MATH3560/GENS2005 HISTORY OF MATHEMATICS
|
|
- Joshua Hensley
- 5 years ago
- Views:
Transcription
1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 HISTORY OF MATHEMATICS (1) TIME ALLOWED 2 Hours (2) TOTAL NUMBER OF QUESTIONS 19 (3) CANDIDATES SHOULD ATTEMPT QUESTIONS WORTH 100 MARKS (4) THE QUESTIONS ARE NOT OF EQUAL VALUE (5) THIS PAPER MAY BE RETAINED BY THE CANDIDATE (6) CALCULATORS WILL BE PROVIDED All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work.
2 OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page 2 Please see over...
3 Please see over... OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page 3 1. [10 marks] i) Use the Babylonian method of finding square roots to obtain first and second rational approximations to 12. ii) Square your approximation to see how close it is to [10 marks] i) Use the Egyptian method of doubling to multiply 37 by 41. ii) Use the Egyptian method to find [10 marks] Let x = i) Express x in (modern) sexagemisal notation x = 0; s 1, s 2. ii) Which ancient civilization used sexagemisal notation? iii) Express x as a sum of unitary fractions. iv) Which ancient civilization represented fractions in this form. 4. [10 marks] Let p n denote the nth pentagonal number. i) Draw a diagram illustrating the first 3 pentagonal numbers. ii) Write down the first 4 pentagonal numbers. iii) Use your diagram to write down a recurrence for p n iv) Assuming that the formula for p n is given by a quadratic p n = an 2 +bn+c, find a, b, c. 5. [10 marks] Cardano ( ) had a method for solving cubics. In this question we shall employ a variant of Cardano s method to solve the cubic x 3 6x = 9. (1) i) Use the substitution x = u + v to convert the cubic equation (1) to the form (u 3 + v 3 ) + 3uv(u + v) = 9 + 6(u + v). (2) ii) Equate terms on both sides of (2) to produce values for u 3 + v 3 and uv. iii) Let α = u 3 and β = v 3. Use the fact that we know the numerical values of α+β and αβ to write down a quadratic which has α and β as its roots. iv) Solve this quadratic. Hence find u and v, and finally, x.
4 Please see over... OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page 4 6. [10 marks] For many centuries people tried to prove Euclid s fifth postulate until it was realised that it truly was independent of Euclid s other postulates. i) State Euclid s fifth postulate, either in the original form, or else in a more recent equivalent form. ii) Write half a page giving some of the history surrounding this postulate. 7. [10 marks] Let ABCDE be a regular pentagon with side 1. i) Calculate the angles EAB and EAD. Let v = 2 cos π 5. ii) Use the cosine rule in DAC to show that and hence calculate cos π 5. iii) Is the angle π 5 constructible? v 3 2v = 0, 8. [10 marks] There is no rational number x for which x 2 = 2. i) Prove this statement. ii) When was this result first proved? Comment on the historical significance of the proof. 9. [10 marks] Pierre de Fermat lived from 1601 to i) Give 3 areas of mathematics to which Fermat made significant contributions. ii) State Fermat s Last Theorem and Fermat s Little Theorem. iii) Did Fermat give proofs of either of these results? 10. [10 marks] Al-Karkhi (ca. 1020) found a family of rational solutions to x 3 + y 3 = z 2. He took x = with n, m natural numbers. n2, y = mx, z = nx, 1 + m3 i) Show that these formulae do indeed give solutions. ii) Find a rational solution (x, y, z) which is not generated by this formula.
5 Please see over... OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page [10 marks] Nasr ad-din al-tusi ( ) is said to have been the first to show that the sum of two odd squares cannot be a square. Let x = 2n + 1 and y = 2m + 1 for two natural numbers n and m. i) Calculate x 2 + y 2 mod 4. ii) Hence or otherwise prove Nasr ad-din al-tusi s result. iii) Give an example of two even squares whose sum is a square. 12. [10 marks] We follow Archimedes in approximating π. i) Inscribe a regular hexagon inside a circle of radius r. What is the perimeter of the hexagon? ii) Do the same with a regular dodecagon. What is its perimeter? Thus, give a lower bound for π. 13. [10 marks] i) Explain the golden rectangle and the golden ratio τ. ii) What are the relationships between τ, τ 2 and 1 τ? iii) How is τ related to the Fibonacci numbers? 14. [10 marks] Suppose a 2 + b 2 = c 2 and let x a y = b. z c i) Show that x 2 + y 2 = z 2 and y x = b a. ii) Starting with (a, b, c) = (5, 12, 13), find two more Pythagorean triads with the difference between the shorter sides equal to [10 marks] i) Define carefully what is meant by Platonic solid. ii) List all the Platonic solids, giving a description of each (give the type of face, number of faces, edges and vertices of each). iii) Explain the duality of the Platonic solids. Which of them is self dual?
6 OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page [10 marks] i) (Roughly) when did Évariste Galois live? ii) How did Galois die? iii) What famous problems did Galois Theory solve? 17. [20 marks] Write a short essay (around 500 words) about Arabic mathematics and its influence on later European mathematical development. 18. [20 marks] Write a short essay (around 500 words) about the mathematics of the Babylonians, Egyptians and Ancient Greeks. You should explain i) how we know about this mathematics. ii) the major differences between the styles of mathematical documents between these cultures. 19. [20 marks] In about 500 words, write a short summary of the ideas in your essay.
MthEd/Math 300 Williams Fall 2011 Midterm Exam 2
Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 2 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first
More informationFoundations of Basic Geometry
GENERAL I ARTICLE Foundations of Basic Geometry Jasbir S Chahal Jasbir S Chahal is Professor of Mathematics at Brigham Young University, Provo, Utah, USA. His research interest is in number theory. The
More informationA Brief History of Algebra
A Brief History of Algebra The Greeks: Euclid, Pythagora, Archimedes Indian and arab mathematicians Italian mathematics in the Renaissance The Fundamental Theorem of Algebra Hilbert s problems 1 Pythagoras,
More informationMSM 707 Number Systems for Middle School Teachers Semester Project
MSM 707 Number Systems for Middle School Teachers Semester Project During the course of the semester, we will discuss some important concepts of Number Theory. The following projects are designed to give
More informationMath Number 842 Professor R. Roybal MATH History of Mathematics 24th October, Project 1 - Proofs
Math Number 842 Professor R. Roybal MATH 331 - History of Mathematics 24th October, 2017 Project 1 - Proofs Mathematical proofs are an important concept that was integral to the development of modern mathematics.
More informationThe Three Ancient Geometric Problems
The Three Ancient Geometric Problems The Three Problems Constructions trisect the angle double the cube square the circle The Three Problems trisecting the angle Given an angle, The Three Problems trisecting
More informationMath 312, Lecture 1. Zinovy Reichstein. September 9, 2015 Math 312
Math 312, Lecture 1 Zinovy Reichstein September 9, 2015 Math 312 Number theory Number theory is a branch of mathematics Number theory Number theory is a branch of mathematics which studies the properties
More informationMath 1230, Notes 2. Aug. 28, Math 1230, Notes 2 Aug. 28, / 17
Math 1230, Notes 2 Aug. 28, 2014 Math 1230, Notes 2 Aug. 28, 2014 1 / 17 This fills in some material between pages 10 and 11 of notes 1. We first discuss the relation between geometry and the quadratic
More informationMathematics (Modular) 43055/2H (Specification B) Module 5
Centre Number Surname Candidate Number For Examiner s Use Other Names Candidate Signature Examiner s Initials General Certificate of Secondary Education Higher Tier June 0 Mathematics (Modular) 43055/H
More informationAQA Level 2 Certificate in FURTHER MATHEMATICS (8365/2)
SPECIMEN MATERIAL AQA Level 2 Certificate in FURTHER MATHEMATICS (8365/2) Paper 2 Specimen 2020 Time allowed: 1 hour 45 minutes Materials For this paper you must have: mathematical instruments You may
More informationChapter 12: Ruler and compass constructions
Chapter 12: Ruler and compass constructions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Spring 2014 M. Macauley (Clemson) Chapter
More informationGrade 6 Math Circles. Ancient Mathematics
Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles October 17/18, 2017 Ancient Mathematics Centre for Education in Mathematics and Computing Have you ever wondered where
More informationHence, the sequence of triangular numbers is given by., the. n th square number, is the sum of the first. S n
Appendix A: The Principle of Mathematical Induction We now present an important deductive method widely used in mathematics: the principle of mathematical induction. First, we provide some historical context
More informationStudy Guide for Exam 1
Study Guide for Exam 1 Math 330: History of Mathematics October 2, 2006. 1 Introduction What follows is a list of topics that might be on the exam. Of course, the test will only contain only a selection
More informationIntroduction to and History of the Fibonacci Sequence
JWBK027-C0[0-08].qxd 3/3/05 6:52 PM Page QUARK04 27A:JWBL027:Chapters:Chapter-0: Introduction to and History of the Fibonacci Sequence A brief look at mathematical proportion calculations and some interesting
More informationTHE INTRODUCTION OF COMPLEX NUMBERS*
THE INTRODUCTION OF COMPLEX NUMBERS* John N. Crossley Monash University, Melbourne, Australia Any keen mathematics student will tell you that complex numbers come in when you want to solve a quadratic
More informationUNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education
UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education *6357012477* CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 Paper 6 (Extended) October/November
More informationBasic Ideas in Greek Mathematics
previous index next Basic Ideas in Greek Mathematics Michael Fowler UVa Physics Department Closing in on the Square Root of 2 In our earlier discussion of the irrationality of the square root of 2, we
More informationHomework 1 from Lecture 1 to Lecture 10
Homework from Lecture to Lecture 0 June, 207 Lecture. Ancient Egyptians calculated product essentially by using additive. For example, to find 9 7, they considered multiple doublings of 7: Since 9 = +
More informationIntroduction to Trigonometry: Grade 9
Introduction to Trigonometry: Grade 9 Andy Soper October 6, 2013 This document was constructed and type-set using P C T E X (a dielect of L A T E X) 1 1 Before you start 1.1 About these notes. These notes
More informationConstructing Trig Values: The Golden Triangle and the Mathematical Magic of the Pentagram
Constructing Trig Values: The Golden Triangle and the Mathematical Magic of the Pentagram A Play in Five Acts 1 "Mathematicians always strive to confuse their audiences; where there is no confusion there
More information1. Peter cuts a square out of a rectangular piece of metal. accurately drawn. x + 2. x + 4. x + 2
1. Peter cuts a square out of a rectangular piece of metal. 2 x + 3 Diagram NOT accurately drawn x + 2 x + 4 x + 2 The length of the rectangle is 2x + 3. The width of the rectangle is x + 4. The length
More informationBrunswick School Department Honors Geometry Unit 6: Right Triangles and Trigonometry
Understandings Questions Knowledge Vocabulary Skills Right triangles have many real-world applications. What is a right triangle? How to find the geometric mean of two numbers? What is the Pythagorean
More informationI.G.C.S.E. Area. You can access the solutions from the end of each question
I.G.C.S.E. Area Index: Please click on the question number you want Question Question Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 You can access the solutions from the
More informationThe Mathematics of Renaissance Europe
The Mathematics of Renaissance Europe The 15 th and 16 th centuries in Europe are often referred to as the Renaissance. The word renaissance means rebirth and describes the renewed interest in intellectual
More informationClasswork 8.1. Perform the indicated operation and simplify each as much as possible. 1) 24 2) ) 54w y 11) wy 6) 5 9.
- 7 - Classwork 8.1 Name Perform the indicated operation and simplify each as much as possible. 1) 4 7) 16+ 5 49 ) 5 4 8) 11 6 81 ) 5 4x 9) 9 x + 49x 4) 75w 10) 6 5 54w y 5) 80wy 11) 15 6 6) 5 9 1) 15x
More informationSome Highlights along a Path to Elliptic Curves
11/8/016 Some Highlights along a Path to Elliptic Curves Part : Conic Sections and Rational Points Steven J Wilson, Fall 016 Outline of the Series 1 The World of Algebraic Curves Conic Sections and Rational
More informationTest 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also.
MATD 0370 ELEMENTARY ALGEBRA REVIEW FOR TEST 4 (1.1-10.1, not including 8.2) Test 4 also includes review problems from earlier sections so study test reviews 1, 2, and 3 also. 1. Factor completely: a 2
More informationSOLUTIONS, what is the value of f(4)?
005 Georgia Tech High School Mathematics Competition Junior-Varsity Multiple-Choice Examination Version A Problem : If f(x) = x4 x +x x SOLUTIONS, what is the value of f(4)? (A) 6 (B) 70 (C) 78 (D) 8 (E)
More informationDiophantine equations
Diophantine equations So far, we have considered solutions to equations over the real and complex numbers. This chapter instead focuses on solutions over the integers, natural and rational numbers. There
More informationMath Round. Any figure shown may not be drawn to scale.
Indiana Academic Super Bowl Math Round 2019 Coaches Practice A Program of the Indiana Association of School Principals Students: Throughout this round we will be pronouncing mathematic symbols and concepts
More informationMATHEMATICS AND ITS HISTORY. Jimmie Lawson
MATHEMATICS AND ITS HISTORY Jimmie Lawson Spring, 2005 Chapter 1 Mathematics of Ancient Egypt 1.1 History Egyptian mathematics dates back at least almost 4000 years ago. The main sources about mathematics
More informationThursday 11 June 2015 Afternoon
Oxford Cambridge and RSA H Thursday 11 June 2015 Afternoon GCSE METHODS IN MATHEMATICS B392/02 Methods in Mathematics 2 (Higher Tier) *4856252055* Candidates answer on the Question Paper. OCR supplied
More informationr=1 Our discussion will not apply to negative values of r, since we make frequent use of the fact that for all non-negative numbers x and t
Chapter 2 Some Area Calculations 2.1 The Area Under a Power Function Let a be a positive number, let r be a positive number, and let S r a be the set of points (x, y) in R 2 such that 0 x a and 0 y x r.
More informationMATHEMATICS National Qualifications - National 5 Paper 1 (Non Calculator) Testing EF and REL
`k N5 Prelim Examination 016 / 17 MATHEMATICS National Qualifications - National 5 Paper 1 (Non Calculator) Testing EF and REL Time allowed - 1 hour Fill in these boxes and read carefully what is printed
More informationz=(r,θ) z 1/3 =(r 1/3, θ/3)
Archimedes and the Archimedean Tradition Thursday April 12 Mark Reeder The topic is Proposition 4 in ook II of Archimedes On the Sphere and the Cylinder (SC), an important result in the Greek and Arabic
More informationPROJECTS. Project work in mathematics may be performed individually by a
PROJECTS Project work in mathematics may be performed individually by a student or jointly by a group of students. These projects may be in the form of construction such as curve sketching or drawing of
More information2015 Manitoba Mathematical Competition Key Thursday, February 24, 2015
015 Manitoba Mathematical Competition Key Thursday, February 4, 015 1. (a) Solve for x: 5 x 7 + x 3x = 3 (b) Find the ratio of x to y, given that x y x + y = 5 Solution: (a) Multiplying through by 3x gives
More informationWhat do you think are the qualities of a good theorem? it solves an open problem (Name one..? )
What do you think are the qualities of a good theorem? Aspects of "good" theorems: short surprising elegant proof applied widely: it solves an open problem (Name one..? ) creates a new field might be easy
More informationMathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions
Mathematics 2260H Geometry I: Euclidean geometry Trent University, Winter 2012 Quiz Solutions Quiz #1. Tuesday, 17 January, 2012. [10 minutes] 1. Given a line segment AB, use (some of) Postulates I V,
More informationMathematics 2018 Practice Paper Paper 3 (Calculator) Higher Tier
Write your name here Surname Other Names Mathematics 2018 Practice Paper Paper 3 (Calculator) Higher Tier Time: 1 hour 30 minutes You must have: Ruler graduated in centimetres and millimetres, protractor,
More informationGlobal Context Statement of Inquiry MYP subject group objectives/assessment
Vertical Planner Subject: Mathematics Year level: MYP 1 Unit Title Key Concept Related Concept Global Context Statement of Inquiry MYP subject group objectives/assessment Number Systems and number properties
More information12.2 Existence proofs: examples
TOPIC 1: IMPOSSIBILITY AND EXISTENCE PROOFS Ref Sc american /gardner, 1.1 Mathematics is perhaps unique in that one may prove that certain situations are impossible. For example, it is not possible, using
More informationGreece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitu
Chapter 3 Greeks Greece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitude of Mediterranean islands. The Greeks
More informationPREFACE. Synergy for Success in Mathematics 9 is designed for Grade 9 students. The textbook
Synergy for Success in Mathematics 9 is designed for Grade 9 students. The textbook contains all the required learning competencies and is supplemented with some additional topics for enrichment. Lessons
More informationInteger-sided equable shapes
Integer-sided equable shapes Shapes with integer side lengths and with equal area and perimeter. Rectangles ab = (a + b) 1 = 1 a + 1 b Trapezia 6 8 14 1 4 3 0 Triangles 6 10 8 P = = 4 13 1 P = = 30 8 10
More informationNumber Theory. Jason Filippou UMCP. ason Filippou UMCP)Number Theory History & Definitions / 1
Number Theory Jason Filippou CMSC250 @ UMCP 06-08-2016 ason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions 06-08-2016 1 / 1 Outline ason Filippou (CMSC250 @ UMCP)Number Theory History & Definitions
More informationThe Computation of π by Archimedes. Bill McKeeman Dartmouth College
The Computation of π by Archimedes Bill McKeeman Dartmouth College 2012.02.15 Abstract It is famously known that Archimedes approximated π by computing the perimeters of manysided regular polygons, one
More informationHistory of Mathematics Workbook
History of Mathematics Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: April 7, 2014 Student: Spring 2014 Problem A1. Given a square ABCD, equilateral triangles ABX
More informationAQA. GCSE Mathematics. Practice Paper 1. Higher Paper 3 Calculator. Summer Time allowed: 1 hour 30 minutes. 8300/MissB/3H
AQA Please write clearly in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature GCSE Mathematics Higher Paper 3 Calculator H Time allowed: 1 hour 30 minutes Materials
More informationComplex numbers. Learning objectives
CHAPTER Complex numbers Learning objectives After studying this chapter, you should be able to: understand what is meant by a complex number find complex roots of quadratic equations understand the term
More informationA π day celebration! Everyone s favorite geometric constant!
A π day celebration! Everyone s favorite geometric constant! Math Circle March 10, 2019 The circumference of a circle is another word for its perimeter. A circle s circumference is proportional to its
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Edexcel International GCSE Mathematics A Paper 4HR Tuesday 21 May 2013 Morning Time: 2 hours Centre Number Candidate Number Higher Tier Paper Reference 4MA0/4HR
More informationMAT 115H Mathematics: An Historical Perspective Fall 2015
MAT 115H Mathematics: An Historical Perspective Fall 2015 Final Student Projects Below are listed 15 projects featuring a famous theorem prominent in the history of mathematics. These theorems, dating
More information648 Index. Axis, 256 horizontal, 256 of symmetry, 340, 343 vertical, 256
Index A Absolute value, 3 in adding integers, 11 12 of zero, 3 Addition in algebraic expressions, 49 of areas, 467 of fractions, 37 38 of integers, 10 12 of polynomials, 377 of rational numbers, 37 38
More informationA WHIRLWIND TOUR BEYOND QUADRATICS Steven J. Wilson, JCCC Professor of Mathematics KAMATYC, Wichita, March 4, 2017
b x1 u v a 9abc b 7a d 7a d b c 4ac 4b d 18abcd u 4 b 1 i 1 i 54a 108a x u v where a 9abc b 7a d 7a d b c 4ac 4b d 18abcd v 4 b 1 i 1 i 54a x u v 108a a //017 A WHIRLWIND TOUR BEYOND QUADRATICS Steven
More informationGrade 9 type questions. GCSE style questions arranged by topic
Write your name here Surname Other names In the style of: Pearson Edecel Level 1/Level 2 GCSE (9-1) Centre Number Mathematics Grade 9 type questions GCSE style questions arranged by topic Candidate Number
More informationSolving Polynomial Equations
Solving Polynomial Equations Introduction We will spend the next few lectures looking at the history of the solutions of polynomial equations. We will organize this examination by the degree of the equations,
More informationThe Three Ancient Problems 1
The Three Ancient Problems 1 Three problems left unsolved by the ancient Greek school challenged later mathematicians, amateur and professional, for two millennia before their resolution. In this brief
More information= 5 2 and = 13 2 and = (1) = 10 2 and = 15 2 and = 25 2
BEGINNING ALGEBRAIC NUMBER THEORY Fermat s Last Theorem is one of the most famous problems in mathematics. Its origin can be traced back to the work of the Greek mathematician Diophantus (third century
More informationLevel 2 Certificate in Further Mathematics FURTHER MATHEMATICS
Please write clearly in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature Level 2 Certificate in Further Mathematics FURTHER MATHEMATICS Level 2 Paper 1 Non-Calculator
More informationPell s Equation Claire Larkin
Pell s Equation is a Diophantine equation in the form: Pell s Equation Claire Larkin The Equation x 2 dy 2 = where x and y are both integer solutions and n is a positive nonsquare integer. A diophantine
More informationMathematics: A Christian Perspective
Mathematics: A Christian Perspective STUDENT VERSION Gino Santa Maria. Image from BigStockPhoto.com. James Bradley, Calvin College Andrew Busch, Fremont High School, Fremont, Michigan David Klanderman,
More informationThe Square, the Circle and the Golden Proportion: A New Class of Geometrical Constructions
Original Paper Forma, 19, 293 313, 2004 The Square, the Circle and the Golden Proportion: A New Class of Geometrical Constructions Janusz KAPUSTA Brooklyn, NY E-mail address: kapusta@earthlink.net (Received
More informationMthEd/Math 300 Williams Fall 2011 Midterm Exam 3
Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 3 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first
More informationIntroduction to Number Theory
Introduction to Number Theory Paul Yiu Department of Mathematics Florida Atlantic University Spring 017 March 7, 017 Contents 10 Pythagorean and Heron triangles 57 10.1 Construction of Pythagorean triangles....................
More informationChapter 0. Introduction. An Overview of the Course
Chapter 0 Introduction An Overview of the Course In the first part of these notes we consider the problem of calculating the areas of various plane figures. The technique we use for finding the area of
More informationSolutions to November 2006 Problems
Solutions to November 2006 Problems Problem 1. A six-sided convex polygon is inscribed in a circle. All its angles are equal. Show that its sides need not be equal. What can be said about seven-sided equal-angled
More informationCreating and Exploring Circles
Creating and Exploring Circles 1. Close your compass, take a plain sheet of paper and use the compass point to make a tiny hole (point) in what you consider to be the very centre of the paper. The centre
More informationPLC Papers. Created For:
PLC Papers Created For: Algebraic argument 2 Grade 5 Objective: Argue mathematically that two algebraic expressions are equivalent, and use algebra to support and construct arguments Question 1. Show that
More informationCALCULUS: Math 21C, Fall 2010 Final Exam: Solutions. 1. [25 pts] Do the following series converge or diverge? State clearly which test you use.
CALCULUS: Math 2C, Fall 200 Final Exam: Solutions. [25 pts] Do the following series converge or diverge? State clearly which test you use. (a) (d) n(n + ) ( ) cos n n= n= (e) (b) n= n= [ cos ( ) n n (c)
More informationGCSE MATHEMATICS (LINEAR) Higher Tier Paper 1. Morning. (NOV H01) WMP/Nov15/4365/1H/E6 4365/1H. Materials. Instructions. Information.
Please write clearly in block capitals. Centre number Candidate number Surname Forename(s) Candidate signature GCSE H MATHEMATICS (LINEAR) Higher Tier Paper 1 Wednesday 4 November 2015 Materials For this
More informationHistory of Mathematics
History of Mathematics A Course for High Schools (Session #132) Chuck Garner, Ph.D. Department of Mathematics Rockdale Magnet School for Science and Technology Georgia Math Conference at Rock Eagle, October
More informationRobert McGee, Professor Emeritus, Cabrini College Carol Serotta, Cabrini College Kathleen Acker, Ph.D.
Robert McGee, Professor Emeritus, Cabrini College Carol Serotta, Cabrini College Kathleen Acker, Ph.D. 1 2 At the end of about 18 pages of discussion of the history of Chinese mathematics, Victor Katz
More informationLecture 1: Axioms and Models
Lecture 1: Axioms and Models 1.1 Geometry Although the study of geometry dates back at least to the early Babylonian and Egyptian societies, our modern systematic approach to the subject originates in
More informationI.31 Now given a circular field, the circumference is 30 bu and the diameter 10 bu. Question: What is the area? Answer:
Chapter 9 Areas of circular regions 9.1 Problems I31 38 1 I.31 Now given a circular field, the circumference is 30 bu and the diameter 10 bu. I.3 Given another circular field, the circumference is 181
More informationAlgebra Mat: Working Towards Year 6
Algebra Mat: Working Towards Year 6 at 3 and adds 3 each time. 5, 10, 15, 20, Use simple formulae. The perimeter of a rectangle = a + a + b + b a = a b = 2, cd = 6, find 2 different pairs of numbers for
More informationStudy Guide for Exam 2
Study Guide for Exam 2 Math 330: History of Mathematics November 7, 2005. 1 Introduction What follows is a list of topics that might be on the exam. Of course, the test will only contain a selection of
More information43603H. (MAR H01) WMP/Mar13/43603H. General Certificate of Secondary Education Higher Tier March Unit H
Centre Number Surname Candidate Number For Examiner s Use Other Names Candidate Signature Examiner s Initials General Certificate of Secondary Education Higher Tier March 2013 Pages 3 4 5 Mark Mathematics
More informationWhy does pi keep popping up? Undergraduate Colloquium, October 2007 I. Definitions and Archimedes. II. Digits and some silliness (and Ramanujan)
Why does pi keep popping up? Undergraduate Colloquium, October 7 I. Definitions and Archimedes II. Digits and some silliness (and Ramanujan III. Antidote: pi is irrational. IV. Pi popping up in factorials.
More informationIndex. Excerpt from "Art of Problem Solving Volume 1: the Basics" 2014 AoPS Inc. / 267. Copyrighted Material
Index Ø, 247 n k, 229 AA similarity, 102 AAS congruence, 100 abscissa, 143 absolute value, 191 abstract algebra, 66, 210 altitude, 95 angle bisector, 94 Angle Bisector Theorem, 103 angle chasing, 133 angle
More informationQUADRATIC EQUATIONS. 4.1 Introduction
70 MATHEMATICS QUADRATIC EQUATIONS 4 4. Introduction In Chapter, you have studied different types of polynomials. One type was the quadratic polynomial of the form ax + bx + c, a 0. When we equate this
More informationAn excursion through mathematics and its history MATH DAY 2013 TEAM COMPETITION
An excursion through mathematics and its history MATH DAY 2013 TEAM COMPETITION A quick review of the rules History (or trivia) questions alternate with math questions Math questions are numbered by MQ1,
More informationSquare-Triangular Numbers
Square-Triangular Numbers Jim Carlson April 26, 2004 Contents 1 Introduction 2 2 Existence 2 3 Finding an equation 3 4 Solutions by brute force 4 5 Speeding things up 5 6 Solutions by algebraic numbers
More informationVibrate geometric forms in Euclidian space
Vibrate geometric forms in Euclidian space CLAUDE ZIAD BAYEH 1, 1 Faculty of Engineering II, Lebanese University EGRDI transaction on mathematics (003) LEBANON Email: claude_bayeh_cbegrdi@hotmail.com NIKOS
More informationGrade 8 Curriculum Map
Grade 8 Curriculum Map 2007-2008 Moving Straight Ahead 25 Days Curriculum Map 2007-2008 CMP2 Investigations Notes Standards 1.2 Finding and Using Rates Walking Rates and Linear Relationships 1.3 Raising
More informationCalifornia Subject Examinations for Teachers
CSET California Subject Eaminations for Teachers TEST GUIDE MATHEMATICS SUBTEST III Sample Questions and Responses and Scoring Information Copyright 005 by National Evaluation Systems, Inc. (NES ) California
More information18-29 mai 2015: Oujda (Maroc) École de recherche CIMPA-Oujda Théorie des Nombres et ses Applications. Continued fractions. Michel Waldschmidt
18-29 mai 2015: Oujda (Maroc) École de recherche CIMPA-Oujda Théorie des Nombres et ses Applications. Continued fractions Michel Waldschmidt We first consider generalized continued fractions of the form
More informationTo manipulate sequences, it is useful to be able to represent them algebraically as a power series known as a generating function.
Sequences In this chapter, we are concerned with infinite sequences of either integers or, more generally, real numbers. Although it is no longer one of the main phyla of questions in the IMO (combinatorics,
More informationMath 1230, Notes 8. Sep. 23, Math 1230, Notes 8 Sep. 23, / 28
Math 1230, Notes 8 Sep. 23, 2014 Math 1230, Notes 8 Sep. 23, 2014 1 / 28 algebra and complex numbers Math 1230, Notes 8 Sep. 23, 2014 2 / 28 algebra and complex numbers Math 1230, Notes 8 Sep. 23, 2014
More information~ 2. Who was Euclid? How might he have been influenced by the Library of Alexandria?
CD Reading Guide Sections 5.1 and 5. 2 1. What was the Museum of Alexandria? ~ 2. Who was Euclid? How might he have been influenced by the Library of Alexandria? --)~ 3. What are the Elements of Euclid?
More informationMathematics Without Calculations It s a Beautiful Thing!
Sacred Heart University DigitalCommons@SHU Mathematics Faculty Publications Mathematics Department 8-4-2016 Mathematics Without Calculations It s a Beautiful Thing! Jason J. Molitierno Sacred Heart University,
More informationTHE KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE (A) 0 (B) 1 (C) 2 (D) 3 (E) 4
THE 007 008 KENNESAW STATE UNIVERSITY HIGH SCHOOL MATHEMATICS COMPETITION PART I MULTIPLE CHOICE For each of the following questions, carefully blacken the appropriate box on the answer sheet with a #
More informationCore Mathematics C12
Write your name here Surname Other names Pearson Edexcel International Advanced Level Centre Number Candidate Number Core Mathematics C12 Advanced Subsidiary Monday 13 January 2014 Morning Time: 2 hours
More informationEgyptian Mathematics
Egyptian Mathematics Dr. Carmen Bruni David R. Cheriton School of Computer Science University of Waterloo November 1st, 2017 Three Part Series Egyptian Mathematics Diophantus and Alexandria Tartaglia,
More informationThe number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72.
ADVANCED SUBSIDIARY GCE UNIT 4752/0 MATHEMATICS (MEI) Concepts for Advanced Mathematics (C2) THURSDAY 7 JUNE 2007 Morning Time: hour 0 minutes Additional materials: Answer booklet (8 pages) Graph paper
More informationSection 0.2 & 0.3 Worksheet. Types of Functions
MATH 1142 NAME Section 0.2 & 0.3 Worksheet Types of Functions Now that we have discussed what functions are and some of their characteristics, we will explore different types of functions. Section 0.2
More informationFigurate Numbers: presentation of a book
Figurate Numbers: presentation of a book Elena DEZA and Michel DEZA Moscow State Pegagogical University, and Ecole Normale Superieure, Paris October 2011, Fields Institute Overview 1 Overview 2 Chapter
More informationCambridge International Examinations Cambridge International General Certificate of Secondary Education
Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 2 0 9 6 7 0 5 6 5 * MATHEMATICS 0580/42 Paper 4 (Extended) October/November 2016 Candidates answer
More information1 Continued Fractions
Continued Fractions To start off the course, we consider a generalization of the Euclidean Algorithm which has ancient historical roots and yet still has relevance and applications today.. Continued Fraction
More information