Principles of Algebra 1A 1-1 1-2 1-3 1-4 1-5 1-6 Expressions and Integers Evaluating Algebraic Expressions Writing Algebraic Expressions Integers and Absolute Value Adding Integers Subtracting Integers Multiplying and Dividing Integers 1B 1-7 1-8 LAB 1-9 Solving Equations Solving Equations by Adding or Subtracting Solving Equations by Multiplying or Dividing Model Two-Step Equations Solving Two-Step Equations KEYWORD: MT8CA Ch1 Firefighters can use algebra to find out how fast
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Cambridge Secondary 1 Mathematics Curriculum Framework Contents Introduction Stage 7 .....................................................................................................1 Welcome to the Cambridge Secondary 1 Mathematics curriculum framework. Stage 8 .....................................................................................................7 Stage 9 ................................................................................................... 14
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135 and 225‚ 225 is larger integer. Using Euclid’s division algorithm‚ [Where 135 is divisor‚ 90 is remainder] 225 = 135 × 1 + 90 Since‚ remainder 90 ≠ 0 ‚ by applying Eudid’s division algorithm to 135 and 90 ∴ 135 = 90 × 1 + 45 Again since‚ remainder 45 ≠ 0 ‚ by applying Eudid’s division algorithm to 90 and 45 ∴ 90 = 45 × 2 + 0 Now‚ the remainder is zero so‚ our procedure stops. Hence‚ HCF of 135 and 225 is 45. (ii) In 196 and 38220‚ 38220 is larger integer. Using Euclid’s division algorithm
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Pre-Calculus—Prerequisite Knowledge &Skills III. Polynomials A. Exponents The expression bn is called a power or an exponential expression. This is read “b to the nth power” The b is the base‚ and the small raised symbol n is called the exponent. The exponent indicates the number of times the base occurs as a factor. Examples—Express each of the following using exponents. a. 5 x 5 x 5 x 5 x 5 x 5 x 5 = b. 8 x 8 x 8 x 8 x 8 x 8 x 8
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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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ANSWERS/HINTS 345 APPENDIX 1 ANSWERS/ HINTS EXERCISE 1.1 1. (i) 45 3. 8 columns 4. An integer can be of the form 3q‚ 3q + 1 or 3q + 2. Square all of these integers. 5. An integer can be of the form 9q‚ 9q + 1‚ 9q + 2‚ 9q + 3‚ . . .‚ or 9q + 8. (ii) 196 (iii) 51 2. An integer can be of the form 6q‚ 6q + 1‚ 6q + 2‚ 6q + 3‚ 6q + 4 or 6q + 5. EXERCISE 1.2 1. 2. 3. (i) 2 × 5 × 7 (iv) 5 × 7 × 11 × 13 (i) LCM = 182; HCF = 13 (i) LCM = 420; HCF = 3 2 (ii) 22 × 3 × 13 (v) 17 × 19 × 23 (ii) LCM = 23460;
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in the above equations -12+13=23 02+13=13 S = {2/3‚ 1/3‚ 5/3‚ 17/3‚ 50/3} 22+13=53 42+13=173 72+13=503 “Q8: Construct a proof that a) If m is odd‚ then m^2 is odd b) for all integers m and n‚ if m is even and n is odd‚ the m+n is odd” a) If m is odd‚ then m=2k + 1 We have to prove that m^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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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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