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The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the "science of restoration and balancing." (The Arabic word for restoration, al-jabru,is the root of the word algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x2 + y2 = z2, and xz = y2.

Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by line segments obtained by…...

...What is Algebra? Algebra is a branch of mathematics that uses mathematical statements to describe relationships between things that vary over time. These variables include things like the relationship between supply of an object and its price. When we use a mathematical statement to describe a relationship, we often use letters to represent the quantity that varies, sisnce it is not a fixed amount. These letters and symbols are referred to as variables. (See the Appendix One for a brief review of constants and variables.) The mathematical statements that describe relationships are expressed using algebraic terms, expressions, or equations (mathematical statements containing letters or symbols to represent numbers). Before we use algebra to find information about these kinds of relationships, it is important to first cover some basic terminology. In this unit we will first define terms, expressions, and equations. In the remaining units in this book we will review how to work with algebraic expressions, solve equations, and how to construct algebraic equations that describe a relationship. We will also introduce the notation used in algebra as we move through this unit. History of algebra The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary...

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...History of algebra The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax2 + bx = c) equations, as well as indeterminate equations such as x2 + y2 = z2, whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They also could solve some indeterminate equations. The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the "science of restoration and balancing." (The Arabic word for restoration, al-jabru,is the root of the word algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x2 + y2 = z2, and xz = y2. Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about...

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...Algebra 2 Honors Name ________________________________________ Test #1 1st 9-weeks September 2, 2011 SHOW ALL WORK to ensure maximum credit. Each question is worth 10 points for a total of 100 points possible. Extra credit is awarded for dressing up. 1. Write the solutions represented below in interval notation. A.) [pic] B.) [pic] 2. Use the tax formula [pic] A.) Solve for I. B.) What is the income, I, when the Tax value, T, is $184? 3. The M&M’s company makes individual bags of M&M’s for sale. In production, the company allows between 20 and 26 m&m’s, including 20 and 26. Write an absolute value inequality describing the acceptable number of m&m’s in each bag. EXPLAIN your reasoning. 4. Solve and graph the solution. [pic] 5. Solve and graph the solution. [pic] 6. Solve. [pic] 7. Solve. [pic] 8. True or False. If false, EXPLAIN why it is false. A.) An absolute value equation always has two solutions. B.) 3 is a solution to the absolute value inequality [pic] C.) 8 is a solution to the compound inequality x < 10 AND x > 0. 9. Solve for w. [pic] 10. You plant a 1.5 foot tall sawtooth oak that grows 3.5 feet per year. You want to know how many years it would take for the tree to outgrow your 20 foot roof. A.) Write an inequality that defines x as the number of years of growth. B.) Determine the number of years, to nearest hundredth,......

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...The relational algebra is a theoretical language with operations that work on one or more relations to define another relation without changing the original relation. Thus, both the operands and the results are relations; hence the output from one operation can become the input to another operation. This allows expressions to be nested in the relational algebra. This property is called closure. Relational algebra is an abstract language, which means that the queries formulated in relational algebra are not intended to be executed on a computer. Relational algebra consists of group of relational operators that can be used to manipulate relations to obtain a desired result. Knowledge about relational algebra allows us to understand query execution and optimization in relational database management system. Role of Relational Algebra in DBMS Knowledge about relational algebra allows us to understand query execution and optimization in relational database management system. The role of relational algebra in DBMS is shown in Fig. 3.1. From the figure it is evident that when a SQL query has to be converted into an executable code, first it has to be parsed to a valid relational algebraic expression, then there should be a proper query execution plan to speed up the data retrieval. The query execution plan is given by query optimizer. Relational Algebra Operations Operations in relational algebra can be broadly classified into set operation and database...

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...Service Description 1. Corporate Finance & Financial Management: Historical performance; Time value of money: compounding, discounting, present values and future values, annuities and perpetuities, etc.; Interest rates and bond valuation; Dividends and stock valuation and dividend policy; Capital budgeting: NPV, IRR, payback period, profitability index, etc.; Risk, return and security market line: beta estimation, CAPM; Cost of capital, financial leverage, capital structure, etc; Cash and liquidity, credit and inventory management; International corporate finance; Risk management and financial engineering; Options and option valuation; Mergers and acquisitions 2. Accounting for Financial Statements: Preparation of income statement, balance sheet and statement of cash flows: Accounting for specialized items: Property, Plant & Equipment, bad debts; provisions; financial instruments; leases; employee benefits; income taxes; revenues,; foreign currency transactions etc.;Accounting for mergers and consolidations; IFRS vs GAAP; Financial statement analysis 3. Cost and Management Accounting: Cost concepts; Job-order costing vs process costing;ABC Costing; Marginal costing vs absorption costing: CVP analysis; Relevant costs: special order, make or buy decisions; ROA, residual income and economic value added; Standard costing and variance analysis; EOQ and linear programming 4. Quantitative Methods and Business Mathematics: Algebra and logarithm; Series...

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...Week four assignment MAT221: introduction to algebra Thurman Solana July 7, 2013 Below we will go through a few equations for this week’s assignment. I will show my knowledge of how to properly find the correct answers to each problem. As well as showing my knowledge of the words: Like terms FOIL Descending Order Dividend and Divisor. Compound semiannually On page 304 problem #90 states “P dollars is invested at annual interest rates r for one year. If the interest rate is compounded semiannually then the polynomial p(1+r2) represents the value of investment after one year. Rewrite the problem without the equation.”(Algebra) For the first equation p will stand for 200 and r will stand for 10%. First I need to turn the interest rate into a decimal. 10%=0.1. Now I can rewrite the equation.2001+0.122. Now that I have my equation written out I can start to solve. I start by dividing 0.1 by 2 to get 0.05. Now I can rewrite 2001+0.052. First I add the 1 and 0.05 giving me 1.05 to square. Any number times itself is called squaring. So now we square (1.05)*(1.05)=(1.1025). Again we rewrite our equation 200*1.1025=220.5. Now we can remove the parentheses leaving us with an answer of 220.5. The answer for this first part of 2001+0.0122=220.5. Second Part On this second part let p stand for 5670 and r will stand for 3.5%. Again I start by turning my percentage into a decimal 3.5%=0.035. Now that we have our decimal we can write out our equation 56701...

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...Algebra III with Statistics Shannon Smith, Ph.D. SSmith3@mcpss.com 1. This course emphasizes the study of functions, graphs, and other skills necessary for the study of Pre-Calculus and the preparation for college. Topics include: Polynomial functions, trigonometric and exponential equations, linear and quadratic equations and inequalities, and an extensive look into statistics. 2. All school rules will be upheld and strictly enforced – see student handbook. 3. Students are expected to bring required materials to class each day. These materials include the following: textbook, paper, pencil (only work done in pencil will be graded), and a scientific calculator, excluding calculator that use math type. GRAPHING CALCULATORS WILL NOT BE ALLOWED! 4. Please be prepared for class when it begins. Have pencils sharpened and materials ready to begin learning before instruction begins. DO NOT talk while the teacher is talking, and remain seated during class. Hall passes will be limited for a necessity or an emergency. Do not ask for a hall pass unnecessarily. 5. If you have permission to attend a field trip, you must realize that you are counted present for that day you miss. Therefore, you will be held accountable for any material you missed. 6. Grading will consist of the following: Assignments: 10% Quizzes: 30% Tests: 60% A comprehensive quarter exam will be...

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...Many people are intimidated and afraid of mathematics and algebra largely due to the fact that upon first glance, certain problems or expressions may seem overwhelmingly large, difficult, or complicated. Along with remembering formulas, this can often times lead to anger, confusion, and frustration. There are several very important key elements and aspects involved within mathematics that helps combat this confusion and frustration and can even help the most intimidated person feel at ease and comfortable with solving these problems. This particular report will demonstrate the importance of understanding certain key mathematical principles and components and show how understanding and utilizing certain mathematical definitions can help limit the amount of confusion and intimidation one may have. These definitions include but are not limited to simplifying, adding like terms, coefficient, distributive property, and removing parenthesis. This report will also demonstrate how much easier and more simplistic mathematics and algebra can be by remembering and utilizing just a few important concepts. Example 1: 2^a(a-5) +4(a-5) This is the first example that will be used. The variable a is used. This particular example has a coefficient of two. Step 1: The distributive property can be utilized to multiply 2a by everything inside of the parenthesis (a-5 in this case) resulting in: 2a^2-10a Step 2: The distributive property is used once again to multiply 4 by everything in...

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...Exam Name___________________________________ MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Using Boolean algebra to simplify the expression Z = AB + A(B + C) + B(B + C), the completed first step would result in the expression: 1) _______ A) Z = AB + ABAC + BB + BC B) Z = AA + AB + AB + AC + BB + BC C) Z = AB + AB + AC + BB + BC D) Z = AB + AB + C + BB + C 2) Which of the following voltage ranges defines acceptable low-level input for a CMOS IC? 2) _______ A) 0 - 2 V B) 2 - 5 V C) 0 - 0.8 V D) 0 - 1.5 V 3) Simplifying logic circuits results in: 3) _______ A) fewer connections. B) fewer gates. C) fewer potential faults . D) all of the above 4) The input combination of A = 1, B = 0, C = 0, and D = 1 would be represented on a Karnaugh map by the square labeled: 4) _______ A) A B) ABCD C) A D D) BCD 5) A logic circuit contains the following: a 3-input NOR gate with inputs A, B, and C, a 2-input NOR gate with inputs C and D, and a 2-input AND gate that is driven by the NOR gate outputs. The correct Boolean expression for the circuit output (X) is: 5) _______ A) X =() () B) X = C) X =(A + B + C) (C + D) D) X = + 6) The logic gate that produces a HIGH output whenever its two...

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...MAPÚA INSTITUTE OF TECHNOLOGY Department of Mathematics COURSE SYLLABUS 1. Course Code: Math 10-3 2. Course Title: Algebra 3. Pre-requisite: none 4. Co-requisite: none 5. Credit: 3 units 6. Course Description: This course covers discussions on a wide range of topics necessary to meet the demands of college mathematics. The course discussion starts with an introductory set theories then progresses to cover the following topics: the real number system, algebraic expressions, rational expressions, rational exponents and radicals, linear and quadratic equations and their applications, inequalities, and ratio, proportion and variations. 7. Student Outcomes and Relationship to Program Educational Objectives Student Outcomes Program Educational Objectives 1 2 (a) an ability to apply knowledge of mathematics, science, and engineering √ (b) an ability to design and conduct experiments, as well as to analyze and interpret from data √ (c) an ability to design a system, component, or process to meet desired needs √ (d) an ability to function on multidisciplinary teams √ √ (e) an ability to identify, formulate, and solve engineering problems √ (f) an understanding of professional and ethical responsibility √ (g) an ability to communicate effectively √ √ (h) the broad education necessary to understand the impact of engineering solutions in the global and societal context √ √ (i) a recognition of the need for, and an ability to...

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...Name: Taylor Harmon_________________________ Score: ______ / ______ Pre-Algebra Midterm Exam Solve the problems below. Show your work when applicable. 1. Write using exponents. (–4)(–4) -4^2 2. Simplify. Show your work. 513 +-3918 5x3+1/3 + -3 9/18 16/3 + -3 9/18 16/3 + -3 ½ 16/3 – 3x2+1/2 16/3 – 7/2 Least common denominator found is 6 16x2/3x2 – 7x3/2x3 32/6 – 21/6 = 11/6 11/6 = 1 5/6 3. What type of measurement would you use to describe the amount of water a pot can hold? Volume – gallons, liters 4. Estimate the sum of 9.327 + 5.72 + 4.132 to one decimal place. 19.2 5. State whether the number 91 is prime, composite, or neither. Composite. It can be divided by 7 or 13 6. What are the mean and the mode of the following set of data: 5, 12, 1, 5, 7 mean: 6 mode: 5 7. To measure the distance from the U.S. to Istanbul, Turkey you would most likely use __________. miles 8. What percent of 67 is 33? Round to the nearest tenth of a percent. 49.3% 9. An adult house cat could be about 1 ___________ high. foot 10. Write a number sentence for the model. Let one white tile equal +1 and one black tile equal –1. There are -14 black tiles and 6 of them become white tiles. -14+6=-8 11. Determine whether the statement is true or false. 94 is divisible by 3. false 12. State whether the number 97 is prime, composite, or neither. prime 13.......

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...Algebra 2 Lesson 5-5 Example 1 Equation with Rational Roots Solve 2x2 – 36x + 162 = 32 by using the Square Root Property. 2x2 – 36x + 162 = 32 Original equation 2(x2 – 18x + 81) = 2(16) Factor out the GCF. x2 – 18x + 81 = 16 Divide each side by 2. (x – 9)2 = 16 Factor the perfect trinomial square. x – 9 = Square Root Property x – 9 = ±4 = 4 x = 9 ± 4 Add 9 to each side. x = 9 + 4 or x = 9 – 4 Write as two equations. x = 13 x = 5 Solve each equation. The solution set is {5, 13}. You can check this result by using factoring to solve the original equation. Example 2 Equation with Irrational Roots Solve x2 + 10x + 25 = 108 by using the Square Root Property. x2 + 10x + 25 = 108 Original equation (x + 5)2 = 108 Factor the perfect square trinomial. x + 5 = Square Root Property x = –5 ±6 Add –5 to each side; = 6 x = –5 + 6 or x = –5 – 6 Write as two equations. x ≈ 5.4 x ≈ –15.4 Use a calculator. The exact solutions of this equation are –5 – 6 and –5 + 6. The approximate solutions are –15.4 and 5.4. Check these results by finding and graphing the related quadratic function. x2 + 10x + 25 = 108 Original equation x2 + 10x – 83 = 0 Subtract 108 from each side. y = x2 + 10x – 83 Related quadratic function. CHECK Use the ZERO function of a graphing calculator. The approximate zeros of...

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...Name: _________________________ Score: ______ / ______ Pre-Algebra Midterm Exam Solve the problems below. Show your work when applicable. 1. Write using exponents. (–4)(–4) 2. Simplify. Show your work. 513 +-3918 3. What type of measurement would you use to describe the amount of water a pot can hold? 4. Estimate the sum of 9.327 + 5.72 + 4.132 to one decimal place. 5. State whether the number 91 is prime, composite, or neither. 6. What are the mean and the mode of the following set of data: 5, 12, 1, 5, 7 mean: mode: 7. To measure the distance from the U.S. to Istanbul, Turkey you would most likely use __________. 8. What percent of 67 is 33? Round to the nearest tenth of a percent. 9. An adult house cat could be about 1 ___________ high. 10. Write a number sentence for the model. Let one white tile equal +1 and one black tile equal –1. 11. Determine whether the statement is true or false. 94 is divisible by 3. 12. State whether the number 97 is prime, composite, or neither. 13. Solve the equation. Show your work. 5d = 17.1 14. A reasonable estimate for the mass of a baby would be 7 ______. 15. Write an equation for: Twelve times h equals one hundred and twenty minus thirty-six. 16. Solve the equation step by step. List each step, along with the property you used during that step. –5 × 1 × 11 × 4. 17. Summarize the order of......

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...Algebra I Quarter 3 Exam Name/Student Number:__________________________ Score:_______/________ Directions: For each question show all work that is required to arrive at the solution. Save this document with your answers and submit as an attachment to be graded. Simplify each expression. Use positive exponents. 1. m3n–6p0 2. a 4 b 3 ab 2 3. (x–2y–4x3) –2 4. Write the explicit formula that represents the geometric sequence -2, 8, -32, 128 5. Evaluate the function f (x) 4 • 7x for x 1 and x = 2. Show your work. 6. Simplify the quotient 4.5 x 103 9 x 107 . Write your answer in scientific notation. Show your work. Simplify the expressions. Show your work. 7. 3x(4x4 – 5x) 8. (5x4 – 3x3 + 6x) – ( 3x3 + 11x2 – 8x) 9. (x – 2) (3x-4) 10. (x + 6)2 Factor each expression. Show your work. 11. r2 + 12r + 27 12. g2 – 9 13. 2p3 + 6p2 + 3p + 9 Solve each quadratic equation. Show your work. 14. (2x – 1)(x + 7) = 0 15. x2 + 3x = 10 16. 4x2 = 100 17. Find the roots of the quadratic equation x2 – 8x = 9 by completing the square. Show your work. 18. Use the discriminant to find the number of real solutions of the equation 3x2 – 5x + 4 = 0. Show your work. A water balloon is tossed into the air with an upward velocity of 25 ft/s. Its height h(t) in ft after t seconds is given by the function h(t) = − 16t2 + 25t + 3. Show your work. 19. After how many seconds will the balloon hit the ground? (hint: Use the......

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...FRESENIUS USA 2008H & 2008K SPARE PARTS MANUAL Part Number 490016 Rev I Click Here for Table of Contents FRESENIUS USA 2637 SHADELANDS DRIVE WALNUT CREEK, CALIF. 94598 (925) 295-0200 (800) 227-2572 Copyright 1994 – 2000 Fresenius USA, Inc. CONTENTS MANUAL UPDATE HISTORY.................................................................................................................................1 STATEMENT: MINIMUM TESTABLE ASSEMBLIES........................................................................................3 1.0.0 CABINET .....................................................................................................................................................5 1.1.0 CABINET WITH ACCESSORIES ...........................................................................................................6 1.1.1 DIALYSATE SAFETY INTERLOCK AND DIALYSATE LINES.........................................................8 1.1.2 CASTER ASSEMBLY............................................................................................................................12 1.2.0 CABINET, REAR, WITH ACCESSORIES (2008H) .............................................................................14 1.2.1 CABINET, REAR, WITH ACCESSORIES (2008K) .............................................................................16 1.3.0 CABINET, REAR, WITH PANELS (2008H...

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