for the analysis of algorithms, each of these values can be stored in a machine word and the total space needed is . If an array index is needed to keep track of the algorithm's position in the input sequence, it doesn't change the overall constant space bound. The algorithm's bit complexity (the space it would need, for instance, on a Turing machine) is higher, the sum of the binary logarithms of the input length and the size of the universe from which the elements are drawn.. Both the random access model and bit complexity analyses only count the working storage of the algorithm, and not the storage for the input sequence itself.
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The requirement of computability reflects on and contrasts with the approach used in analytic number theory to prove the results. It for example brings into question any use of Landau notation and its implied constants: are assertions pure existence theorems for such constants, or can one recover a version in which 1000 (say) takes the place of the implied constant? In other words if it were known that there was M > N with a change of sign and such that :M = O(G(N)) for some explicit function G, say built up from powers, logarithms and exponentials, that means only :M < A.G(N) for some absolute constant A. The value of A, the so-called implied constant, may also need to be made explicit, for computational purposes. One reason Landau notation was a popular introduction is that it hides exactly what A is.
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The Jesuits introduced to China Western science and mathematics which was undergoing its own revolution. "Jesuits were accepted in late Ming court circles as foreign literati, regarded as impressive especially for their knowledge of astronomy, calendar-making, mathematics, hydraulics, and geography."Patricia Buckley Ebrey, p 212 In 1627, the Jesuit Johann Schreck produced the first book to present Western mechanical knowledge to a Chinese audience, Diagrams and explanations of the wonderful machines of the Far West.Ricci roundtable This influence worked in both directions: Jan Mikołaj Smogulecki (1610–1656) is credited with introducing logarithms to China, while Sabatino de Ursis (1575–1620) worked with Matteo Ricci on the Chinese translation of Euclid's Elements, published books in Chinese on Western hydraulics, and by predicting an eclipse which Chinese astronomers had not anticipated, opened the door to the reworking of the Chinese calendar using Western calculation techniques.
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If A can be written in this form, it is called diagonalizable. More generally, and applicable to all matrices, the Jordan decomposition transforms a matrix into Jordan normal form, that is to say matrices whose only nonzero entries are the eigenvalues λ to λ of A, placed on the main diagonal and possibly entries equal to one directly above the main diagonal, as shown at the right. Given the eigendecomposition, the n power of A (that is, n-fold iterated matrix multiplication) can be calculated via :A = (VDV) = VDV'VDV...VDV = VD'V and the power of a diagonal matrix can be calculated by taking the corresponding powers of the diagonal entries, which is much easier than doing the exponentiation for A instead. This can be used to compute the matrix exponential e, a need frequently arising in solving linear differential equations, matrix logarithms and square roots of matrices.
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In mathematics, in the field of tropical analysis, the log semiring is the semiring structure on the logarithmic scale, obtained by considering the extended real numbers as logarithms. That is, the operations of addition and multiplication are defined by conjugation: exponentiate the real numbers, obtaining a positive (or zero) number, add or multiply these numbers with the ordinary "linear" operations on real numbers, and then take the logarithm to reverse the initial exponentiation. As usual in tropical analysis, the operations are denoted by ⊕ and ⊗ to distinguish them from the usual addition + and multiplication × (or ⋅). These operations depend on the choice of base for the exponent and logarithm ( is a choice of logarithmic unit), which corresponds to a scale factor, and are well-defined for any positive base other than 1; using a base is equivalent to using a negative sign and using the inverse .
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Around this time, Wright also lectured mathematics to merchant seamen, and from 1608 or 1609 was mathematics tutor to the son of James I, the heir apparent Henry Frederick, Prince of Wales, until the latter's very early death at the age of 18 in 1612. A skilled designer of mathematical instruments, Wright made models of an astrolabe and a pantograph, and a type of armillary sphere for Prince Henry. In the 1610 edition of Certaine Errors he described inventions such as the "sea-ring" that enabled mariners to determine the magnetic variation of the compass, the sun's altitude and the time of day in any place if the latitude was known; and a device for finding latitude when one was not on the meridian using the height of the pole star. Apart from a number of other books and pamphlets, Wright translated John Napier's pioneering 1614 work which introduced the idea of logarithms from Latin into English.
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Gervas Portal's mother Rose Leslie Portal née Napier was the granddaughter of General Sir William Napier and his wife Caroline Amelia Fox. General Napier and his brothers, Generals Sir Charles James Napier and Sir George Thomas Napier (respectively Commanders-in-Chief of the British Armies in India and in the Cape Colony), were sons of George Napier (a sixth-generation descendant, via the Lords Napier, of John Napier, the inventor of logarithms) and his second wife Lady Sarah Lennox. Caroline Amelia Fox was the daughter of General Henry Edward Fox, younger brother of prominent Whig politician Charles James Fox; they were the sons of politician Henry Fox, 1st Baron Holland and his wife Lady Caroline Lennox. Lady Caroline Lennox and Lady Sarah Lennox were two of the five famous Lennox sisters, daughters of the 2nd Duke of Richmond, son of the 1st Duke of Richmond, illegitimate son of King Charles II and his mistress Louise de Kérouaille, Duchess of Portsmouth.
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Digit sums are also a common ingredient in checksum algorithms to check the arithmetic operations of early computers.. Earlier, in an era of hand calculation, suggested using sums of 50 digits taken from mathematical tables of logarithms as a form of random number generation; if one assumes that each digit is random, then by the central limit theorem, these digit sums will have a random distribution closely approximating a Gaussian distribution.. The digit sum of the binary representation of a number is known as its Hamming weight or population count; algorithms for performing this operation have been studied, and it has been included as a built-in operation in some computer architectures and some programming languages. These operations are used in computing applications including cryptography, coding theory, and computer chess. Harshad numbers are defined in terms of divisibility by their digit sums, and Smith numbers are defined by the equality of their digit sums with the digit sums of their prime factorizations.
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He stated that his method could be expanded for the case of four variables: "The formulas will be more complicated, while the problems leading to such equations are rare in analysis". Also of interest is the integration of differential equations in Lexell's paper "On reducing integral formulas to rectification of ellipses and hyperbolae", which discusses elliptic integrals and their classification, and in his paper "Integrating one differential formula with logarithms and circular functions", which was reprinted in the transactions of the Swedish Academy of Sciences. He also integrated a few complicated differential equations in his papers on continuum mechanics, including a four-order partial differential equation in a paper about coiling a flexible plate to a circular ring. There is an unpublished Lexell paper in the archive of the Russian Academy of Sciences with the title "Methods of integration of some differential equations", in which a complete solution of the equation x=y\phi(x')+\psi(x'), now known as the Lagrange-d'Alembert equation, is presented.
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It was described, in the late 18th century, as "one of the most entertaining narratives in our language", in particular for the historical portrayal it leaves of men like John Dee, Simon Forman, John Booker, Edward Kelley, including a whimsical first meeting of John Napier and Henry Briggs, respective co-inventors of the logarithm and Briggsian logarithms,David Stuart & John Minto in "Account of the Life of John Napier of Merchiston," The Edinburgh Magazine, or Literary Miscellany (1787) Vol.6 and for its curious tales about the effects of crystals and the appearance of Queen Mab. In it, Lilly describes the friendly support of Oliver Cromwell during a period in which he faced prosecution for issuing political astrological predictions. He also writes about the 1666 Great Fire of London, and how he was brought before the committee investigating the cause of the fire, being suspected of involvement because of his publication of images, 15 years earlier, which depicted a city in flames surrounded by coffins.
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It was an anticipation, as far as publication was concerned, of an extended memoir, which had been read by Hamilton before the Royal Irish Academy on 24 November 1833, On Conjugate Functions or Algebraic Couples, and subsequently published in the seventeenth volume of the Transactions of the Royal Irish Academy. To this memoir were prefixed A Preliminary and Elementary Essay on Algebra as the Science of Pure Time, and some General Introductory Remarks. In the concluding paragraphs of each of these three papers Hamilton acknowledges that it was "in reflecting on the important symbolical results of Mr. Graves respecting imaginary logarithms, and in attempting to explain to himself the theoretical meaning of those remarkable symbolisms", that he was conducted to "the theory of conjugate functions, which, leading on to a theory of triplets and sets of moments, steps, and numbers" were foundational for his own work, culminating in the discovery of quaternions. For many years Graves and Hamilton maintained a correspondence on the interpretation of imaginaries.
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Before he left Padua, Gregory published Vera Circuli et Hyperbolae Quadratura (1667) in which he approximated the areas of the circle and hyperbola with convergent series: :[James Gregory] cannot be denied the authorship of many curious theorems on the relation of the circle to inscribed and circumscribed polygons, and their relation to each other. By means of these theorems he gives with infinitely less trouble than by the usual calculations, … the measure of the circle and hyperbola (and consequently the construction of logarithms) to more than twenty decimal places. Following the example of Huygens, he also gave constructions of straight lines equal to the arcs of the circle, and whose error is still less.Jean Montucla (1873) History of the Quadrature of the Circle, J. Babin translator, William Alexander Myers editor, page 23, link from HathiTrust "The first proof of the fundamental theorem of calculus and the discovery of the Taylor series can both be attributed to him."W.
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Bernice Weldon Sargent was born in Williamsburg, Ontario, on 24 September 1906, the son of Henry Sargent, a farmer, and his wife Ella Dillabough. He attended Chesterville High School and Morrisburg Collegiate Institute. He was awarded a Prince of Wales Entrance Scholarship and a Carter Scholarship, and entered Queen's University, where he earned a Bachelor of Arts degree with honours in mathematics and physics in 1926, and a Master of Arts degree the following year. In 1928, Sargent was awarded an 1851 Research Fellowship, which allowed him to travel to England to study at the Cavendish Laboratory at the University of Cambridge under Ernest Rutherford. His 1932 doctoral thesis, written under the supervision of Rutherford and Charles Drummond Ellis, on "The Disintegration Electrons", subsequently published in the Proceedings of the Royal Society, described relationship between the radioactive disintegration constants of beta particle-emitting radioisotopes and corresponding logarithms of their maximum beta particle energies.
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In computational complexity, an NP-complete (or NP-hard) problem is weakly NP- complete (or weakly NP-hard), if there is an algorithm for the problem whose running time is polynomial in the dimension of the problem and the magnitudes of the data involved (provided these are given as integers), rather than the base-two logarithms of their magnitudes. Such algorithms are technically exponential functions of their input size and are therefore not considered polynomial. For example, the NP-hard knapsack problem can be solved by a dynamic programming algorithm requiring a number of steps polynomial in the size of the knapsack and the number of items (assuming that all data are scaled to be integers); however, the runtime of this algorithm is exponential time since the input sizes of the objects and knapsack are logarithmic in their magnitudes. However, as Garey and Johnson (1979) observed, “A pseudo- polynomial-time algorithm … will display 'exponential behavior' only when confronted with instances containing 'exponentially large' numbers, [which] might be rare for the application we are interested in.
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The case-control studies clearly showed a close link between smoking and lung cancer, but were criticized for not showing causality. Follow-up large prospective cohort studies in the early 1950s showed clearly that smokers died faster, and were more likely to die of lung cancer, cardiovascular disease, and a list of other diseases which lengthened as the studies continued The British Doctors Study, a longitudinal study of some 40,000 doctors, began in 1951. By 1954 it had evidence from three years of doctors' deaths, based on which the government issued advice that smoking and lung cancer rates were related (the British Doctors Study last reported in 2001, by which time there were ≈40 linked diseases). The British Doctors Study demonstrated that about half of the persistent cigarette smokers born in 1900–1909 were eventually killed by their addiction (calculated from the logarithms of the probabilities of surviving from 35–70, 70–80, and 80–90) and about two thirds of the persistent cigarette smokers born in the 1920s would eventually be killed by their addiction.
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After a copy of "The Library of Babel" itself, as translated into English by Andrew Hurley, The Unimaginable Mathematics of Borges' Library of Babel has seven chapters on its mathematics. The first chapter, on combinatorics, repeats the calculation above, of the number of books in the library, putting it in context with the size of the known universe and with other huge numbers, and uses this material as an excuse to branch off into a discussion of logarithms and their use in estimation. The second chapter concerns a line in the story about the existence of a library catalog for the library, using information theory to prove that such a catalog would necessarily equal in size the library itself, and touching on topics including the prime number theorem. The third chapter considers the mathematics of the infinite, and the possibility of books with infinitely many, infinitely thin pages, connecting these topics both to a footnote in "The Library of Babel" and to another Borges story, "The Book of Sand", about such an infinite book.
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Graves devised also a pure-triplet system founded on the roots of positive unity, simultaneously with his brother Charles Graves, the bishop of Limerick. He afterwards stimulated Hamilton to the study of polyhedra, and was told of the discovery of the icosian calculus. Graves contributed also to the Philosophical Magazine for April 1836 a paper On the lately proposed Logarithms of Unity in reply to Professor De Morgan, and in the London and Edinburgh Philosophical Magazine for the same year a "postscript" entitled Explanation of a Remarkable Paradox in the Calculus of Functions, noticed by Mr. Babbage. To the same periodical he contributed in September 1838 A New and General Solution of Cubic Equations; in 1839 a paper On the Functional Symmetry exhibited in the Notation of certain Geometrical Porisms, when they are stated merely with reference to the arrangement of points; and in April 1845 a paper on the Connection between the General Theory of Normal Couples and the Theory of Complete Quadratic Functions of Two Variables.
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Planskoy was an inventor of technologies relating to cinematography and photography and registered patents in France, UK and USA for production of composite motion pictures (1930);Grant US1959498A Planskoy Leonti A R L Metra Soc Priority 1930-07-10, Filing 1931-04-22, Grant 1934-05-22, Publication 1934-05-22 for the production of composite images (1935);Grant US2130777A Planskoy Leonti Planskoy Leonti; Priority 1934-12-21; Filing 1935-12-20; Grant 1938-09-20; Publication 1938-09-20 photographic development to a predetermined value of contrast (1938)–in effect, an automated means of 'development by inspection';”Leonti Planskoy, Paris, Photographic development up to a certain y-values. Through two fields to be developed in the film two beams of light are sent during the development, the intensities of which were chosen before the illumination so that the difference between the logarithms of these intensities the difference between the n. Dev is estimated to achieve equivalent blackening of the two fields . After passing through the film, the intensities of the beams are measured until they are equal.
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Since 2006 Schneider, who has a BSc in mathematics, has composed using a Non-Pythagorean scale of his own invention based on logarithms, incorporated prime numbers and the sieve of Eratosthenes in both a composition for bell towers and in the score for a play by mathematician Andrew Granville and playwright Jennifer Granville that debuted at the Institute for Advanced Study on December 12, 2009, has written a plan for an electronic composition based on prime numbers lasting millions of years, and has engaged in a number of other experimental music projects taking inspiration from mathematical concepts. Since September, 2010, Schneider has performed compositions in an experimental notation (including his score "Composition for Two Hemispheres" and a score by Jeff Mangum of Neutral Milk Hotel) for his Teletron Mind-Controlled Interface for Analog Synthesizers, a mind-controlled control voltage generator made from a circuit-bent Mattel MindFlex electronic toy, scored for one "conductor" wearing an EEG sensor with Schneider and experimental musician and visual artist Robert Beatty controlling the filters of Moog synthesizers. Other experimental musicians have subsequently built Teletron units from an instructional video Schneider released online. In 2012 Schneider announced he was stopping touring; whether this hiatus is temporary or permanent is unclear.
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