MIDTERM STUDY GUIDE
MIDTERM STUDY GUIDE 4109
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This 9 page Study Guide was uploaded by Kassandra Walker on Monday October 3, 2016. The Study Guide belongs to 4109 at University of North Carolina - Charlotte taught by Dr. Kim Harris in Fall 2016. Since its upload, it has received 179 views. For similar materials see History of Mathematics in Math at University of North Carolina - Charlotte.
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Date Created: 10/03/16
History ofMath –Test #1 Filled in study guide below with Moodle Questions and answers. Topics Describe how the Egyptian hieroglyphic system was used to represent numbers. Be able to go back and forth between our numeration system and the Egyptian system. Know the symbols for 1 through 1000. o The Egyptians had a writing system based on hieroglyphs from around 3000 BC. The Egyptian numeration system was a base 10, non-place value system of hieroglyphs for numerals. Egyptian numeration system up to 1,000: What is an algorithm? o According to the Oxford dictionary, an algorithm is a process or set of rules to be followed in calculations or other problem-solving operations. Characterize the Egyptians’ use of fractions. o The Egyptians used only unit fractions and 2/3. o A unit fraction is a fraction writt1/nthe numerator is always 1, and the denominator is a natural number. The ancient Egyptians wrote all fractions as sums of unit fractions except 2/3. What is the Rhind Mathematical Papyrus and what is its significance? o A 6 m x 1/3 m papyrus contains 87 problems concerned with practical problems. Showed the method of false position, multiplication and division procedures, and fractions. o This gave us the majority of our knowledge about ancient Egyptian mathematics. Characterize Egyptian geometry. o Areas of rectangles, triangles, trapezoids, and circles. They Egyptians made it seem like they knew the Pythagorean theorem, but this is an empty claim. Characterize the Babylonian numeration system. o The Babylonian numeration system was a positional base 60 system. Translate numerals between the Babylonian numeral system and our system (numbers less than one as well as greater than one). You do not need to know the symbols for 1 and 10 in this system; I will use Hindu-Arabic numerals and commas and semicolons to separate the places. o See Example on Next Page From where do we get our information about Babylonian mathematics? o Cuneiform & clay tablets Who was Thales, and what were his contributions to the history of mathematics? o The first mathematician on record. o In some circles, he is known as the “Father of Geometry,” and he is credited with the phrase, “Know thyself.” He is credited with several propositions and theorems. The value of Thales’ work is not in the statements themselves, but in the logical reasoning that he used to support them. Explain: “number is the substance of all things.” o This statement is contributed to the Pythagorean Doctrine. What was the Pythagorean crisis in mathematics? o Discovering irrational numbers. This challenged the Pythagorean people to redefine their view on numbers. What are the figurate numbers? Recursive vs. Closed formulas o Odd and Even Numbers o Triangular, Square, Pentagonal, Oblong, etc. See Problem Set #4 for Table o Recursive Formulas rely on previous data. Closed formulas rely simply on what term it is in the series. What is the Pythagorean Theorem? Be able to state and apply the theorem. o “In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.” - PPT What are prime and composite numbers? What are perfect numbers? o Prime numbers only have factors of themselves and 1. Composite numbers are the “opposite” of prime numbers; they are factorable by numbers outside of 1 and themselves. A number is called a perfect number if it is exactly equal to the sum of its properties factors. What were the three construction problems of antiquity? o Squaring a circle The problem asks us to find a square whose side length is ???? with an area of ???? where ???? = ???????? . The result is a side is equal √o ???? √…???? is irrational. o Duplicating a cube o Trisecting an angle Describe the contents of Euclid’s Elements. o Collection of 465 theorems from plane and solid geometry and number theory organized into 13 books. Describe Euclid’s method in putting together the Elements. (Know axiomatic system.) o In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. Today we use the words “postulate” and “axiom” interchangeably. They both refer to statements that we assume to be true and do not require a proof. Euclid used the word “proposition,” but today we would call these statements “theorems.” State Euclid’s Fifth Postulate. o That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. o Also called Playfair’s Axiom/Postulate (given P and L, there is one & only one line through P that is parallel to L) and Angle-Sum Axiom, Parallel Postualte. What is the Fundamental Theorem of Arithmetic? o Also called the Unique Factorization Theorem, the theorem says every natural number greater than 1 can be expressed as a product of prime numbers in a unique way apart from the order in which the prime factors might be written. In other words, there is one prime factorization of 24 which is 2 ∙ 3. Who was Eratosthenes, and what were some of his contributions to the history of mathematics? o See Mathematicians section below. Who was Archimedes and what were some of his contributions to the history of mathematics? o See Mathematicians section below. o “Give me a leaver and I can move the Earth”. What method did Archimedes use to approximate ????? o Determined the perimeters of inscribed and circumscribed polygons in a circle. What method did Archimedes use to determine the area of a circle? o method of exhaustion What is the method, reductio ad absurdum? o Method of Proof by Contradiction – From Euclid What were the Indian contributions related to numeration systems and number sets? o Place value and base 10 system. What are the three stages of algebra? Characterize each stage, and identify key people or civilizations in each. o Rhetorical: No symbols were used; everything was written out in words. Ex. Egyptians, Babylonians, Greeks. No equations, formulas – all descriptive. o Syncopated: Abbreviations were used. Diophantus’ book Arithmetica introduced syncopated algebra. o Symbolic: Our modern algebra with symbols for meaning. Mathematicians Thales o First mathematician on record. o When: 625 – 547 BCE o Where: Greek Empire o Contributions: In some circles, he is known as the “Father of Geometry,” and he is credited with the phrase, “Know thyself.” He is credited with several propositions and theorems. The value of Thales’ work is not in the statements themselves, but in the logical reasoning that he used to support them. An angle inscribed in a semi-circle is a right angle. Pythagoras o When: 580 – 500 BCE o Where: Greek Empire o Contributions: Pythagoras founded a “school” in southern Italy that had many aspects of what we would call a cult today. The school was well-known for the rules that governed all aspects of the Pythagoreans’ lives. The motto was “ALL IS NUMBER”. They thought of numbers in a very visual way, and a study of “figurate numbers” provides beautiful examples of both geometric and numerical patterns. Pythagorean Proposition (Theorem), Different types of Numbers (Perfect, Deficient, Abundant, Triangle, Square, etc.) Euclid o When: 325 - 265 BCE o Where: Greek Empire o Contributions: Euclid’s Elements, an organized way to look at Euclidean geometry. Used as THE geometry curriculum for 2 centuries. Eratosthenes o When: 276 – 194 BCE o Where: Alexandria o Contributions: Sieve of Eratosthenes, great geographer, renaissance man before it’s time. Developed a mechanical solution for the problem of doubling the cube and approximated the circumference of the Earth. Wrote Geographica, which we have little of today. Archimedes o “Greatest” of the ancient mathematicians. o When: 287 – 212 BCE o Where: Syracuse, Sicily, Italy o Contributions: Inventions such as the Archimedean screw and possibly several other war like inventions (the claw, catapults, the burning mirror). Most famous for the golden crown problem (displacement), his several “books”, and for using the method of exhaustion. Introduced new propositions since Euclid. Approximation of Pi using polygons inscribed in circles. Ptolemy o When: 85 – 165 CE o Where: Alexandria (Greek ethnicity, Roman Citizen) o Contributions: He developed the geocentric theory (earth-centered) of planetary motion in a form that prevailed for 1400 years and wrote Almagest. Influenced by Babylonian sciences. Diophantus o “Father of Algebra” o When: 200 – 284 CE o Where: Alexandria o Contributions: Wrote Arithmetica and introduced syncopated algebra. Hypatia o When: 375 – 415 CE o Where: Alexandria, Greek Empire o Contributions: Teacher in Alexandria whose contributions consisted of commentary on previous Greek mathematicians works that described complex mathematical ideas clearly and precisely. Many inventions for astronomy: Hydroscope, astrolabe Brahmagupta o When: 598 – 670 AD o Where: Indian Subcontient, Ujjain school o Contributions: Contributed one of the most significant contributions to the development of the numbers system through negative numbers and zero. Also contributed to understanding of integer solutions to indeterminate equations and to interpolation formulas that were invented to aid the computation of the sine tables. Books Rhind Papyrus o Who: Egyptians o What: A 6 m x 1/3 m papyrus contains 87 problems concerned with practical problems. Showed the method of false position, multiplication and division procedures, and fractions. o When: 1650 B.C. o Significance: Teaches how ancient Egyptian mathematics worked. Euclid’s Elements o Who: Euclid of the Greek Empire o What: Collection of 465 theorems from plane and solid geometry and number theory organized into 13 books; Relatively few were original proofs by Euclid; Genius was in his clear, thorough, and logical organization of the mathematics; First printed edition appeared in 1482; Second only to the Bible in the number of editions; Copernicus, Kepler, Galileo, and Newton were all influenced by the Elements; THE geometry text for over 2000 years! o When: ~300 BCE o Significance: He created a deductive system based on definitions and axioms from which theorems could be proved. For over 2000 years, the first six books of the Elements WERE the curriculum goals, standards, and textbook for geometry students. When it comes to geometry, this is the real “common core.” Measurement of a Circle o Who: Archimedes o What: Contained 3 propositions. o Significance: The propositions gave ways to calculate the area and circumference of a circle and approximate Pi. Arithmetica o Who: Diophantus o What: First book entirely devoted to algebra. Assortment of 189 problems. o Significance: First documented use of syncopated algebra. Moving from just rhetorical to symbolic (not quite there, but closer). Almagest o Who: Ptolemy o What: Introduced trigonometry and the Ptolemaic Model of the universe (geocentric). o Significance: Was not questioned for 1400 years. Gave theory of the motions of the sun, moon, and the planets. Used trigonometry. Nine Chapters on the Mathematical Art o A.K.A. Jiuzhang suanshu o Who: Chinese o What: Practical handbook of mathematics consisting of 246 problems intended to provide methods to be used to solve everyday problems of engineering, surveying, trade, and taxation. o Significance: Played a fundamental role in the development of Mathematics in China. Similar to how Euclid’s Elements played a role in the Greek Empire. Moodle Questions & Answers 1. Which of the following numeration systems had a symbol for zero? a. Mayan 2. What number is the base for a decimal numeration system? a. 10 3. What does the Roman numeral MDC represent in Hindu-Arabic numerals? a. 1600 4. What type of system did the Babylonians use to write their numbers? a. Sexagesimal 5. Which system employed a subtractive principle in which the symbol for a smaller number is placed in front of the symbol for a larger number? a. Roman 6. Which of the following equations best explains the procedure of doubling and adding to multiply 13 by 34? a. 13 x 24 = (1 + 4 + 8) x 24 = 1x24 + 4x24 + 8x24 (1, 4, and 8 are all powers of 2) 7. How would the ancient Egyptians have written the fraction, 3/4 (three-fourths)? a. 1/2 + 1/4 8. What is a unit fraction? a. A fraction whose numerator is 1. 9. What was the most important rule that the Egyptians used when they wrote sums of unit fractions? a. They could never use the same fraction twice. 10. Here's a problem: Suppose a quantity and its one-half give me 18. What is the quantity? If I try the number 6, I add the number 6 and its half (3) to get 9. What should I multiply 6 by to get the correct answer to my problem? a. 2 11. The Babylonians knew how to solve quadratic equations of the form, x^2+bx=c . What were the restrictions on the values of b and c? a. b and c had to be positive rational numbers 12. The mathematics of the Babylonians was more advanced than the mathematics of the Egyptians. Which of the following was known to the Babylonians but not the Egyptians? a. How to solve certain cubic equations 13. What was the system of writing used by the Babylonians? a. Cuneiform 14. The Plimpton 322 tablet is an important artifact in our knowledge of Babylonian mathematics. Which of the following is demonstrated in the Plimpton 322 Tablet? Hint: We usually associate this with the ancient Greeks. a. The Pythagorean relationship 15. What was the main emphasis of Babylonian geometry? a. Measurement 16. What do we call an angle whose vertex is on a circle and whose sides intersect the same circle? a. An inscribed angle 17. Whose paradox challenged the way mathematicians thought about infinity? a. Zeno 18. What are the only two tools that are allowed to trisect an angle? a. Compass and straightedge 19. What is true about an angle that is inscribed in a semi-circle? In other words, the sides of the inscribed angle go through the opposite ends of a diameter of the circle. a. The angle is a right angle 20. What is a perfect number? a. A number that is equal to the sum of its proper factors. 21. What type of proof did Euclid use to prove that the square root of 2 is irrational? a. Proof by contradiction 22. What does the fundamental theorem of arithmetic say about the number 100? a. 100 can be written in the form 2x2x5x5. 23. What is the purpose of the Euclidean Algorithm? a. To find the greatest common divisor of two numbers. 24. What are the tools that Euclid used to carry out the constructions in The Elements? a. A compass and a straight edge 25. According to Euclid's Fifth Postulate, what is true if a transversal intersects two straight lines such that the sum of the measures of the interior angles on the same side is less than 180 degrees? a. The two straight lines meet on the same side as the interior angles. 26. Which city was the center of learning in the ancient world? a. Alexandria 27. Who wrote Measurement of a Circle? a. Archimedes 28. In which city did Hypatia live and work? a. Alexandria 29. Why is the book, Arithmetica, significant in the history of algebra? a. Its use of abbreviations to represent mathematical symbols and operations. 30. In the Ptolemaic model of the universe, what heavenly body was at its center? a. Earth 31. In India, which field of science was most closely related to the development of mathematics? a. Astronomy 32. What is the most significant contribution of Indian mathematics to the development of the history of mathematics? a. The base 10 numeration system 33. Which civilization had a place value system before the Indian base ten system? a. Babylonian 34. Which Indian mathematician introduced negative numbers? a. Brahmagupta 35. In Indian mathematics, the Pythagorean Theorem was applied to what geometric shapes/figures instead of right triangles? a. The diagonal and sides of a rectangle
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