1.
Pythagorean triple
–
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written, and an example is. If is a Pythagorean triple, then so is for any integer k. A primitive Pythagorean triple is one in which a, b and c are coprime, a right triangle whose sides form a Pythagorean triple is called a Pythagorean triangle. However, right triangles with non-integer sides do not form Pythagorean triples, for instance, the triangle with sides a = b =1 and c = √2 is right, but is not a Pythagorean triple because √2 is not an integer. Moreover,1 and √2 do not have a common multiple because √2 is irrational. There are 16 primitive Pythagorean triples with c ≤100, Note, for example, each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot. The formula states that the integers a = m 2 − n 2, b =2 m n, c = m 2 + n 2 form a Pythagorean triple. The triple generated by Euclids formula is primitive if and only if m and n are coprime, every primitive triple arises from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples and this relationship of a, b and c to m and n from Euclids formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclids formula does not produce all triples—for example and this can be remedied by inserting an additional parameter k to the formula. That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra, many formulas for generating triples with particular properties have been developed since the time of Euclid. A proof of the necessity that a, b, c be expressed by Euclids formula for any primitive Pythagorean triple is as follows, all such triples can be written as where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime, as a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd, from a 2 + b 2 = c 2 we obtain c 2 − a 2 = b 2 and hence = b 2. Since b is rational, we set it equal to m n in lowest terms, thus b = n m, as being the reciprocal of b. As m n is fully reduced, m and n are coprime, and they cannot be both even. If they were odd, the numerator of m 2 − n 22 m n would be a multiple of 4
2.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
3.
Equation
–
In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity equation is true for all values of the variable, a conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation, each member will contain one or more terms. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw
4.
Integer
–
An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain
5.
Zero of a function
–
In other words, a zero of a function is an input value that produces an output of zero. A root of a polynomial is a zero of the polynomial function. If the function maps real numbers to real numbers, its zeroes are the x-coordinates of the points where its graph meets the x-axis, an alternative name for such a point in this context is an x-intercept. Every equation in the unknown x may be rewritten as f =0 by regrouping all terms in the left-hand side and it follows that the solutions of such an equation are exactly the zeros of the function f. Every real polynomial of odd degree has an odd number of roots, likewise. Consequently, real odd polynomials must have at least one real root, the fundamental theorem of algebra states that every polynomial of degree n has n complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs, vietas formulas relate the coefficients of a polynomial to sums and products of its roots. Computing roots of functions, for polynomial functions, frequently requires the use of specialised or approximation techniques. However, some functions, including all those of degree no greater than 4. In topology and other areas of mathematics, the set of a real-valued function f, X → R is the subset f −1 of X. Zero sets are important in many areas of mathematics. One area of importance is algebraic geometry, where the first definition of an algebraic variety is through zero-sets. For instance, for each set S of polynomials in k, one defines the zero-locus Z to be the set of points in An on which the functions in S simultaneously vanish, that is to say Z =. Then a subset V of An is called an algebraic set if V = Z for some S. These affine algebraic sets are the building blocks of algebraic geometry. Zero Pole Fundamental theorem of algebra Newtons method Sendovs conjecture Mardens theorem Vanish at infinity Zero crossing Weisstein, Eric W. Root
6.
Algebraic curve
–
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables. More generally an algebraic curve is similar but may be embedded in a dimensional space or defined over some more general field. For example, the circle is a real algebraic curve. Various technical considerations result in the complex zeros of a polynomial being considered as belonging to the curve, the points of the curve with coordinates in k are the k-points of the curve and, all together, are the k part of the curve. For example, is a point of the curve defined by x2 + y2 −1 =0, the term unit circle may refer to all the complex points as well as to only the real points, the exact meaning usually clear from the context. The equation x2 + y2 +1 =0 defines an algebraic curve, more generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve, the simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and this leads to the most general definition of an algebraic curve, In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a polynomial equation p =0. This equation is called the implicit equation of the curve. Given a curve given by such an equation, the first problems that occur is to determine the shape of the curve. These problems are not as easy to solve as in the case of the graph of a function, the fact that the defining equation is a polynomial implies that the curve has some structural properties that may help to solve these problems. Every algebraic curve may be decomposed into a finite number of smooth monotone arcs connected by some points sometimes called remarkable points. A smooth monotone arc is the graph of a function which is defined. In each direction, an arc is either unbounded or has an end point which is either a point or a point with a tangent parallel to one of the coordinate axes. For example, for the Tschirnhausen cubic of the figure, there are two arcs having the origin as end point. This point is the singular point of the curve. There are two arcs having this singular point as one end point and having a second end point with a horizontal tangent
7.
Lattice (group)
–
In geometry and group theory, a lattice in R n is a subgroup of R n which is isomorphic to Z n, and which spans the real vector space R n. In other words, for any basis of R n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, a lattice may be viewed as a regular tiling of a space by a primitive cell. Lattices have many significant applications in mathematics, particularly in connection to Lie algebras, number theory. More generally, lattice models are studied in physics, often by the techniques of computational physics, a lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, as a group a lattice is a finitely-generated free abelian group, and thus isomorphic to Z n. A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e. g, a simple example of a lattice in R n is the subgroup Z n. More complicated examples include the E8 lattice, which is a lattice in R8, the period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics, it generalises to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras, for example, a typical lattice Λ in R n thus has the form Λ = where is a basis for R n. Different bases can generate the lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ. If one thinks of a lattice as dividing the whole of R n into equal polyhedra and this is why d is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular, minkowskis theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in an all of whose vertices are elements of the lattice is described by the polytopes Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well, Lattice basis reduction is the problem of finding a short and nearly orthogonal lattice basis. The Lenstra-Lenstra-Lovász lattice basis reduction algorithm approximates such a basis in polynomial time, it has found numerous applications. There are five 2D lattice types as given by the crystallographic restriction theorem, below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, and Coxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, a full list of subgroups is available. For example below the hexagonal/triangular lattice is given twice, with full 6-fold, if the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. For the classification of a lattice, start with one point
8.
Greek mathematics
–
Greek mathematics, as the term is used in this article, is the mathematics written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture, Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics. The word mathematics itself derives from the ancient Greek μάθημα, meaning subject of instruction, the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented, the earliest advanced civilizations in Greece and in Europe were the Minoan and later Mycenaean civilization, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, though no direct evidence is available, it is generally thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, despite this, it is generally agreed that Thales is the first of the seven wise men of Greece. The two earliest mathematical theorems, Thales theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle and it is for this reason that Thales is often hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is also thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed, another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras also traveled to Egypt and Babylon, then under the rule of Nebuchadnezzar, Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order. And since in antiquity it was customary to give all credit to the master, aristotle for one refused to attribute anything specifically to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a basis for the conduct of life. Indeed, the philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements and it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclids Elements. The reason it is not clear exactly what either Thales or Pythagoras actually did is that almost no documentation has survived. The only evidence comes from traditions recorded in such as Proclus’ commentary on Euclid written centuries later. Some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments
9.
Diophantus
–
Diophantus of Alexandria, sometimes called the father of algebra, was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations and this led to tremendous advances in number theory, and the study of Diophantine equations and of Diophantine approximations remain important areas of mathematical research. Diophantus coined the term παρισότης to refer to an approximate equality and 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, in modern use, Diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation, little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between AD200 and 214 to 284 or 298, much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the states, Here lies Diophantus, the wonder behold. Alas, the child of master and sage After attaining half the measure of his fathers life chill fate took him. After consoling his fate by the science of numbers for four years and this puzzle implies that Diophantus age x can be expressed as x = x/6 + x/12 + x/7 +5 + x/2 +4 which gives x a value of 84 years. However, the accuracy of the information cannot be independently confirmed, the Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations, of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources and it should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus, “Our author not the slightest trace of a general, comprehensive method is discernible, each problem calls for some special method which refuses to work even for the most closely related problems. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the modern world, copied by. In addition, some portion of the Arithmetica probably survived in the Arab tradition. ”Arithmetica was first translated from Greek into Latin by Bombelli in 1570, however, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander, the best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins. I have a marvelous proof of this proposition which this margin is too narrow to contain. ”Fermats proof was never found
10.
Alexandria
–
Alexandria is the second largest city and a major economic centre in Egypt, extending about 32 km along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it vulnerable to rising sea levels. Alexandria is Egypts largest seaport, serving approximately 80% of Egypts imports and exports and it is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also an important tourist destination, Alexandria was founded around a small Ancient Egyptian town c.331 BC by Alexander the Great. Alexandria was the second most powerful city of the ancient world after Rome, Alexandria is believed to have been founded by Alexander the Great in April 331 BC as Ἀλεξάνδρεια. Alexanders chief architect for the project was Dinocrates, Alexandria was intended to supersede Naucratis as a Hellenistic center in Egypt, and to be the link between Greece and the rich Nile valley. The city and its museum attracted many of the greatest scholars, including Greeks, Jews, the city was later plundered and lost its significance. Just east of Alexandria, there was in ancient times marshland, as early as the 7th century BC, there existed important port cities of Canopus and Heracleion. The latter was rediscovered under water. An Egyptian city, Rhakotis, already existed on the shore also and it continued to exist as the Egyptian quarter of the city. A few months after the foundation, Alexander left Egypt and never returned to his city, after Alexanders departure, his viceroy, Cleomenes, continued the expansion. Although Cleomenes was mainly in charge of overseeing Alexandrias continuous development, the Heptastadion, inheriting the trade of ruined Tyre and becoming the center of the new commerce between Europe and the Arabian and Indian East, the city grew in less than a generation to be larger than Carthage. In a century, Alexandria had become the largest city in the world and and it became Egypts main Greek city, with Greek people from diverse backgrounds. Alexandria was not only a center of Hellenism, but was home to the largest urban Jewish community in the world. The Septuagint, a Greek version of the Tanakh, was produced there, in AD115, large parts of Alexandria were destroyed during the Kitos War, which gave Hadrian and his architect, Decriannus, an opportunity to rebuild it. On 21 July 365, Alexandria was devastated by a tsunami, the Islamic prophet, Muhammads first interaction with the people of Egypt occurred in 628, during the Expedition of Zaid ibn Haritha. He sent Hatib bin Abi Baltaeh with a letter to the king of Egypt and Alexandria called Muqawqis In the letter Muhammad said, I invite you to accept Islam, Allah the sublime, shall reward you doubly. But if you refuse to do so, you bear the burden of the transgression of all the Copts
11.
Algebra
–
Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
12.
Srinivasa Ramanujan
–
Srinivasa Iyengar Ramanujan FRS was an Indian mathematician and autodidact who lived during the British Raj. Though he had almost no training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series. Ramanujan initially developed his own research in isolation, it was quickly recognized by Indian mathematicians. When his skills became obvious and known to the mathematical community, centred in Europe at the time. The Cambridge professor realized that Srinivasa Ramanujan had produced new theorems in addition to rediscovering previously known ones, during his short life, Ramanujan independently compiled nearly 3,900 results. Nearly all his claims have now been proven correct and his original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired a vast amount of further research. The Ramanujan Journal, a scientific journal, was established to publish work in all areas of mathematics influenced by Ramanujan. Deeply religious, Ramanujan credited his substantial mathematical capacities to divinity, An equation for me has no meaning, he once said, the name Ramanujan means younger brother of the god Rama. Iyengar is a caste of Hindu Brahmins of Tamil origin whose members follow the Visishtadvaita philosophy propounded by Ramanuja, Ramanujan was born on 22 December 1887 into a Tamil Brahmin Iyengar family in Erode, Madras Presidency, at the residence of his maternal grandparents. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and his mother, Komalatammal, was a housewife and also sang at a local temple. They lived in a traditional home on Sarangapani Sannidhi Street in the town of Kumbakonam. The family home is now a museum, when Ramanujan was a year and a half old, his mother gave birth to a son, Sadagopan, who died less than three months later. In December 1889, Ramanujan contracted smallpox, but unlike the thousands in the Thanjavur district who died of the disease that year and he moved with his mother to her parents house in Kanchipuram, near Madras. His mother gave birth to two children, in 1891 and 1894, but both died in infancy. On 1 October 1892, Ramanujan was enrolled at the local school, after his maternal grandfather lost his job as a court official in Kanchipuram, Ramanujan and his mother moved back to Kumbakonam and he was enrolled in the Kangayan Primary School. When his paternal grandfather died, he was sent back to his maternal grandparents and he did not like school in Madras, and tried to avoid attending. His family enlisted a local constable to make sure the boy attended school, within six months, Ramanujan was back in Kumbakonam. Since Ramanujans father was at work most of the day, his mother took care of the boy as a child and he had a close relationship with her
13.
G. H. Hardy
–
Godfrey Harold G. H. Hardy FRS was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, Hardy is known for the Hardy–Weinberg principle, a principle of population genetics. In addition to his research, Hardy is remembered for his 1940 essay on the aesthetics of mathematics and he was the mentor of the Indian mathematician Srinivasa Ramanujan. Starting in 1914, Hardy was the mentor of the Indian mathematician Srinivasa Ramanujan, Hardy almost immediately recognised Ramanujans extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was and he called their collaboration the one romantic incident in my life. G. H. Hardy was born on 7 February 1877, in Cranleigh, Surrey, England and his father was Bursar and Art Master at Cranleigh School, his mother had been a senior mistress at Lincoln Training College for teachers. Hardys own natural affinity for mathematics was perceptible at an early age, when just two years old, he wrote numbers up to millions, and when taken to church he amused himself by factorising the numbers of the hymns. After schooling at Cranleigh, Hardy was awarded a scholarship to Winchester College for his mathematical work, in 1896 he entered Trinity College, Cambridge. After only two years of preparation under his coach, Robert Alfred Herman, Hardy was fourth in the Mathematics Tripos examination. Years later, he sought to abolish the Tripos system, as he felt that it was becoming more an end in itself than a means to an end, while at university, Hardy joined the Cambridge Apostles, an elite, intellectual secret society. In 1900 he passed part II of the tripos and was awarded a fellowship, in 1903 he earned his M. A. which was the highest academic degree at English universities at that time. From 1906 onward he held the position of a lecturer where teaching six hours per week left him time for research, in 1919 he left Cambridge to take the Savilian Chair of Geometry at Oxford in the aftermath of the Bertrand Russell affair during World War I. Hardy spent the academic year 1928–1929 at Princeton in an exchange with Oswald Veblen. Hardy gave the Josiah Willards Gibbs lecture for 1928, Hardy left Oxford and returned to Cambridge in 1931, where he was Sadleirian Professor until 1942. The Indian Clerk is a novel by David Leavitt based on Hardys life at Cambridge, including his discovery of, Hardy is credited with reforming British mathematics by bringing rigour into it, which was previously a characteristic of French, Swiss and German mathematics. British mathematicians had remained largely in the tradition of applied mathematics, from 1911 he collaborated with John Edensor Littlewood, in extensive work in mathematical analysis and analytic number theory. This led to progress on the Warings problem, as part of the Hardy–Littlewood circle method. In prime number theory, they proved results and some notable conditional results and this was a major factor in the development of number theory as a system of conjectures, examples are the first and second Hardy–Littlewood conjectures
14.
Fermat's Last Theorem
–
In number theory, Fermats Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n =1 and n =2 have been known to have many solutions since antiquity. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, the unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. The Pythagorean equation, x2 + y2 = z2, has an number of positive integer solutions for x, y, and z. Around 1637, Fermat wrote in the margin of a book that the general equation an + bn = cn had no solutions in positive integers. Although he claimed to have a proof of his conjecture, Fermat left no details of his proof. His claim was discovered some 30 years later, after his death and this claim, which came to be known as Fermats Last Theorem, stood unsolved in mathematics for the following three and a half centuries. The claim eventually became one of the most notable unsolved problems of mathematics, attempts to prove it prompted substantial development in number theory, and over time Fermats Last Theorem gained prominence as an unsolved problem in mathematics. With the special case n =4 proved, it suffices to prove the theorem for n that are prime numbers. Over the next two centuries, the conjecture was proved for only the primes 3,5, and 7, in the mid-19th century, Ernst Kummer extended this and proved the theorem for all regular primes, leaving irregular primes to be analyzed individually. Around 1955, Japanese mathematicians Goro Shimura and Yutaka Taniyama suspected a link might exist between elliptic curves and modular forms, two different areas of mathematics. Known at the time as the Taniyama–Shimura-Weil conjecture, and as the modularity theorem, it stood on its own and it was widely seen as significant and important in its own right, but was widely considered completely inaccessible to proof. In 1984, Gerhard Frey noticed an apparent link between the modularity theorem and Fermats Last Theorem and this potential link was confirmed two years later by Ken Ribet, who gave a conditional proof of Fermats Last Theorem that depended on the modularity theorem. On hearing this, English mathematician Andrew Wiles, who had a fascination with Fermats Last Theorem. In 1993, after six years working secretly on the problem, Wiless paper was massive in size and scope. A flaw was discovered in one part of his paper during peer review and required a further year and collaboration with a past student, Richard Taylor. As a result, the proof in 1995 was accompanied by a second smaller joint paper to that effect
15.
Andrew Wiles
–
Sir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is most notable for proving Fermats Last Theorem, for which he received the 2016 Abel Prize, Wiles has received numerous other honours. Wiles was born in 1953 in Cambridge, England, the son of Maurice Frank Wiles, the Regius Professor of Divinity at the University of Oxford and his father worked as the Chaplain at Ridley Hall, Cambridge, for the years 1952–55. Wiles attended Kings College School, Cambridge, and The Leys School, Wiles states that he came across Fermats Last Theorem on his way home from school when he was 10 years old. He stopped by his local library where he found a book about the theorem. Fascinated by the existence of a theorem that was so easy to state that he, a ten-year-old, could understand it, Wiles earned his bachelors degree in mathematics in 1974 at Merton College, Oxford, and a PhD in 1980 at Clare College, Cambridge. After a stay at the Institute for Advanced Study in New Jersey in 1981, in 1985–86, Wiles was a Guggenheim Fellow at the Institut des Hautes Études Scientifiques near Paris and at the École Normale Supérieure. From 1988 to 1990, Wiles was a Royal Society Research Professor at the University of Oxford and he rejoined Oxford in 2011 as Royal Society Research Professor. Wiless graduate research was guided by John Coates beginning in the summer of 1975, together these colleagues worked on the arithmetic of elliptic curves with complex multiplication by the methods of Iwasawa theory. He further worked with Barry Mazur on the conjecture of Iwasawa theory over the rational numbers. The modularity theorem involved elliptic curves, which was also Wiless own specialist area, the conjecture was seen by contemporary mathematicians as important, but extraordinarily difficult or perhaps impossible to prove. Despite this, Wiles, with his fascination with Fermats Last Theorem, decided to undertake the challenge of proving the conjecture. In June 1993, he presented his proof to the public for the first time at a conference in Cambridge and he gave a lecture a day on Monday, Tuesday and Wednesday with the title Modular Forms, Elliptic Curves and Galois Representations. There was no hint in the title that Fermats last theorem would be discussed, finally, at the end of his third lecture, Dr. Wiles concluded that he had proved a general case of the Taniyama conjecture. Then, seemingly as an afterthought, he noted that that meant that Fermats last theorem was true, in August 1993, it was discovered that the proof contained a flaw in one area. Wiles tried and failed for over a year to repair his proof, according to Wiles, the crucial idea for circumventing, rather than closing this area, came to him on 19 September 1994, when he was on the verge of giving up. Together with his former student Richard Taylor, he published a paper which circumvented the problem. Both papers were published in May 1995 in a volume of the Annals of Mathematics
16.
Pell's equation
–
These solutions may be used to accurately approximate the square root of n by rational numbers of the form x/y. His Brahma Sphuta Siddhanta was translated into Arabic in 773 and was translated into Latin in 1126. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pells equation and other quadratic indeterminate equations, the name of Pells equation arose from Leonhard Eulers mistakenly attributing Lord Brounckers solution of the equation to John Pell. Indeed, if x and y are positive integers satisfying this equation, then x/y is an approximation of √2. Similarly, Baudhayana discovered that x =17, y =12 and x =577, later, Archimedes approximated the square root of 3 by the rational number 1351/780. Although he did not explain his methods, this approximation may be obtained in the same way, Archimedes cattle problem involves solving a Pellian equation. It is now accepted that this problem is due to Archimides. Around AD250, Diophantus considered the equation a 2 x 2 + c = y 2 and this equation is different in form from Pells equation but equivalent to it. Diophantus solved the equation for equal to, and, al-Karaji, a 10th-century Persian mathematician, worked on similar problems to Diophantus. In Indian mathematics, Brahmagupta discovered that =2 − N2 =2 − N2, using this, he was able to compose triples and that were solutions of x 2 − N y 2 = k, to generate the new triples and. For instance, for N =92, Brahmagupta composed the triple with itself to get the new triple, dividing throughout by 64 gave the triple, which when composed with itself gave the desired integer solution. Brahmagupta solved many Pell equations with this method, in particular he showed how to obtain solutions starting from a solution of x 2 − N y 2 = k for k = ±1, ±2. The first general method for solving the Pell equation was given by Bhaskara II in 1150, called the chakravala method, it starts by composing any triple with the trivial triple to get the triple, which can be scaled down to. When m is chosen so that / k is an integer, among such m, the method chooses one that minimizes / k, and repeats the process. This method always terminates with a solution, Bhaskara used it to give the solution x =1766319049, y =226153980 to the notorious N =61 case. Several European mathematicians rediscovered how to solve Pells equation in the 17th century, Fermat found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to Digby, Bernard Frénicle de Bessy said that Fermat found the smallest solution for N up to 150, both Wallis and Lord Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker. Pells connection with the equation is that he revised Thomas Brankers translation of Johann Rahns 1659 book Teutsche Algebra into English, euler mistakenly thought that this solution was due to Pell, as a result of which he named the equation after Pell
17.
John Pell
–
John Pell was an English mathematician. He was born at Southwick in Sussex and his father, also named John Pell, was from Southwick, and his mother was Mary Holland, from Halden in Kent. He was the second of two sons, and by the age of six he was an orphan, his father dying in 1616, John Pell senior had a fine library and this proved valuable to the young Pell as he grew up. He was educated at Steyning Grammar School, and entered Trinity College, Cambridge, during his university career he became an accomplished linguist, and even before he took his B. A. degree corresponded with Henry Briggs and other mathematicians. He received his M. A. in 1630, and taught in the short-lived Chichester Academy, on 3 July 1632 he married Ithamaria Reginald, sister of Bathsua Makin. They went on to have four sons and four daughters, ithumaria died in 1661, and some time before 1669 he remarried. Pell spent much of the 1630s working under Hartlibs influence, on a variety of topics in the area of pedagogy, encyclopedism and pansophy, combinatorics, by 1638 he had formulated a proposal for a universal language. In mathematics, he concentrated on expanding the scope of algebra in the theory of equations, as part of a joint lobbying effort with Hartlib to find himself support to continue as a researcher, he had his short Idea of Mathematics printed in October 1638. The campaign brought interested responses from Johann Moriaen and Marin Mersenne, from 1644 he worked on a polemical work, against Longomontanus. It finally appeared as Controversy with Longomontanus concerning the Quadrature of the Circle, Pell moved in 1646, on the invitation of Frederick Henry, Prince of Orange, to Breda, and remained teaching at the University there until 1652. Pell realised that war between the English and the Dutch was imminent and that he would be in a difficult position in Breda. He returned to England before the outbreak of the First Anglo-Dutch War in July 1652, after his return, Pell was appointed by Oliver Cromwell to a post teaching mathematics in London. From 1654 to 1658 Pell acted as Cromwells political agent in Zurich to the Protestant cantons of Switzerland, he cooperated with Samuel Morland, the English resident at Geneva. Pell was described in Zurich by the English traveller Sir John Reresby in about 1656 as an unknown person, not unsuiting the people he was sent to. They are here so strict in their religion, they suffer not the Venetian ambassador to hear mass in his own house, Cromwell wanted to split the Protestant cantons of Switzerland off to join a Protestant League, with England at its head. However Pells negotiations were drawn out and he returned to England to deliver his report only shortly before Cromwells death. He was unable to report as he waited in vain for an audience with the ailing Cromwell, a mathematical pupil and disciple in Switzerland, from 1657, was Johann Heinrich Rahn, known as Rhonius. This book by Rahn also contained what would become known as the Pell equation, diophantine equations was a favourite subject with Pell, he lectured on them at Amsterdam
18.
Brahmagupta
–
Brahmagupta was an Indian mathematician and astronomer. He is the author of two works on mathematics and astronomy, the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka. According to his commentators, Brahmagupta was a native of Bhinmal, Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in verse in Sanskrit. As no proofs are given, it is not known how Brahmaguptas results were derived, Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala during the reign of the Chapa dynasty ruler Vyagrahamukha and he was the son of Jishnugupta. He was a Shaivite by religion, even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a part of his life. Prithudaka Svamin, a commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Multan region or the Abu region and it was also a center of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, in the year 628, at an age of 30, he composed Brāhmasphuṭasiddhānta which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he has incorported a great deal of originality to his revision, the book consists of 24 chapters with 1008 verses in the ārya meter. Later, Brahmagupta moved to Ujjain, which was also a centre for astronomy. At the mature age of 67, he composed his next well known work Khanda-khādyaka and he is believed to have died in Ujjain. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, the division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmaguptas case, the disagreements stemmed largely from the choice of astronomical parameters, the historian of science George Sarton called him one of the greatest scientists of his race and the greatest of his time. Brahmaguptas mathematical advances were carried on to further extent by Bhāskara II, a descendant in Ujjain. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations, lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka
19.
Leonhard Euler
–
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy, Euler was one of the most eminent mathematicians of the 18th century, and is held to be one of the greatest in history. He is also considered to be the most prolific mathematician of all time. His collected works fill 60 to 80 quarto volumes, more than anybody in the field and he spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Eulers influence on mathematics, Read Euler, read Euler, Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, a pastors daughter. He had two sisters, Anna Maria and Maria Magdalena, and a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Paul Euler was a friend of the Bernoulli family, Johann Bernoulli was then regarded as Europes foremost mathematician, and would eventually be the most important influence on young Leonhard. Eulers formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, aged thirteen, he enrolled at the University of Basel, during that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupils incredible talent for mathematics. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono, at that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition, Pierre Bouguer, who became known as the father of naval architecture, won and Euler took second place. Euler later won this annual prize twelve times, around this time Johann Bernoullis two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727 and he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration. Euler mastered Russian and settled life in Saint Petersburg. He also took on a job as a medic in the Russian Navy. The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia, as a result, it was made especially attractive to foreign scholars like Euler
20.
Noam Elkies
–
Noam David Elkies is an American mathematician and chess master. Along with A. O. L. Atkin, he extended Schoofs algorithm to create the Schoof–Elkies–Atkin algorithm, in 1993, when he was 26 years old, he became the youngest full professor in the history of Harvard University. He was a Putnam Fellow two more times during his undergraduate years, in 1987, he proved that an elliptic curve over the rational numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Eulers sum of powers conjecture for fourth powers and his work on these and other problems won him recognition and a position as an associate professor at Harvard in 1990. In 1993, he was made a full, tenured professor at the age of 26 and this made him the youngest full professor in the history of Harvard. Elkies, along with A. O. L. Atkin, in 1994 he was an invited speaker at the International Congress of Mathematicians in Zurich. In 2004 he received a Lester R. Ford Award, Elkies also studies the connections between music and mathematics. He sits on the Advisory Board of the Journal of Mathematics and he has discovered many new patterns in Conways Game of Life and has studied the mathematics of still life patterns in that cellular automaton rule. Elkies is a fellow at Harvards Lowell House, Elkies is a composer and solver of chess problems. He holds the title of National Master from the United States Chess Federation, but he no longer plays competitively
21.
Greatest common divisor
–
In mathematics, the greatest common divisor of two or more integers, when at least one of them is not zero, is the largest positive integer that is a divisor of both numbers. For example, the GCD of 8 and 12 is 4, the greatest common divisor is also known as the greatest common factor, highest common factor, greatest common measure, or highest common divisor. This notion can be extended to polynomials and other commutative rings, in this article we will denote the greatest common divisor of two integers a and b as gcd. What is the greatest common divisor of 54 and 24, the number 54 can be expressed as a product of two integers in several different ways,54 ×1 =27 ×2 =18 ×3 =9 ×6. Thus the divisors of 54 are,1,2,3,6,9,18,27,54, similarly, the divisors of 24 are,1,2,3,4,6,8,12,24. The numbers that these two share in common are the common divisors of 54 and 24,1,2,3,6. The greatest of these is 6 and that is, the greatest common divisor of 54 and 24. The greatest common divisor is useful for reducing fractions to be in lowest terms, for example, gcd =14, therefore,4256 =3 ⋅144 ⋅14 =34. Two numbers are called relatively prime, or coprime, if their greatest common divisor equals 1, for example,9 and 28 are relatively prime. For example, a 24-by-60 rectangular area can be divided into a grid of, 1-by-1 squares, 2-by-2 squares, 3-by-3 squares, 4-by-4 squares, therefore,12 is the greatest common divisor of 24 and 60. A 24-by-60 rectangular area can be divided into a grid of 12-by-12 squares, in practice, this method is only feasible for small numbers, computing prime factorizations in general takes far too long. Here is another example, illustrated by a Venn diagram. Suppose it is desired to find the greatest common divisor of 48 and 180, first, find the prime factorizations of the two numbers,48 =2 ×2 ×2 ×2 ×3,180 =2 ×2 ×3 ×3 ×5. What they share in common is two 2s and a 3, Least common multiple =2 ×2 × ×3 ×5 =720 Greatest common divisor =2 ×2 ×3 =12. To compute gcd, divide 48 by 18 to get a quotient of 2, then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, note that we ignored the quotient in each step except to notice when the remainder reached 0, signalling that we had arrived at the answer. Formally the algorithm can be described as, gcd = a gcd = gcd, in this sense the GCD problem is analogous to e. g. the integer factorization problem, which has no known polynomial-time algorithm, but is not known to be NP-complete. Shallcross et al. showed that a problem is NC-equivalent to the problem of integer linear programming with two variables, if either problem is in NC or is P-complete, the other is as well
22.
Coprime integers
–
In number theory, two integers a and b are said to be relatively prime, mutually prime, or coprime if the only positive integer that divides both of them is 1. That is, the common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1, the numerator and denominator of a reduced fraction are coprime. In addition to gcd =1 and =1, the notation a ⊥ b is used to indicate that a and b are relatively prime. For example,14 and 15 are coprime, being divisible by only 1. The numbers 1 and −1 are the only integers coprime to every integer, a fast way to determine whether two numbers are coprime is given by the Euclidean algorithm. The number of integers coprime to an integer n, between 1 and n, is given by Eulers totient function φ. A set of integers can also be called if its elements share no common positive factor except 1. A set of integers is said to be pairwise coprime if a and b are coprime for every pair of different integers in it, a number of conditions are individually equivalent to a and b being coprime, No prime number divides both a and b. There exist integers x and y such that ax + by =1, the integer b has a multiplicative inverse modulo a, there exists an integer y such that by ≡1. In other words, b is a unit in the ring Z/aZ of integers modulo a, the least common multiple of a and b is equal to their product ab, i. e. LCM = ab. As a consequence of the point, if a and b are coprime and br ≡ bs. That is, we may divide by b when working modulo a, as a consequence of the first point, if a and b are coprime, then so are any powers ak and bl. If a and b are coprime and a divides the product bc and this can be viewed as a generalization of Euclids lemma. In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6/π2, which is about 61%, two natural numbers a and b are coprime if and only if the numbers 2a −1 and 2b −1 are coprime. As a generalization of this, following easily from the Euclidean algorithm in base n >1, a set of integers S = can also be called coprime or setwise coprime if the greatest common divisor of all the elements of the set is 1. For example, the integers 6,10,15 are coprime because 1 is the positive integer that divides all of them. If every pair in a set of integers is coprime, then the set is said to be pairwise coprime, pairwise coprimality is a stronger condition than setwise coprimality, every pairwise coprime finite set is also setwise coprime, but the reverse is not true
23.
Chinese remainder theorem
–
This theorem has this name because it is a theorem about remainders and was first discovered in the 3rd century AD by the Chinese mathematician Sunzi in Sunzi Suanjing. The Chinese remainder theorem is true over every principal ideal domain and it has been generalized to any commutative ring, with a formulation involving ideals. What amounts to an algorithm for solving this problem was described by Aryabhata, special cases of the Chinese remainder theorem were also known to Brahmagupta, and appear in Fibonaccis Liber Abaci. The result was later generalized with a solution called Dayanshu in Qin Jiushaos 1247 Mathematical Treatise in Nine Sections. The notion of congruences was first introduced and used by Gauss in his Disquisitiones Arithmeticae of 1801, Gauss introduces a procedure for solving the problem that had already been used by Euler but was in fact an ancient method that had appeared several times. Nk be integers greater than 1, which are often called moduli or divisors, Let us denote by N the product of the ni. The Chinese remainder theorem asserts that if the ni are pairwise coprime and this may be restated as follows in term of congruences, If the ni are pairwise coprime, and if a1. Ak are any integers, then there exists an x such that x ≡ a 1 ⋮ x ≡ a k. This means that for doing a sequence of operations in Z / N Z, one may do the same computation independently in each Z / n i Z. This may be faster than the direct computation if N. This is widely used, under the name multi-modular computation, for linear algebra over the integers or the rational numbers, the theorem can also be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family. The existence and the uniqueness of the solution may be proven independently, however, the first proof of existence, given below, uses this uniqueness. Suppose that x and y are both solutions to all the congruences, as x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N divides also x − y, If x and y are supposed to be non negative and less than N, then their difference may be a multiple of N only if x = y. The map x ↦ maps congruence classes modulo N to sequences of congruence classes modulo ni, the proof of uniqueness shows that this map is injective. As the domain and the codomain of this map have the number of elements, the map is also surjective. This proof is simple but does not provide any direct way for computing a solution. Moreover, it cannot be generalized to situations where the following proof can
24.
System of linear equations
–
In mathematics, a system of linear equations is a collection of two or more linear equations involving the same set of variables. A solution to a system is an assignment of values to the variables such that all the equations are simultaneously satisfied. A solution to the system above is given by x =1 y = −2 z = −2 since it all three equations valid. The word system indicates that the equations are to be considered collectively, in mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject which is used in most parts of modern mathematics. Computational algorithms for finding the solutions are an important part of linear algebra, and play a prominent role in engineering, physics, chemistry, computer science. A system of equations can often be approximated by a linear system. For solutions in an integral domain like the ring of the integers, or in other structures, other theories have been developed. Integer linear programming is a collection of methods for finding the best integer solution, gröbner basis theory provides algorithms when coefficients and unknowns are polynomials. Also tropical geometry is an example of linear algebra in a more exotic structure, the simplest kind of linear system involves two equations and two variables,2 x +3 y =64 x +9 y =15. One method for solving such a system is as follows, first, solve the top equation for x in terms of y, x =3 −32 y. Now substitute this expression for x into the equation,4 +9 y =15. This results in an equation involving only the variable y. Solving gives y =1, and substituting this back into the equation for x yields x =3 /2. Here x 1, x 2, …, x n are the unknowns, a 11, a 12, …, a m n are the coefficients of the system, and b 1, b 2, …, b m are the constant terms. Often the coefficients and unknowns are real or complex numbers, but integers and rational numbers are seen, as are polynomials. One extremely helpful view is that each unknown is a weight for a vector in a linear combination. X1 + x 2 + ⋯ + x n = This allows all the language, If every vector within that span has exactly one expression as a linear combination of the given left-hand vectors, then any solution is unique. This is important because if we have m independent vectors a solution is guaranteed regardless of the right-hand side, and otherwise not guaranteed
25.
Matrix (mathematics)
–
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 ×3, the individual items in an m × n matrix A, often denoted by ai, j, where max i = m and max j = n, are called its elements or entries. Provided that they have the size, two matrices can be added or subtracted element by element. The rule for multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field, a major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f = 4x. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations, if the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a transformation is obtainable from the matrixs eigenvalues. Applications of matrices are found in most scientific fields, in computer graphics, they are used to manipulate 3D models and project them onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions, Matrices are used in economics to describe systems of economic relationships. A major branch of analysis is devoted to the development of efficient algorithms for matrix computations. Matrix decomposition methods simplify computations, both theoretically and practically, algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory, a simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is an array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is an array of scalars each of which is a member of F. Most of this focuses on real and complex matrices, that is, matrices whose elements are real numbers or complex numbers. More general types of entries are discussed below, for instance, this is a real matrix, A =
26.
Unimodular matrix
–
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is a matrix that is invertible over the integers. Thus every equation Mx = b, where M and b are both integer, and M is unimodular, has an integer solution, the unimodular matrices of order n form a group, which is denoted G L n. Unimodular matrices form a subgroup of the linear group under matrix multiplication. This follows since det = q p, where p and q are the dimensions of A and B, a totally unimodular matrix need not be square itself. From the definition it follows that any totally unimodular matrix has only 0, the opposite is not true, i. e. a matrix with only 0, +1 or −1 entries is not necessarily unimodular. Totally unimodular matrices are extremely important in combinatorics and combinatorial optimization since they give a quick way to verify that a linear program is integral. Specifically, if A is TU and b is integral, then linear programs of forms like or have integral optima, hence if A is totally unimodular and b is integral, every extreme point of the feasible region is integral and thus the feasible region is an integral polyhedron. The unoriented incidence matrix of a graph, which is the coefficient matrix for bipartite matching, is totally unimodular. More generally, in the appendix to a paper by Heller and Tompkins, A. J. Hoffman, let A be an m by n matrix whose rows can be partitioned into two disjoint sets B and C. The converse is valid for signed graphs without half edges, the constraints of maximum flow and minimum cost flow problems yield a coefficient matrix with these properties. Thus, such network flow problems with bounded integer capacities have an optimal value. Note that this does not apply to multi-commodity flow problems, in which it is possible to have fractional optimal value even with bounded integer capacities. The consecutive-ones property, if A is a 0-1 matrix in which for every row, the 1s appear consecutively, then A is TU.4. The rows of a network matrix correspond to a tree T= and this and several other if-and-only-if characterizations are proven in Schrijver. Hoffman and Kruskal proved the following theorem, suppose G is a directed graph without 2-dicycles, P is the set of all dipaths in G, and A is the 0-1 incidence matrix of V versus P. Then A is totally unimodular if and only if every simple arbitrarily-oriented cycle in G consists of alternating forwards and backwards arcs, suppose a matrix has 0- entries and in each column, the entries are non-decreasing from top to bottom. Fujishige showed that the matrix is TU iff every 2-by-2 submatrix has determinant in 0, ±1, seymour proved a full characterization of all TU matrices, which we describe here only informally
27.
Integer programming
–
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to linear programming, in which the objective function. A special case, 0-1 integer linear programming, in which unknowns are binary, the blue lines together with the coordinate axes define the polyhedron of the LP relaxation, which is given by the inequalities without the integrality constraint. The goal of the optimization is to move the black dotted line as far upward while still touching the polyhedron, the optimal solutions of the integer problem are the points and which both have an objective value of 2. The unique optimum of the relaxation is with objective value of 2.8, note that if the solution of the relaxation is rounded to the nearest integers, it is not feasible for the ILP. The following is a reduction from minimum vertex cover to integer programming that will serve as the proof of NP-hardness, let G = be an undirected graph. The first constraint implies that at least one end point of every edge is included in this subset, therefore, the solution describes a vertex cover. Additionally given some vertex cover C, y v can be set to 1 for any v ∈ C, thus we can conclude that if we minimize the sum of y v we have also found the minimum vertex cover. Mixed integer linear programming problems in which only some of the variables, x i, are constrained to be integers. Zero-one linear programming problems in which the variables are restricted to be either 0 or 1. Note that any bounded integer variable can be expressed as a combination of binary variables, there are two main reasons for using integer variables when modeling problems as a linear program, The integer variables represent quantities that can only be integer. For example, it is not possible to build 3.7 cars, the integer variables represent decisions and so should only take on the value 0 or 1. These considerations occur frequently in practice and so integer linear programming can be used in many applications areas, mixed integer programming has many applications in industrial production, including job-shop modelling. One important example happens In agricultural production planning involves determining production yield for crops that can share resources. A possible objective is to maximize the production, without exceeding the available resources. In some cases, this can be expressed in terms of a linear program and these problems involve service and vehicle scheduling in transportation networks. For example, a problem may involve assigning buses or subways to individual routes so that a timetable can be met, here binary decision variables indicate whether a bus or subway is assigned to a route and whether a driver is assigned to a particular train or subway. The goal of these problems is to design a network of lines to install so that a set of communication requirements are met
28.
Inequation
–
In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, some examples of inequations are, a < b, x + y + z ≤1, n >1, x ≠0. Some authors apply the term only to inequations in which the inequality relation is, specifically, a shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain 0 ≤ a < b ≤1 is shorthand for 0 ≤ a a n d a < b a n d b ≤1. Similar to equation solving, inequation solving means finding what values fulfill a condition stated in the form of an inequation or a conjunction of several inequations. These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled, to be precise, what is sought are often not necessarily actual values, but, more in general, mathematical expressions. Often, an additional objective expression is given that is to be minimized by an optimal solution, see Linear programming#Example for a larger example. Computer support in solving inequations is described in constraint programming, in particular, the programming language Prolog III supports solving algorithms for particular classes of inequalities as a basic language feature, see constraint logic programming. F < g ⇔ { f ≥0 g >0 f <2 Equation Equals sign Inequality Relational operator
29.
Projective space
–
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a line, intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the space is R3 with the camera entrance pupil at the origin. Projective spaces can be studied as a field in mathematics. Geometric objects, such as points, lines, or planes, as a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be more consistent. For example, in the standard Euclidean geometry for the plane, in a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points. Other mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, as outlined above, projective space is a geometric object that formalizes statements like Parallel lines intersect at infinity. For concreteness, we give the construction of the projective plane P2 in some detail. There are three equivalent definitions, The set of all lines in R3 passing through the origin, every such line meets the sphere of radius one centered in the origin exactly twice, say in P = and its antipodal point. P2 can also be described as the points on the sphere S2, for example, the point is identified with, etc. The usual way to write an element of the projective plane, the last formula goes under the name of homogeneous coordinates. In homogeneous coordinates, any point with z ≠0 is equivalent to, so there are two disjoint subsets of the projective plane, that consisting of the points = for z ≠0, and that consisting of the remaining points. The latter set can be subdivided similarly into two disjoint subsets, with points and, in the last case, x is necessarily nonzero, because the origin was not part of P2. This last point is equivalent to, geometrically, the first subset, which is isomorphic to R2, is in the image the yellow upper hemisphere, or equivalently the lower hemisphere. The second subset, isomorphic to R1, corresponds to the line, or, again. Finally we have the red point or the equivalent light red point and we thus have a disjoint decomposition P2 = R2 ⊔ R1 ⊔ point
30.
Rational number
–
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
31.
Modular arithmetic
–
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, a familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7,00 now, then 8 hours later it will be 3,00. Usual addition would suggest that the time should be 7 +8 =15. Likewise, if the clock starts at 12,00 and 21 hours elapse, then the time will be 9,00 the next day, because the hour number starts over after it reaches 12, this is arithmetic modulo 12. According to the definition below,12 is congruent not only to 12 itself, Modular arithmetic can be handled mathematically by introducing a congruence relation on the integers that is compatible with the operations on integers, addition, subtraction, and multiplication. For a positive n, two integers a and b are said to be congruent modulo n, written, a ≡ b. The number n is called the modulus of the congruence, for example,38 ≡14 because 38 −14 =24, which is a multiple of 12. The same rule holds for negative values, −8 ≡72 ≡ −3 −3 ≡ −8. Equivalently, a ≡ b mod n can also be thought of as asserting that the remainders of the division of both a and b by n are the same, for instance,38 ≡14 because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that 38 −14 =24 is a multiple of 12. A remark on the notation, Because it is common to consider several congruence relations for different moduli at the same time, in spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been if the notation a ≡n b had been used. The properties that make this relation a congruence relation are the following, if a 1 ≡ b 1 and a 2 ≡ b 2, then, a 1 + a 2 ≡ b 1 + b 2 a 1 − a 2 ≡ b 1 − b 2. The above two properties would still hold if the theory were expanded to all real numbers, that is if a1, a2, b1, b2. The next property, however, would fail if these variables were not all integers, the notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is referred to as the modulo operation. For example, the remainder of the division of 14 by 12 is denoted by 14 mod 12, as this remainder is 2, we have 14 mod 12 =2