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From Wikipedia
In mathematics, a proof is a convincing demonstration (within the accepted standards of the field) that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single exception. An unproven proposition that is believed to be true is known as a conjecture.
The statement that is proved is often called a theorem. Once a theorem is proved, it can be used as the basis to prove further statements. A theorem may also be referred to as a lemma, especially if it is intended for use as a stepping stone in the proof of another theorem.
Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasiempiricism in mathematics, and socalled folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
History and etymology
The word Proof comes from the Latin probare meaning "to test". Related modern words are the English "probe", "proboscisâ€�, "probation", and "probability", the Spanish "probar" (to smell or taste, or (lesser use) touch or test),, Italian "provare" (to try), and the German "probieren" (to try). The early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony.
Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, which originally meant the same as "land measurement". The development of mathematical proof is primarily the product of ancient Greek mathematics, and one of its greatest achievements. Thales (624â€“546 BCE) proved some theorems in geometry. Eudoxus (408â€“355 BCE) and Theaetetus (417â€“369 BCE) formulated theorems but did not prove them. Aristotle (384â€“322 BCE) said definitions should describe the concept being defined in terms of other concepts already known. Mathematical proofs were revolutionized by Euclid (300 BCE), who introduced the axiomatic method still in use today, starting with undefined terms and axioms (propositions regarding the undefined terms assumed to be selfevidently true from the Greek â€œaxiosâ€� meaning â€œsomething worthyâ€�), and used these to prove theorems using deductive logic. His book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. In addition to the familiar theorems of geometry, such as the Pythagorean theorem, the Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.
Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician AlHashimi provided general proofs for numbers (rather than geometric demonstrations) as he considered multiplication, division, etc. for â€�lines.â€� He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the AlFakhri (1000) by AlKaraji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclideanparallel postulate.
Modern proof theory treats proofs as inductively defined data structures. There is no longer an assum
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing. Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics. The tradition of algebraic statistics In the past, statisticians have used algebra to advance research in statistics. Some algebraic statistics led to the development of new topics in algebra and combinatorics, such as association schemes. Design of experiments For example, Ronald A. Fisher, Henry B. Mann, and Rosemary A. Bailey applied Abelian groups to the design of experiments. Experimental designs were also studied with affine geometry over finite fields and then with the introduction of association schemes by R. C. Bose. Orthogonal arrays were introduced by C. R. Rao also for experimental designs. Algebraic analysis and abstract statistical inference Invariant measures on locally compact groups have long been used in statistical theory, particularly in multivariate analysis. Beurling's factorization theorem and much of the work on (abstract) harmonic analysis sought better understanding of the Wold decomposition of stationary stochastic processes, which is important in time series statistics. Encompassing previous results on probability theory on algebraic structures, Ulf Grenander developed a theory of "abstract inference". Grenander's abstract inference and his theory of patterns are useful for spatial statistics and image analysis; these theories rely on lattice theory. Partially ordered sets and lattices Partially ordered vector spaces and vector lattices are used throughout statistical theory. Garrett Birkhoff metrized the positive cone using Hilbert's projective metric and proved Jentsch's theorem using the contraction mapping theorem. Birkhoff's results have been used for maximum entropy estimation (which can be viewed as linear programming in infinite dimensions) by Jonathan Borwein and colleagues. Vector lattices and conical measures were introduced into statistical decision theory by Lucien Le Cam. Recent work using commutative algebra and algebraic geometry In recent years, the term "algebraic statistics" has been used more restrictively, to label the use of algebraic geometry and commutative algebra to study problems related to discrete random variables with finite state spaces. Commutative algebra and algebraic geometry have applications in statistics because many commonly used classes of discrete random variables can be viewed as algebraic varieties. Introductory example Consider a random variable X which can take on the values 0, 1, 2. Such a variable is completely characterized by the three probabilities p_i=\mathrm{Pr}(X=i),\quad i=0,1,2 and these numbers clearly satisfy \sum_{i=0}^2 p_i = 1 \quad \mbox{and}\quad 0\leq p_i \leq 1. Conversely, any three such numbers unambiguously specify a random variable, so we can identify the random variable X with the tuple (p0,p1,p2)âˆˆR3. Now suppose X is a Binomial random variable with parameter p = 1 âˆ’ q and n = 2, i.e. X represents the number of successes when repeating a certain experiment two times, where each experiment has an individual success probability of q. Then p_i=\mathrm{Pr}(X=i)={2 \choose i}q^i (1q)^{2i} and it is not hard to show that the tuples (p0,p1,p2) which arise in this way are precisely the ones satisfying 4 p_0 p_2p_1^2=0.\ The latter is a polynomial equation defining an algebraic variety (or surface) in R3, and this variety, when intersected with the simplex given by \sum_{i=0}^2 p_i = 1 \quad \mbox{and}\quad 0\leq p_i \leq 1, yields a piece of an algebraic curve which may be identified with the set of all 3state Bernoulli variables. Determining the parameter q amounts to locating one point on this curve; testing the hypothesis that a given variable X is Bernoulli amounts to testing whether a certain point lies on that curve or not.
Combinatorics is a branch of mathematics concerning the study of finite or countablediscretestructures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as incombinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatoricsandcombinatorial optimization), and studying combinatorial structures arising in analgebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).
Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century however powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.
A mathematician who studies combinatorics is called a combinatorialist.
History of combinatorics
Basic combinatorial concepts and enumerative results have appeared throughout the ancient world. In 6th century BC, physicianSushruta asserts in Sushruta Samhita that 63 combinations can be made out of 6 different tastes, taken one at a time, two at a time, etc., thus computing all 2^{6}1 possibilities. RomanhistorianPlutarch discusses an argument between Chrysippus (3rd century BC) and Hipparchus (2nd century BC) of a rather delicate enumerative problem, which was later shown to be related to SchrÃ¶der numbers. In the Ostomachion,Archimedes (3rd century BC) considers a tiling puzzle.
In the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. Notably, an Indian mathematician Mahavira (c. 850) provided the general formulae for the number of permutations and combinations. The philosopher and astronomer Rabbi Abraham ibn Ezra (c. 1140) established the symmetry of binomial coefficients, while a closed formula was obtained later by the talmudist and mathematicianLevi ben Gerson (better known as Gersonides), in 1321. The arithmetical triangleâ€” a graphical diagram showing relationships among the binomial coefficientsâ€” was presented by mathematicians in treatises dating as far back as the 10th century, and would eventually become known as Pascal's triangle. Later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations.
During the Renaissance, together with the rest of mathematics and the sciences, combinatorics enjoyed a rebirth. Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In the modern times, the works by
The Kumon method, developed by educator Toru Kumon, is a math and reading educational method that is practiced in Kumon's own learning centers. The Kumon Method advocates several values of learning that include speed, accuracy and mastery of material before a student is able to move on to the next lesson. As of 2009, over 4 million students were studying under the Kumon Method at more than 26,000 Kumon Centers in 46 countries.
History
In 1954, Toru Kumon, a Japanese high school mathematics teacher, began to teach his eldest son due to his problems in mathematics at school. Kumon developed the Kumon Method. In 1956, Kumon opened the first Kumon Center in Osaka, Japan with the help of parents that were very interested in the Method. In 1958, he founded the Kumon Institute of Education, after which Kumon Centers began to open around the world. Since 1956, some 20 million students have been enrolled in Kumon. Today, there are around 4 million Kumon students worldwide. At present there are 1500 Kumon Centers in the USA, and there are a total of 26,000 Kumon Centers in 44 countries. This list of countries includes the USA, Canada, Mexico, Brazil, Argentina, United Kingdom, Spain, Germany, South Africa, Australia, New Zealand, Japan, South Korea, the Philippines, Singapore, Malaysia, Indonesia, India, Thailand, and Hong Kong.
The programs
Kumon is a math and reading enrichment program. Students do not work together as a class, but progress through the curriculum at their own pace, moving on to the next level when they have achieved mastery of the previous level. Mastery is defined as speed (using a standard completion time) and accuracy. They take an achievement test at the end of each level.
Company value
The Kumon family, led by his wife Teiko, own 60 percent of the company. Forbesmagazine estimated in March 2009 that the entire company was currently worth over $650 million.
Mathematics curriculum
 Level 7A: Counting to 10
 Level 6A: Counting to 30
 Level 5A: Line drawing, number puzzles to 50
 Level 4A: Reciting and writing numbers up to 220
 Level 3A: Adding with numbers up to 5
 Level 2A: Adding with numbers up to 10, subtracting with numbers up to 9
 Level A: Horizontal addition and subtraction of larger numbers
 Level B: Vertical addition and subtraction
 Level C: Multiplication, division
 Level D: Long multiplication, long division, introduction to fractions
 Level E: Fractions
 Level F: Four operations of fractions, decimals
 Level G: Positive/negative numbers, exponents, introduction to algebra
 Level H: Linear/simultaneous equations, inequalities, functions, graphs high school level math
 Level I: Factorization, square roots, quadratic equations, Pythagorean theorem
 Level J: Advanced algebra
 Level K: Functions: Quadratic, fractional, irrational, exponential
 Level L: Logarithms, basic limits, derivatives, integrals, and its applications
 Level M: Trigonometry, straight lines, equation of circles.
 Level N: Loci, limits of functions, sequences, differentiation
 Level O: Advanced differentiation, integration, applications of calculus, differential equations.
 Level X is an elective level; also, there are no tests available yet on this level.
 Level XTâ€”Triangles: Sine, cosine theorems, application of trigonometry in area
 Level XVâ€”Vectors: Vectors, inner products, equations of lines, planes, and figures in 3space.
 Level XMâ€”Linear algebra: Matrices, determinant, mapping, linear transformation
 Level XPâ€”Probability: Permutations, combinations, probability, independent trials, expected value
 Level XSâ€”Statistics: Binomial and normal distributions, probability density functions, confidence intervals, hypothesis testing
Reading curriculum
 Prereading skills
 Phonics
 Vocabulary building
 Grammar and punctuation
 Reading comprehension
Variation of programs throughout the world
The Kumon language program varies regionally. For example, the Chinese reading program in Mainland China is different from the Chinese reading program in Hong Kong and in Singapore, and the English program in the U.S., Canada, and the Philippines varies from the English program in the United Kingdom. Additionally, Kumon Korea has other subjects, such as science, calligraphy, Korean, and Chinese characters, which are not available elsewhere.
The Math program also differs. The math program for most countries goes up to Level O and Level X. However, in Japan, the math program is available up to Level V.
From Encyclopedia
Algebra is a branch of mathematics that uses variables to solve equations. When solving an algebraic problem, at least one variable will be unknown. Using the numbers and expressions that are given, the unknown variable(s) can be determined. The history of algebra began in ancient Egypt and Babylon. The Rhind Papyrus, which dates to 1650 b.c.e., provides insight into the types of problems being solved at that time. The Babylonians are credited with solving the first quadratic equation . Clay tablets that date to between 1800 and 1600 b.c.e. have been found that show evidence of a procedure similar to the quadratic equation. The Babylonians were also the first people to solve indeterminate equations, in which more than one variable is unknown. The Greek mathematician Diophantus continued the tradition of the ancient Egyptians and Babylonians into the common era. Diophantus is considered the "father of algebra," and he eventually furthered the discipline with his book Arithmetica. In the book he gives many solutions to very difficult indeterminate equations. It is important to note that, when solving equations, Diophantus was satisfied with any positive number whether it was a whole number or not. By the ninth century, an Egyptian mathematician, Abu Kamil, had stated and proved the basic laws and identities of algebra. In addition, he had solved many problems that were very complicated for his time. Medieval Algebra. During medieval times, Islamic mathematicians made great strides in algebra. They were able to discuss high powers of an unknown variable and could work out basic algebraic polynomials . All of this was done without using modern symbolism. In addition, Islamic mathematicians also demonstrated knowledge of the binomial theorem . An important development in algebra was the introduction of symbols for the unknown in the sixteenth century. As a result of the introduction of symbols, Book III of La gÃ©ometrie by RenÃ© Descartes strongly resembles a modern algebra text. Descartes's most significant contribution to algebra was his development of analytical algebra. Analytical algebra reduces the solution of geometric problems to a series of algebraic ones. In 1799, German mathematician Carl Friedrich Gauss was able to prove Descartes's theory that every polynomial equation has at least one root in the complex plane . Following Gauss's discovery, the focus of algebra began to shift from polynomial equations to studying the structure of abstract mathematical systems. The study of the quaternion became extensive during this period. The study of algebra went on to become more interdisciplinary as people realized that the fundamental principles could be applied to many different disciplines. Today, algebra continues to be a branch of mathematics that people apply to a wide range of topics. Current Status of Algebra. Today, algebra is an important daytoday tool; it is not something that is only used in a math course. Algebra can be applied to all types of realworld situations. For example, algebra can be used to figure out how many right answers a person must get on a test to achieve a certain grade. If it is known what percent each question is worth, and what grade is desired, then the unknown variable is how many right answers one must get to reach the desired grade. Not only is algebra used by people all the time in routine activities, but many professions also use algebra just as often. When companies figure out budgets, algebra is used. When stores order products, they use algebra. These are just two examples, but there are countless others. Just as algebra has progressed in the past, it will continue to do so in the future. As it is applied to more disciplines, it will continue to evolve to better suit peoples' needs. Although algebra may be something not everyone enjoys, it is one branch of mathematics that is impossible to ignore. see also Descartes and His Coordinate System; Mathematics, Very Old. Brook E. Hall Amdahl, Kenn, and Jim Loats. Algebra Unplugged. Broomfield, CO: Clearwater Publishing Co., 1995. History of Algebra. Algebra.com.
From Yahoo Answers
Answers:This is just a straightforward application of the binomial theorem. The first term is x , the second x , third x , fourth x . So we need to find the coefficient. The binomial theorem states: (a + b) = ... + 5C2 a b + ... Here, a = x and b = 2. (Note that 5C2 = 10). Substituting: 10 x * 2 = 10x * 8 = 80x
Answers:Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps or linear transformations and are often represented by matrices. Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to the solution of a systems of linear equations in several unknowns. More advanced applications are ubiquitous, in areas as diverse as abstract algebra and functional analysis. Linear algebra has a concrete representation in analytic geometry and is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences. Nonlinear mathematical models can often be approximated by linear ones.
Answers:The Pythagorean theorem states that the square on the hypotenuse of a right triangle has an area equal of the sum of squares of the other two sides. The Pythagorean theorem was a mathematical fact that the Babylonians knew and used. A triangle with sides a, b, and c (longest) is a right triangle if and only if a + b = c . Hence we know how the sides are related if is a right triangle. Common Pythagorean triple are: 3, 4, 5 5, 12, 13 7, 24, 25 9, 40, 41 and 6, 8, 10 All but the last triple are primitive, the last is called a multiple. A regular polygon has all sides equilateral and all angles equiangular. In a triangle, these cannot occur independently. The resulting triangle with sides in the ratio 1:1:1 and angles 60 60 60 is discussed. The three most important right triangles are: 345, the isosceles right (45 45 90 ), and 30 60 90 triangle. 345 triangle has angle measure of about 37 53 90 . Watch especially for these special angles and triangles. The isosceles (2 or more sides equal) right triangle can be thought of as having legs (the shorter side of a right triangle) of 1. Thus the hypotenuse is square root of 2. The 30 60 90 triangle can thought as a bisected equilateral triangle. Thus one side might be 1, hypotenuse then is 2 and the other side by square root of 3. These side length ratio must be memorized and will be seen often in trigonometry which is the study of triangle, but primarily involved triangle sides length ratio. Note: if a + b < c , the triangle is obtuse; if a + b > c , the triangle is acute. The most important application of Pythagorean theorem is for finding the distance between points in a plane. Distance = [ (x  x ) + (y  y ) ] With the above formula, Pythagoras (and other famous mathematicians) is able to create more formulae, each suitable for its purposes. Pythagoras's Identities: sin x + cos x = 1, tan x + 1 = sec x, cot x + 1 = cosec x Sine rule: a/sinA = b/sinB = c/sinB Cosine rule: a = b + c  2bc cosA, b = a + c  2ac cosB, c = a + b  2ab cosC Heron's formula: A = [ s(s  a)(s  b)(s  c) ] Pythagoras was a great Greek philosopher responsible for important development in mathematics, astronomy, and the theory of matter. He left Samos because of the tyrant who ruled there and went to southern Italy about 532 BC. He founded a philosophical and religious school in Croton that had many followers. Of his actual work nothing is known. His school practiced secrecy and communalism making it hard to distinguish between the work of Pythagoras and of his followers. His school made outstanding contribution to the foundation in mathematics.
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