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Diophantus II.VIII
The eighth problem of the second book of '' Arithmetica'' by Diophantus () is to divide a square into a sum of two squares. The solution given by Diophantus Diophantus takes the square to be 16 and solves the problem as follows: To divide a given square into a sum of two squares. To divide 16 into a sum of two squares. Let the first summand be x^2, and thus the second 16-x^2. The latter is to be a square. I form the square of the difference of an arbitrary multiple of ''x'' diminished by the root f16, that is, diminished by 4. I form, for example, the square of 2''x'' − 4. It is 4x^2+16-16x. I put this expression equal to 16-x^2. I add to both sides x^2+16x and subtract 16. In this way I obtain 5x^2=16x, hence x=16/5. Thus one number is 256/25 and the other 144/25. The sum of these numbers is 16 and each summand is a square. Geometrical interpretation Geometrically, we may illustrate this method by drawing the circle ''x''2 + ''y''2 = 42 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantus 1 Jpg
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the author of a series of books called '' Arithmetica'', many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations ("Diophantine geometry") and Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as ''adaequalitas'' in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetica
''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations. Summary Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the ''Arithmetica'' problems lead to quadratic equations. In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form 4n + 3 cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the author of a series of books called ''Arithmetica'', many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations ("Diophantine geometry") and Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as ''adaequalitas'' in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantus And Diophantine Equations
''Diophantus and Diophantine Equations'' is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russian by Isabella Bashmakova, and published by Nauka in 1972 under the title ''Диофант и диофантовы уравнения''. It was translated into German by Ludwig Boll as ''Diophant und diophantische Gleichungen'' (Birkhäuser, 1974) and into English by Abe Shenitzer as ''Diophantus and Diophantine Equations'' (Dolciani Mathematical Expositions 20, Mathematical Association of America, 1997). Topics In the sense considered in the book, a Diophantine equation is an equation written using polynomials whose coefficients are rational numbers. These equations are to be solved by finding rational-number values for the variables that, when plugged into the equation, make it become true. Although there is also a well-developed theory of integer (rather than rational) solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isabella Bashmakova
Isabella Grigoryevna Bashmakova (russian: Изабелла Григорьевна Башмакова, 1921–2005) was a Russian historian of mathematics. In 2001, she was a recipient of the Alexander Koyré́ Medal of the International Academy of the History of Science. Education and career Bashmakova was born on January 3, 1921, in Rostov-on-Don, to a family of Armenian descent. Her father, Grigory Georgiyevich Bashmakov, was a lawyer. Her family moved to Moscow in 1932. She began studies in the Faculty of Mechanics and Mathematics at Moscow State University in 1938, but was evacuated from Moscow during World War II, during which she served as a nurse in Samarkand... She completed a Ph.D. in 1948, under the supervision of Sofya Yanovskaya. She continued at Moscow State as an assistant professor, and in 1949 was promoted to associate professor. In 1950 her husband, mathematician Andrei I. Lapin, was arrested for his opposition to Lysenkoism, but in part due to Bashmakova's effo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantus 2
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the author of a series of books called '' Arithmetica'', many of which are now lost. His texts deal with solving algebraic equations. Diophantine equations ("Diophantine geometry") and Diophantine approximations are important areas of mathematical research. Diophantus coined the term παρισότης (parisotes) to refer to an approximate equality. This term was rendered as ''adaequalitas'' in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, Diophantine equations are usually algebraic equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Platonic Sequence
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis. The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend the sine a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clearing Denominators
In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. Example Consider the equation : \frac x 6 + \frac y = 1. The smallest common multiple of the two denominators 6 and 15''z'' is 30''z'', so one multiplies both sides by 30''z'': : 5xz + 2y = 30z. \, The result is an equation with no fractions. The simplified equation is not entirely equivalent to the original. For when we substitute and in the last equation, both sides simplify to 0, so we get , a mathematical truth. But the same substitution applied to the original equation results in , which is mathematically meaningless. Description Without loss of generality, we may assume that the right-hand side of the equation is 0, since an equation may equivalently be rewritten in the form . So let the equation have the form :\sum_^n \frac = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Triple
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle. The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=\sqrt2 is a right triangle, but (1,1,\sqrt2) is not a Pythagorean triple because \sqrt2 is not an integer. Moreover, 1 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Tannery
Paul Tannery (20 December 1843 – 27 November 1904) was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose ''Notions Mathématiques'' he contributed an historical chapter. Though Tannery's career was in the tobacco industry, he devoted his evenings and his life to the study of mathematicians and mathematical development. Life and career Tannery was born in Mantes-la-Jolie on 20 December 1843, to a deeply Catholic family. He attended private school in Mantes, followed by the Lycées in Le Mans and Caen. He then entered the École Polytechnique, on whose entrance exam he excelled. His curriculum included mathematics, the sciences, and the classics, all of which would be represented in his future academic work. Tannery's life of public service began as he then entered the École d'Applications des Tabacs as an apprentice engineer. As an assistant engineer, Tannery spent two years in the state tobacco factory a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
