Math assignment show all work . 10. Solve by the method of false position: A quantity and its 2/3 are added together and from the sum 1/3 of the sum is subtracted, and 10 remains. What is the quantity? (problem 28 of the Rhind Mathematical Papyrus) 16. Some scholars have conjectured that the area calculated in problem 10 of the Moscow Mathematical Papyrus is that of a semicylinder rather than a hemisphere. Show that the calculation in that problem does give the correct surface area of a semicylinder of diameter and height both equal to 41. 2 17. 28 .Solve the problem from the Old Babylonian tablet BM 13901: The sum of the areas of two squares is 1525. The side of the second square is 2/3 that of the first plus 5. Find the sides of each square. (Katz 29) 34. Solve the following problem from tablet YBC 6967: A number exceeds its reciprocal by 7. Find the number and the reciprocal. (In this case, that two numbers are \"reciprocals\" means that their product is 60.) 38. Given a circle of circumference 60 and a chord of length 12, what is the perpendicular distance from the chord to the circumference? (This problem is from tablet BM 85194.) Chapter 2 page 47 2 .Represent 8/9 as a sum of distinct unit fractions. Express the result in the Greek notation. Note that the answer to this problem is not unique. 8. Show that the nth triangular number is represented algebraically as T = n(n+1) and therefore that an oblong num- ber is double a triangular number 10. Show using dots that eight times any triangular number plus 1 makes a square. Conversely, show that any odd square diminished by 1 becomes eight times a triangular number. Show these results algebraically as well. 20. Construct five Pythagorean triples using the formula (n, n21, n2+1), where n is odd. Construct five different ones Chapter 3 page 91 6. Prove Proposition I-32, that the three interior angles of any triangle are equal to two right angles. Show that the proof depends on I-29 and therefore on postulate 5. 12. Prove Proposition III-3, that if a diameter of a circle bisects a chord, then it is perpendicular to the chord. And if a diameter is perpendicular to a chord, then it bisects the chord. 18. Prove that the last nonzero remainder in the Euclidean algorithm applied to the numbers a, b, is in fact the greatest common divisor of a and b. 32. Find the two mean proportionals between two cubes guaranteed by Proposition VIII-12. 36. Use Euclid's criterion in Proposition IX-36 to find the next perfect number after 8128. 40. Solve the equations of Proposition 86 of the Data alge- braically. Show that the two hyperbolas defined by the equa- tions each have their axes as the asymptotes of the other. Math. Please name each assignment for example discussion number 1 or assignment 1 with proper name. Discussion Forum 1 In Discussion Forum 1, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. A good amount of mathematical theorems were developed by the ancient Greeks beginning around 600 BCE. Choose a Greek mathematician of this time period and discuss a theorem he is most widely known for. In addition, provide an example of the mathematician's theorem as it is used in anything you have learned so far in the subject. Please try to use different theorems compared to your classmates. Use the Add a new discussion topic button to start your first posting for this activity. The written assignment draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 1: 10, 16, 28, 34, 38 Chapter 2: 2, 8, 10, 12, 20 Chapter 3: 6, 12, 18, 32, 36, 40 The written assignment draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 4: 2, 10, 16, 24, 30 Chapter 5: 2, 8, 14, 18, 22, 26, 36 Chapter 6: 4, 6, 8, 12, 16, 24 Discussion Forum 2 In Discussion Forum 2, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. As you have been seeing so far, even though many mathematical theorems were derived in Greece, other parts of the world have had their fair share of theorems as well. Based on the theorems you have learned in this module, choose one that interests you the most. Describe the theorem in more depth. In addition, as like the first discussion, provide an example of the theorem as it is used in anything you have learned so far in the subject. The written assignment 3 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 7: 2, 8, 10, 14, 20 Chapter 8: 4, 8, 12, 14, 16, 30 The written assignment 4 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 9: 4, 8, 12, 18, 22, 32 Chapter 10: 2, 6, 10, 16, 20, 28, 36 Chapter 11: 2, 8 Discussion Forum 3 In Discussion Forum 3, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. Until the 17th century, there was no such thing as Calculus. This is thanks to Newton and Leibniz. Discuss how Calculus changed the face of mathematics. Then, choose a Calculus theorem and discuss it in more depth. Use the Add a new discussion topic button to start your first posting for this activity. The written assignment 5 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 12: 2, 6, 8, 12, 14, 18, 20, 26, 30 Chapter 13: 4, 8, 14, 18, 20 The written assignment 6 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 14: 6, 10, 20, 28, 30 Chapter 15: 4, 12, 24, 26 Chapter 16: 2, 4, 16, 18, 24, 26 Discussion Forum 4 In Discussion Forum 4, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. Most of the theorems you remember came from the eighteenth and nineteenth centuries. Pick one of the many theorems mentioned in this module and describe it in detail. In addition, create one example based on your theory of choice. Use the Add a new discussion topic button to start your first posting for Written Assignment 7 The written assignment 7 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 17: 4, 10, 14, 20, 26 Chapter 18: 2, 12, 14, 18, 22 Chapter 19: 2, 4, 6, 8, 14, 26 Chapter 20: 4, 8, 12, 14 Written Assignment 8 The written assignment draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 21: 2, 6, 10, 16, 32, 38, 42 Chapter 22: 2, 10, 16, 20, 34 Chapter 23: 2, 4, 6, 8 Chapter 24: 2, 6, 8, 14, 18, 22 MAT301: HISTORY OF MATHEMATICS Midterm Paper You are required to write a midterm paper instead of taking a proctored midterm exam. This paper will focus on the first two modules of the course. As a reminder, these two modules deal with ancient and Medieval mathematics. Choose a mathematician you learned about in modules 1 or 2. Write a biography on the mathematician. In addition, share the theorems that made the mathematician famous. Include any realword examples based on those theorems. Please consult the Course Calendar for the due date for the midterm paper. GUIDELINES FOR THE MIDTERM PAPER The paper must be 68 pages long (double spaced, 12 size Times New Roman font) excluding references page or any appendix. Includes at least three references other than the textbook. Uses the MLA format for this paper. Grading Rubric WRITING RESOURCES The following links provide online writing aids to help you with your paper assignments. OWL (Online Writing Lab) at Purdue University Writer's Handbook , the Writing Center at the University of Wisconsin- Madison Documentation Rules and Citation Styles MAT301: HISTORY OF MATHEMATICS Final Paper You are required to write a final paper instead of taking a proctored final exam. This paper will focus on the last two modules of the course. As a reminder, these two modules deal with modern mathematics from the Renaissance to the Nineteenth Century. Choose a mathematical topic listed below. Discuss at least five different theorems for each topic. You must indicate the following: 1. The mathematician who created the theorem. 2. The idea behind the theorem. 3. An example you would use to show students, if you are teaching a lesson on this topic. The Mathematical Topics Algebraic Symbolism in Elementary Algebra Algebraic Symbolism in Intermediate Algebra Solving Polynomial Equations Complex Numbers Basic Probability Laws and Calculations Statistical Inference and Hypothesis Testing Conic Sections Theory of Equations in Precalculus Limits in Calculus Derivatives Integrals Fundamental Theorem of Calculus Power Series Vector Spaces Solutions of Ordinary Differential Equations Groups in Abstract Algebra Algebraic Numbers and Fields in Abstract Algebra Projective Geometry Set Theory in Advanced Calculus Final Paper Outline You are required to submit a 2page outline of your final paper that describes the mathematical topic you will research on. You should also include the name of the mathematician, the theorem, and at least two resources you have found. The outline will count 20% of your grade of the final paper (or 6% of your final grade). GUIDELINES FOR THE FINAL PAPER AND ITS OUTLINE The outline should be 2 pages long (double spaced, 12 size Times New Roman font) and be submitted earlier. The paper must be 7-10 pages long (double spaced, 12 size Times New Roman font) excluding references page or any appendix. Includes at least three references other than the textbook. Uses the MLA format for this paper. Grading Rubric WRITING RESOURCES The following links provide online writing aids to help you with your paper assignments. OWL (Online Writing Lab) at Purdue University Writer's Handbook, the Writing Center at the University of Wisconsin-Madison Math. Show all work Please name each assignment for example discussion number 1 or assignment 1 with proper me. this is the text book we use for the assignments.If you don't have access to it I will send you the questions/ and pages requires CHAPTER . Discussion Forum 1 In Discussion Forum 1, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. A good amount of mathematical theorems were developed by the ancient Greeks beginning around 600 BCE. Choose a Greek mathematician of this time period and discuss a theorem he is most widely known for. In addition, provide an example of the mathematician's theorem as it is used in anything you have learned so far in the subject. Please try to use different theorems compared to your classmates. Use the Add a new discussion topic button to start your first posting for this activity. The written assignment 1 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 1: 10, 16, 28, 34, 38 Chapter 2: 2, 8, 10, 12, 20 Chapter 3: 6, 12, 18, 32, 36, 40 The written assignment draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 4: 2, 10, 16, 24, 30 Chapter 5: 2, 8, 14, 18, 22, 26, 36 Chapter 6: 4, 6, 8, 12, 16, 24 Discussion Forum 2 In Discussion Forum 2, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. As you have been seeing so far, even though many mathematical theorems were derived in Greece, other parts of the world have had their fair share of theorems as well. Based on the theorems you have learned in this module, choose one that interests you the most. Describe the theorem in more depth. In addition, as like the first discussion, provide an example of the theorem as it is used in anything you have learned so far in the subject. The written assignment 3 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 7: 2, 8, 10, 14, 20 Chapter 8: 4, 8, 12, 14, 16, 30 The written assignment 4 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 9: 4, 8, 12, 18, 22, 32 Chapter 10: 2, 6, 10, 16, 20, 28, 36 Chapter 11: 2, 8 Discussion Forum 3 In Discussion Forum 3, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. Until the 17th century, there was no such thing as Calculus. This is thanks to Newton and Leibniz. Discuss how Calculus changed the face of mathematics. Then, choose a Calculus theorem and discuss it in more depth. Use the Add a new discussion topic button to start your first posting for this activity. The written assignment 5 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 12: 2, 6, 8, 12, 14, 18, 20, 26, 30 Chapter 13: 4, 8, 14, 18, 20 The written assignment 6 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 14: 6, 10, 20, 28, 30 Chapter 15: 4, 12, 24, 26 Chapter 16: 2, 4, 16, 18, 24, 26 Discussion Forum 4 In Discussion Forum 4, post your response to the following discussion question. Reply to at least two classmates' responses by the date indicated in the course Calendar. Most of the theorems you remember came from the eighteenth and nineteenth centuries. Pick one of the many theorems mentioned in this module and describe it in detail. In addition, create one example based on your theory of choice. Use the Add a new discussion topic button to start your first posting for Written Assignment 7 The written assignment 7 draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 17: 4, 10, 14, 20, 26 Chapter 18: 2, 12, 14, 18, 22 Chapter 19: 2, 4, 6, 8, 14, 26 Chapter 20: 4, 8, 12, 14 Written Assignment 8 The written assignment draws on even-numbered exercises from the textbook. Answer all assigned exercises, and show all work. Chapter 21: 2, 6, 10, 16, 32, 38, 42 Chapter 22: 2, 10, 16, 20, 34 Chapter 23: 2, 4, 6, 8 Chapter 24: 2, 6, 8, 14, 18, 22 MAT301: HISTORY OF MATHEMATICS Midterm Paper You are required to write a midterm paper instead of taking a proctored midterm exam. This paper will focus on the first two modules of the course. As a reminder, these two modules deal with ancient and Medieval mathematics. Choose a mathematician you learned about in modules 1 or 2. Write a biography on the mathematician. In addition, share the theorems that made the mathematician famous. Include any realword examples based on those theorems. Please consult the Course Calendar for the due date for the midterm paper. GUIDELINES FOR THE MIDTERM PAPER The paper must be 68 pages long (double spaced, 12 size Times New Roman font) excluding references page or any appendix. Includes at least three references other than the textbook. Uses the MLA format for this paper. Grading Rubric WRITING RESOURCES The following links provide online writing aids to help you with your paper assignments. OWL (Online Writing Lab) at Purdue University Writer's Handbook , the Writing Center at the University of Wisconsin-Madison Documentation Rules and Citation Styles MAT301: HISTORY OF MATHEMATICS Final Paper You are required to write a final paper instead of taking a proctored final exam. This paper will focus on the last two modules of the course. As a reminder, these two modules deal with modern mathematics from the Renaissance to the Nineteenth Century. Choose a mathematical topic listed below. Discuss at least five different theorems for each topic. You must indicate the following: 1. The mathematician who created the theorem. 2. The idea behind the theorem. 3. An example you would use to show students, if you are teaching a lesson on this topic. The Mathematical Topics Algebraic Symbolism in Elementary Algebra Algebraic Symbolism in Intermediate Algebra Solving Polynomial Equations Complex Numbers Basic Probability Laws and Calculations Statistical Inference and Hypothesis Testing Conic Sections Theory of Equations in Precalculus Limits in Calculus Derivatives Integrals Fundamental Theorem of Calculus Power Series Vector Spaces Solutions of Ordinary Differential Equations Groups in Abstract Algebra Algebraic Numbers and Fields in Abstract Algebra Projective Geometry Set Theory in Advanced Calculus Final Paper Outline You are required to submit a 2page outline of your final paper that describes the mathematical topic you will research on. You should also include the name of the mathematician, the theorem, and at least two resources you have found. The outline will count 20% of your grade of the final paper (or 6% of your final grade). GUIDELINES FOR THE FINAL PAPER AND ITS OUTLINE The outline should be 2 pages long (double spaced, 12 size Times New Roman font) and be submitted earlier. The paper must be 7-10 pages long (double spaced, 12 size Times New Roman font) excluding references page or any appendix. Includes at least three references other than the textbook. Uses the MLA format for this paper. Grading Rubric WRITING RESOURCES The following links provide online writing aids to help you with your paper assignments. OWL (Online Writing Lab) at Purdue University Writer's Handbook, the Writing Center at the University of Wisconsin-Madison Written assignment 1 EXERCISES show all work 10. Solve by the method of false position: A quantity and its 2/3 are added together and from the sum 1/3 of the sum is subtracted, and 10 remains. What is the quantity? (problem 28 of the Rhind Mathematical Papyrus) 16. Some scholars have conjectured that the area calculated in problem 10 of the Moscow Mathematical Papyrus is that of a semicylinder rather than a hemisphere. Show that the calculation in that problem does give the correct surface area of a semicylinder of diameter and height both equal to 4 1/2 . 28 .Solve the problem from the Old Babylonian tablet BM 13901: The sum of the areas of two squares is 1525. The side of the second square is 2/3 that of the first plus 5. Find the sides of each square. 34 solve the following problem from tablet YBC 6967: A number exceeds its reciprocal by 7. Find the number and the reciprocal. (In this case, that two numbers are \"reciprocals\" means that their product is 60.) 38. Given a circle of circumference 60 and a chord of length 12, what is the perpendicular distance from the chord to the circumference? (This problem is from tablet BM 85194.) CHAPTER 2 forcing mathematicians to the present day to think carefully about their assumptions in dealing with the concepts of the infinite or the infinitely small. And in Greek times they were probably a significant factor in the development of the distinction between continuous magnitude and discrete number so important to Aristotle and ultimately to Euclid. EXERCISES 2. Represent 8/9 as a sum of distinct unit fractions. Express the result in the Greek notation. Note that the answer to this problem is not unique. 8. Show that the nth triangular number is represented algebraically as T = n(n+1) and therefore that an oblong num- ber is double a triangular number. 10. Show using dots that eight times any triangular number plus 1 makes a square. Conversely, show that any odd square diminished by 1 becomes eight times a triangular number. Show these results algebraically as well. 12. Construct five Pythagorean triples using the formula (n, n21, n2+1), where n is odd. Construct five different ones 22 using the formula (m, (m)2 1, (m)2 + 1), where m is 22 even. 20- Give an example of each of the four rules of inference discussed in the text. Chapter 3 6. Prove Proposition I-32, that the three interior angles of any triangle are equal to two right angles. Show that the proof depends on I-29 and therefore on postulate 5. . 12. Prove Proposition III-3, that if a diameter of a circle bisects a chord, then it is perpendicular to the chord. And if a diameter is perpendicular to a chord, then it bisects the chord. . 18. Prove that the last nonzero remainder in the Euclidean algorithm applied to the numbers a, b, is in fact the greatest common divisor of a and b. 32. Find the two mean proportionals between two cubes guaranteed by Proposition VIII-12. 36. Use Euclid's criterion in Proposition IX-36 to find the next perfect number after 8128 40. Solve the equations of Proposition 86 of the Data alge- braically. Show that the two hyperbolas defined by the equa- tions each have their axes as the asymptotes of the other Written assignment 2 Chapter 4 xercises 127 in that book. Other theorems there enable one to show that a conic also solves the four-line locus problem, to find the locus of a point such that the product of its distances to one pair of lines is in a constant ratio to the product of its distances to the other pair. In later Greek times, an attempt was made, without great success, to find the locus with regard to greater numbers of lines. It was this problem that Descartes and Fermat both demonstrated they could solve through their new method of analytic geometry in the seventeenth century, a method whose germ came from a careful reading of Apollonius's work. Descartes in fact was able to derive the equations of curves that satisfied analogous conditions for various numbers of lines and to classify the solutions. As should be evident from our description of many of the Greek problems in modern notation, the Greek tradition of geometric problem solving, which was carried on in the Islamic world long after its demise in the Hellenic world, ultimately led to new advances in mathematical technique, advances that finally reduced much of this kind of Greek mathematics to mere textbook exercises. EXERCISES 2. I faweightof8kgisplaced10mfromthefulcrumofa lever and a weight of 12 kg is placed 8 m from the fulcrum in the opposite direction, toward which weight will the lever incline? . 10. Use calculus to prove Archimedes' result from The Method that the volume of the segment of the cylinder described in the text is 1/6 the volume of the rectangular parallelepiped circumscribing the cylinder. 16. There is a story about Archimedes that he used a \"burning mirror\" in the shape of a paraboloid of revolution to set fire to enemy ships in the harbor. What would be the equation of the parabola that one would rotate to form the appropriate 24. Prove analytically Proposition VII-12, that in any ellipse the sum of the squares on any two of its conjugate diameters is equal to the sum of the squares on its two axes. (In Figure 4.32, this means that P G2 + DK2 = AE2 + BL2.) 30. Prove Proposition III-46: Under the same assumptions as in III-45, angle ACF = angle DCG and angle CDF = angle BDG (Fig. 4.34). Chapter 5 EXERCISES . 2- Calculate crd(120), crd(150), crd(165), and crd(172 1 ) 2 usingHipparchus'sformulaforcrd(180 ). Sun at noon on winter solstice 2. Calculate crd(120), crd(150), crd(165), and crd(172 1 ) 2 usingHipparchus'sformulaforcrd(180 ). 3. calculate 4500 to two sexagesimal Use Theon's method to 8. Calculate, using Ptolemy's methods, the length of a noon shadow of a pole of length 60 at the vernal equinox at a place of latitude 40. 14. Suppose that the maximum length of day at a particular location is known to be 15 hours. Calculate the latitude of that location and the position of the sun at sunrise on the summer and winter solstices. ( 18. Calculate the sun's maximal northerly sunrise point for latitude 36 degree 22. Calculate the area of a triangle with sides of lengths 4, 7, and 10 using both of Heron's methods 26. Derive Heron's formula for the volume 1 3 octahedron of edge length a. ? 36. Chapter 6 Look up in an astronomy work the \"equation of time,\" and discuss why the times of sunrise and sunset calculated via the methods in the text are likely to be incorrect by several minutes. 4. Nicomachus defined a subcontrary proportion, which oc- 4 curs when in three terms the greatest is to the smallest as the difference of the smaller terms is to the difference of the greater. Show that 3, 5, 6, are in the subcontrary proportion. Find two other sets of three terms that are in subcontrary proportion 6 . Nicomachus defined a \"fifth proportion\" to exist whenever among three terms the middle term is to the lesser as their difference is to the difference between the greater and the mean. Show that 2, 4, 5, are in fifth proportion. Find two more triples in this proportion. . 8. Solve Diophantus's Problem I-27 by the method of I-28: To find two numbers such that their sum and product are given. Diophantus gives the sum as 20 and the product as 96. 12. Solve Diophantus's Problem B-8: To find two numbers such that their difference and the difference of their cubes are equal to two given numbers. (Write the equations as xy=a, x3y3=b. Diophantus takes a=10, b=2120.) Derive necessary conditions on a and b that ensure a rational solution. 16. Solve Diophantus's Problem V-10 for the two given num- bers 3, 9. 24. Solve Epigram 130: Of the four spouts, one filled the whole tank in a day, the second in two days, the third in three days, and the fourth in four days. What time will all four take to fill it