early civilizations in Egypt‚ Greece‚ Babylon‚ and India did. Their efforts have provided the basic fundamentals for mathematics that are used today. Prime Numbers A prime number is “any integer other than a 0 or + 1 that is not divisible without a remainder by any other integers except + 1 and + the integer itself (Merriam-Webster‚ 1996). These numbers were first studied in-depth by ancient Greek mathematicians who looked to numbers for their mystical and numerological properties‚ seeking perfect
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Chapter 1.5 Word Problems The product of two consecutive even integers. 1. Find two consecutive even integers whose product is 168 Sides of a Square 2. The length of each side of a square is 3 in. more than the length of each side of a smaller square. The sum of the areas of the squares is 149 in2. Find the lengths of the sides of the two squares. Uniform Strip 3. Cynthia Besch wants to buy a rug for a room that is 12 ft wide and 15 ft long. She wants to leave a uniform strip
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of Euclid’s Lemma 13.3. The Lindemann-Zermelo Inductive Proof of FTA References 1 4 5 6 7 10 11 13 14 16 20 21 23 23 24 25 25 1. Introduction Principle of Mathematical Induction for sets Let S be a subset of the positive integers. Suppose that: (i) 1 ∈ S‚ and (ii) ∀ n ∈ Z+ ‚ n ∈ S =⇒ n + 1 ∈ S. Then S = Z+ . The intuitive justification is as follows: by (i)‚ we know that 1 ∈ S. Now apply (ii) with n = 1: since 1 ∈ S‚ we deduce 1 + 1 = 2 ∈ S. Now apply (ii) with n = 2:
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Diminishing returns From Wikipedia‚ the free encyclopedia Jump to: navigation‚ search In economics‚ diminishing returns (also called diminishing marginal returns) refers to how the marginal production of a factor of production starts to progressively decrease as the factor is increased‚ in contrast to the increase that would otherwise be normally expected. According to this relationship‚ in a production system with fixed and variable inputs (say factory size and labor)‚ each additional unit of
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then x0 = 5 -2y for every y‚ which is not possible. 8. Determine whether the following argument is valid: pr qr (p q) ________ r Ans: Not valid: p false‚ q false‚ r true 9. Prove that the following is true for all positive integers n: n is even if and only if 3n2 8 is even. Ans: If n is even‚ then n 2k. Therefore 3n2 8 3(2k)2 8 12k2 8 2(6k2 4)‚ which is even. If n is odd‚ then n 2k 1. Therefore 3n2 8 3(2k 1)2 8 12k2 12k
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CLASS VII CBSE-i Introduction to Rational Numbers nt’s Section Stude Shiksha Kendra‚ 2‚ Community Centre‚ Preet Vihar‚Delhi-110 092 India UNIT-3 CLASS VII UNIT-3 CBSE-i Mathematics Introduction to Rational Numbers Shiksha Kendra‚ 2‚ Community Centre‚ Preet Vihar‚Delhi-110 092 India The CBSE-International is grateful for permission to reproduce and/or translate copyright material used in this publication. The acknowledgements have been included
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Quadratic Equation For the equation ax2 + bx + c = 0‚ –b ± √ b2 – 4ac x = –––––––––––––– . 2a Binomial Theorem (a + b)n = an + (n) a 1 n – 1b + (n) a 2 n – 2 b2 +…+ ( nr ) a n – r br + … + b n‚ where n is a positive integer and –––––––– ( nr ) = (n –n! r! . r)! 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. Formulae for ∆ ABC c b a –––– = –––– = –––– . sin A sin B sin C a2 = b 2 + c2 – 2bc cos A. 1 ∆ = – bc sin A
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Let’s Discuss (p. 1.30) The solution obtained by using the factor method is the exact value of the root. However‚ the solution obtained by using the graphical method is an approximation only. Classwork Classwork (p. 1.8) (a) Integer (b) Natural number (c) Negative integer (d) Terminating decimal (e) Recurring decimal (f) Fraction (g) Irrational number Classwork (p. 1.11) 1. (a) When x = 3‚ L.H.S. == 0 R.H.S. = 0 Since L.H.S. = R.H.S.‚ 3 is a root of the equation. (b) When x = 6
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OLYMPIC TIN HỌC HUFLIT 2014 BÀI 11 (NC) UNION-FIND DISJOINT SETS THEORY In computing‚ a disjoint-set data structure‚ also called a union–find data structure or merge–find set‚ is a data structure that keeps track of a set of elements partitioned into a number of disjoint (nonoverlapping) subsets. It supports two useful operations: o Find: Determine which subset a particular element is in. Find typically returns an item from this set that serves as its "representative"; by comparing the
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September 25‚ 2010 Revised June 1‚ 2011 A number of graph coloring problems have their roots in a communication problem known as the channel assignment problem. The channel assignment problem is the problem of assigning channels (non-negative integers) to the stations in an optimal way such that interference is avoided‚ see Hale [4]. The radio coloring of a graph is a special type of channel assignment problem. Here we develop a technique to find an upper bound for radio number of an arbitrary
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