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Archimedes' Cattle Problem
Archimedes's cattle problem (or the or ) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by (1845–1916), headmaster of the ' ( Gymnasium of the Holy Cross) in Dresden, Germany. Using logarithmic tables, he calculated the first digits of the smallest solution, showing that it is about 7.76 \times 10^ cattle, far more than could fit in the observable universe. The decimal form is too long for humans to calculate exactly, but multiple-precision ari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes Cattle Problem
Archimedes's cattle problem (or the or ) is a problem in Diophantine equation#Diophantine analysis, Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of Helios, the sun god from a given set of restrictions. The problem was discovered by Gotthold Ephraim Lessing in a Greek manuscript containing a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. The problem remained unsolved for a number of years, due partly to the difficulty of computing the huge numbers involved in the solution. The general solution was found in 1880 by (1845–1916), headmaster of the ' (Gymnasium (school), Gymnasium of the Holy Cross) in Dresden, Germany. Using logarithmic tables, he calculated the first digits of the smallest solution, showing that it is about 7.76 \times 10^ cattle, far more than could fit in the observable universe. The decimal form is to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhäuser
Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particularly: history of science, geosciences, computer science) and mathematics books and journals under the Birkhäuser imprint (with a leaf logo) sometimes called Birkhäuser Science. * Birkhäuser Verlag – an architecture and design publishing company was (re)created in 2010 when Springer sold its design and architecture segment to ACTAR. The resulting Spanish-Swiss company was then called ActarBirkhäuser. After a bankruptcy, in 2012 Birkhäuser Verlag was sold again, this time to De Gruyter. Additionally, the Reinach-based printer Birkhäuser+GBC operates independently of the above, being now owned by ''Basler Zeitung''. History The original Swiss publishers program focused on regional literature. In the 1920s the sons of Emil Birkh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IBM 1620
The IBM 1620 was announced by IBM on October 21, 1959, and marketed as an inexpensive scientific computer. After a total production of about two thousand machines, it was withdrawn on November 19, 1970. Modified versions of the 1620 were used as the CPU of the IBM 1710 and IBM 1720 Industrial Process Control Systems (making it the first digital computer considered reliable enough for real-time process control of factory equipment). Being variable-word-length decimal, as opposed to fixed-word-length pure binary, made it an especially attractive first computer to learn on and hundreds of thousands of students had their first experiences with a computer on the IBM 1620. Core memory cycle times were 20 microseconds for the (earlier) Model I, 10 microseconds for the Model II (about a thousand times slower than typical computer main memory in 2006). The Model II was introduced in 1962. Architecture Memory The IBM 1620 Model I was a variable "word" length decimal ( BCD) computer us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IBM 7040
The IBM 7040 was a historic but short-lived model of transistor computer built in the 1960s. History It was announced by IBM in December 1961, but did not ship until April 1963. A later member of the IBM 700/7000 series of scientific computers, it was a scaled-down version of the IBM 7090. It was not fully compatible with the 7090. Some 7090 features, including index registers, character instructions and floating point, were extra-cost options. It also featured a different input/output architecture, based on the IBM 1414 data synchronizer, allowing more modern IBM peripherals to be used. A model designed to be compatible with the 7040 with more performance was announced as the 7044 at the same time. Peter Fagg headed the development of the 7040 under executive Bob O. Evans. A number of IBM 7040 and 7044 computers were shipped, but it was eventually made obsolete by the IBM System/360 family, announced in 1964. The schedule delays caused by IBM's multiple incompatible archite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh C
Hugh may refer to: *Hugh (given name) Noblemen and clergy French * Hugh the Great (died 956), Duke of the Franks * Hugh Magnus of France (1007–1025), co-King of France under his father, Robert II * Hugh, Duke of Alsace (died 895), modern-day France * Hugh of Austrasia (7th century), Mayor of the Palace of Austrasia * Hugh I, Count of Angoulême (1183–1249) * Hugh II, Count of Angoulême (1221–1250) * Hugh III, Count of Angoulême (13th century) * Hugh IV, Count of Angoulême (1259–1303) * Hugh, Bishop of Avranches (11th century), France * Hugh I, Count of Blois (died 1248) * Hugh II, Count of Blois (died 1307) * Hugh of Brienne (1240–1296), Count of the medieval French County of Brienne * Hugh, Duke of Burgundy (d. 952) * Hugh I, Duke of Burgundy (1057–1093) * Hugh II, Duke of Burgundy (1084–1143) * Hugh III, Duke of Burgundy (1142–1192) * Hugh IV, Duke of Burgundy (1213–1272) * Hugh V, Duke of Burgundy (1294–1315) * Hugh Capet (939–996), King of France * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Waterloo
The University of Waterloo (UWaterloo, UW, or Waterloo) is a public research university with a main campus in Waterloo, Ontario Waterloo is a city in the Canadian province of Ontario. It is one of three cities in the Regional Municipality of Waterloo (formerly Waterloo County). Waterloo is situated about west-southwest of Toronto. Due to the close proximity of the ci ..., Canada. The main campus is on of land adjacent to "Uptown" Waterloo and Waterloo Park. The university also operates three satellite campuses and four affiliated school, affiliated university colleges. The university offers academic programs administered by six faculties and thirteen faculty-based schools. Waterloo operates the largest post-secondary co-operative education program in the world, with over 20,000 undergraduate students enrolled in the university's co-op program. Waterloo is a member of the U15 Group of Canadian Research Universities, U15, a group of research-intensive universities in Canada. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell Equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the trivial solution with ''x'' = 1 and ''y'' = 0. Joseph Louis Lagrange proved that, as long as ''n'' is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of ''n'' by rational numbers of the form ''x''/''y''. This equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to 92x^2 + 1 = y^2 in his ''Brāhmasphuṭasiddhānta'' circa 628. Bhaskara II in the 12th century and Narayana Pandit i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pell's Equation
Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever the curve passes through a point whose ''x'' and ''y'' coordinates are both integers, such as the trivial solution with ''x'' = 1 and ''y'' = 0. Joseph Louis Lagrange proved that, as long as ''n'' is not a perfect square, Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately approximate the square root of ''n'' by rational numbers of the form ''x''/''y''. This equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to 92x^2 + 1 = y^2 in his ''Brāhmasphuṭasiddhānta'' circa 628. Bhaskara II in the 12th century and Narayana Pandit i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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