BINOMIAL THEOREM OBJECTIVES Recognize patterns in binomial expansions. Evaluate a binomial coefficient. Expand a binomial raised to a power. Find a particular term in a binomial expansion Understand the principle of mathematical induction. Prove statements using mathematical induction. Definition: BINOMIAL THEOREM Patterns in Binomial Expansions A number of patterns‚ as follows‚ begin to appear when we write the binomial expansion of a b n‚ where n is a positive integer
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Mathematics in medieval Islam 1 Mathematics in medieval Islam In the history of mathematics‚ mathematics in medieval Islam‚ often termed Islamic mathematics or Arabic mathematics‚ is the mathematics developed in the Islamic world between 622 and 1600‚ during what is known as the Islamic Golden Age‚ in that part of the world where Islam was the dominant religion. Islamic science and mathematics flourished under the Islamic caliphate (also known as the Islamic Empire) established across the
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levitation. In the past‚ magnetic levitation was attempted by using permanent magnets. Earnshaw’s theorem however‚ proves that this is mathematically impossible. There exists no arrangement of static magnets of charges that can stably levitate an object. There are however means of circumventing this theorem by altering its basic assumptions. The following conditions are exceptions to Earnshaw’s theorem: • Diamagnetism: occurs in materials which have a relative permeability less than one. The result
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m∠ABC + m∠CBD = m∠ABD Linear Pair Postulate: ∠1 and ∠2 are linear pair → ∠1 and ∠2 are supplementary Theorems: Vertical Angles Theorem (V.A.T. or V.A.C): ∠1 and ∠2 are vertical angles → ∠1 ∠2 Right Angle Congruence Theorem (R.A.C): ∠1 and ∠2 are right angles → ∠1 ∠2 Congruent Supplements Theorem (C.S.T.): ∠1 and ∠2 are supp and ∠1 and ∠3 are supp → ∠2 ∠3 Congruent Complements Theorem (C.C.T): ∠1 and ∠2 are comp and ∠1 and ∠3 are comp → ∠2 ∠3 Corresponding ∠ Thm (C.A.T.): if a ⃦ b
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Algebra is one of the broad parts of mathematics‚ together with number theory‚ geometry and analysis. For historical reasons‚ the word "algebra" has several related meanings in mathematics‚ as a single word or with qualifiers. • As a single word without article‚ "algebra" names a broad part of mathematics (see below). • As a single word with article or in plural‚ "algebra" denotes a specific mathematical structure. See algebra (ring theory) and algebra over a field. • With a qualifier‚ there
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Cambridge (1916)‚ Government Arts College‚ Kumbakonam (1904–1906)‚ Town Higher Secondary School (1904)‚ Pachaiyappa’s College‚ University of Works & Achievements: Ramanujan constant‚ Ramanujan prime‚ Ramanujan theta function‚ Ramanujan’s master theorem‚ Mock theta functions‚ Ramanujan conjecture‚ Ramanujan-Soldner constant‚ Ramanujan’s sum. ------------------------------------------------- Top of Form Bottom of Form Rightly regarded as ’natural genius’ by the English mathematician G.H. Hardy
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treatment is unified in the sense that these probabilistic quantities are well defined for both ergodic and nonergodic Markov chains‚ and all results are valid for both approximation methods. Our proofs are simple in the sense that they only depend on one theorem from analysis: Weierstrass’ M-test. Our results include the convergence of stationary probability distributions when the Markov chain is ergodic. This work was directly motivated by the need to compute stationary probabilities for infinite-state Markov
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Measures of Risk Aversion Financial Economics Martín Solá October 2010 Martín Solá (FE) Measures of Risk Aversion 08/10 1 / 41 Introduction In this …rst stage we will study the individual decisions of optimal portfolio choice under uncertainty and its consequences in the valuation of risky assets. In short‚ the Financial Theory rests on the no-arbitrage principle. The idea behind this principle is that it is not possible to make pro…ts without risk‚ without initial investment
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4/24/2013 Microeconomics Introduction to General Equilibrium and Welfare Economics 1 Welfare Economics • Are market allocations of resources socio-economically efficient? • Welfare Economics: “The branch of economic theory concerned with the social desirability of alternative economic states.” • The theory is used to distinguish circumstances under which markets can be expected to perform well • It also helps define some circumstances under which we might want government
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Digital Logic •Introduction‚ Truth Tables and Boolean expressions •Logic Gates •Combination of Logic Gates •Circuit tracing and Boolean expressions •Creating Truth Table from a circuit •Creating a Circuit from a Truth Table •Basic Boolean Algebra •Using Boolean Algebra for Circuit Simplification •Practical Considerations Digital Circuit versus an analog circuit •Digital systems operate based on discrete information or signals‚ were as an analog system relies on continuous information
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