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Viète's Formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the Multiplicative inverse, reciprocal of the mathematical constant pi, : \frac2\pi = \frac2 \cdot \frac2 \cdot \frac2 \cdots It can also be represented as \frac2\pi = \prod_^ \cos \frac. The formula is named after François Viète, who published it in 1593. As the first formula of European mathematics to represent an infinite process, it can be given a rigorous meaning as a Limit (mathematics), limit expression and marks the beginning of mathematical analysis. It has linear convergence and can be used for calculations of , but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses and as a motivating example for the concept of statistical independence. The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximations Of π
Approximation#Mathematics, Approximations for the mathematical constant pi () in the history of mathematics reached an accuracy within 0.04% of the true value before the beginning of the Common Era. In Chinese mathematics, this was improved to approximations correct to what corresponds to about seven decimal digits by the 5th century. Further progress was not made until the 14th century, when Madhava of Sangamagrama developed approximations correct to eleven and then thirteen digits. Jamshīd al-Kāshī achieved sixteen digits next. Early modern mathematicians reached an accuracy of 35 digits by the beginning of the 17th century (Ludolph van Ceulen), and 126 digits by the 19th century (Jurij Vega). The record of manual approximation of is held by William Shanks, who calculated 527 decimals correctly in 1853. Since the middle of the 20th century, the approximation of has been the task of electronic digital computers (for a comprehensive account, see chronology of computation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergence (math)
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. The limit inferior and limit superior provide generalizations of the concept of a limit which are particularly relevant when the limit at a point may not exist. Notation In formulas, a limit of a function is usually written as : \lim_ f(x) = L, and is read as "the limit of of as approaches equals ". This means that the value of the function can be made arbitrarily close to , by choosing sufficiently close to . Alternatively, the fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or \ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rademacher System
In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form: : \. The Rademacher system is stochastically independent, and is closely related to the Walsh system. Specifically, the Walsh system can be constructed as a product of Rademacher functions. To see that the Rademacher system is an incomplete orthogonal system and not an orthonormal basis, consider the function on the unit interval defined by the following equation: f(t) = 4 \left, x-\frac 12\ - 1 This function is orthogonal to all the functions in the Rademacher system, yet is nonzero. References * * * * External links Rademacher systemin the ''Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries cover ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersion Relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the dispersion relation, one can calculate the frequency-dependent phase velocity and group velocity of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching Kramers–Kronig relations describe the frequency-dependence of wave propagation and attenuation. Dispersion may be caused either by geometric boundary conditions ( waveguides, shallow water) or by interaction of the waves with the transmitting medium. Elementary particles, considered as matter waves, have a nontrivial dispersion relation, even in the absence of geometric constraints and other media. In the presence of dispersion, a wave does not propagate with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ludolph Van Ceulen
Ludolph van Ceulen (, ; 28 January 1540 – 31 December 1610) was a German- Dutch mathematician from Hildesheim. He emigrated to the Netherlands. Biography Van Ceulen moved to Delft most likely in 1576 to teach fencing and mathematics and in 1594 opened a fencing school in Leiden. In 1600 he was appointed the first professor of mathematics at the Engineering School, Duytsche Mathematique, established by Maurice, Prince of Orange, at the relatively new Leiden University. He shared this professorial level at the school with the surveyor and cartographer, , which shows that the intention was to promote practical, rather than theoretical instruction. The curriculum for the new Engineering School was devised by Simon Stevin who continued to act as the personal advisor to the Prince. At first the professors at Leiden refused to accept the status of Van Ceulen and Van Merwen, especially as they taught in Dutch rather than Latin. Theological professors generally believed that p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sexagesimal
Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinate system, geographic coordinates. The number 60, a superior highly composite number, has twelve divisors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions involving sexagesimal numbers are simplified. For example, one hour can be divided evenly into sections of 30 minutes, 20 minutes, 15 minutes, 12 minutes, 10 minutes, 6 minutes, 5 minutes, 4 minutes, 3 minutes, 2 minutes, and 1 minute. 60 is the smallest number that is divisible by every number from 1 to 6; that is, it is the lowest common multiple of 1, 2, 3, 4, 5, and 6. ''In this article, all sexagesimal digits are represented as decimal numbers, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jamshīd Al-Kāshī
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxiana) was a Persian astronomer and mathematician during the reign of Tamerlane. Much of al-Kāshī's work was not brought to Europe and still, even the extant work, remains unpublished in any form. Biography Al-Kashi was born in 1380, in Kashan, in central Iran, to a Persian family. This region was controlled by Tamerlane, better known as Timur. The situation changed for the better when Timur died in 1405, and his son, Shah Rokh, ascended into power. Shah Rokh and his wife, Goharshad, a Turkish princess, were very interested in the sciences, and they encouraged their court to study the various fields in great depth. Consequently, the period of their power became one of many scholarly accomplishments. This was the perfect environment for al-Kashi to begin his career as one of the world's greatest mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics In Medieval Islam
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of the period include extension of the place-value system to include decimal fractions, the systematised study of algebra and advances in geometry and trigonometry. The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of algebra, influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal Digit
A numerical digit (often shortened to just digit) or numeral is a single symbol used alone (such as "1"), or in combinations (such as "15"), to represent numbers in positional notation, such as the common base 10. The name "digit" originates from the Latin ''digiti'' meaning fingers. For any numeral system with an integer radix, base, the number of different digits required is the absolute value of the base. For example, decimal (base 10) requires ten digits (0 to 9), and Binary number, binary (base 2) requires only two digits (0 and 1). Bases greater than 10 require more than 10 digits, for instance hexadecimal (base 16) requires 16 digits (usually 0 to 9 and A to F). Overview In a basic digital system, a numeral system, numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a positional notation, place value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jonathan Borwein
Jonathan Michael Borwein (20 May 1951 – 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia. He was a close associate of David H. Bailey, and they have been prominent public advocates of experimental mathematics. Borwein's interests spanned pure mathematics (analysis), applied mathematics (optimization), computational mathematics (numerical and computational analysis), and high performance computing. He authored ten books, including several on experimental mathematics, a monograph on convex functions, and over 400 refereed articles. He was a co-founder in 1995 of software company MathResources, consulting and producing interactive software primarily for school and university mathematics. He was not associated with MathResources at the time of his death. Borwein was also an expert on the number pi and especially its computation. Early life and education Borwein was born in St. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
