DOCSLIB.ORG

On Constructible Numbers and Angles Original Notes Adopted from March 5, 2002 (Week 21)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Constructible Numbers and Angles Original Notes Adopted from March 5, 2002 (Week 21) More on Constructible Numbers and Angles Original Notes adopted from March 5, 2002 (Week 21) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Lemma: If x0 is a root of a polynomial with coefficients in F(r), then x0 is a root of a polynomial with coefficients in F. n n-1 Proof: Given (an + bn √r) x0 + (an-1 + bn-1 √r) x0 + ... +(a1 + b1 √r)x0 + a0 + b0 √r =0 with ai , bi ∈ F. n By Dividing through by an + bn √r, we can assume it is poly monic. (ie. coefficients of x0 is 1). n n-1 x0 (an-1 + bn-1 √r) x0 + ... +(a1 + b1 √r)x0 + a0 + b0 √r =0 Then, n n-1 n-1 x0 + an-1 x0 +....+ a1x0 + a0 = √r (bn-1x0 + ... + b1x0 + b0 ) Square both Sides n n-1 2 n-1 2 (x0 + an-1 x0 +....+ a1x0 + a0 ) - r (bn-1x0 + ... + b1x0 + b0 ) =0 Recall: x0 algebraic if p(x0)= 0 some polynomial p with rational coefficients. Theorem: Every constructible number is algebraic. Proof: Suppose x0 is constructible. There exists Q = F0 ⊂ F1 ⊂ F2 ⊂......⊂ Fk with Fi = Fi-1 (√ri ) some ri ∈ Fi-1 , ri = 0, √ri ∉ Fi-1 such that x0 ∈ Fk Fk = Fk-1 ( ¥ rk ) x0 = a + b ¥ rk, a, b ∈ Fk x0 root of x – ( a + b ¥U k ) = 0. Lemma ⇒ x0 root of polynomial with coefficients in Fk-1 ⇒ x0 root of polynomial with coefficients in Fk-2 ∴ x0 root of polynomial with coefficients in Q. Corollary: The set of constructible numbers is countable. Proof: It's contained in set of algebraic numbers which we proved countable. Corollary: Angle θ can't be constructed with straightedge and compass if cosθ is transcendental Eg. 370 We know that cos (37/3) is transcendental ⇒ (37/3) 0 is not constructible. Can you trisect 37 degrees if it is just given ( We presume you are given the angle , it is irrelevant if you can construct it or not) Theorem: If cosθ is transcendental, then θ can't be trisected with straightedge and compass. Proof: Can construct number cosθ from θ given: Let F0 = {p(c)/q(c) , p, q polynomials with rational coefficients, q(c) ≠ 0} where c = cosθ. F0 number field. Before we started with Q. Here, we start with F0, which is the smallest field containing Q & c. Here, with straightedge and compass , we can make towers starting with F0 F0 ⊂ F0( ¥ r0 ) ⊂ (F0( ¥ r0 )) ( ¥ r0 ) cos3δ = 4cos3δ - 3cosδ. if 3δ = θ, δ = 1/3θ. Show: cosδ is not in any such tower. If it were, cos3δ = c, so c = 4x3 – 3x would have a solution in a tower. Preceding as before(if root of such a cubic in F(¥U LWVFRQMXJDWHLVD lso a root, go down until set root in F0. 3 ∴it suffices to prove: 4x – 3x – c =0 has no root in F0. Suppose x = p(c)/q(c) is a root with p, q polynomials with coefficients in Q. 4 (p(c)) 3 - 3p(c) - c =0 (q(c)) 3 q(c) 4 (p(c)) 3 - 3p(c) (q(c)) 2 - c(q(c)) 3 =0 Polynomial in c, rational coefficients = 0, contradicts c being transcendental unless the polynomial(after collecting coefficients) has all its coefficients 0; eg. 2(π3- π) + 2π - 2π3 = 0. But this can't happen: 1) if degree p > degree q, 4 (p(c)) 3 has only terms of highest degree 2) if degree p <= degree q. c(q(c)) 3 has only term of highest degree. ∴ Leading coefficients of the polynomial (after collecting terms) is not 0. Corollary: Only countable numbers of angles can be trisected. Showing we can Trisect ANY angle given we have a "marked" straightedge. It is clear from the diagram that x is a third of θ. x + x + w = π w + y = π y = 2x 2x + 2x + z = π z = π - 4x x+ π - 4x + θ = π θ = 3x x = θ/3 x + z + θ = π Polyhedron: solid, faces are polygons. A regular polyhedron(platonic solid) is a polyhedron all of whose faces are regular polygons with same number of sides as each other and all of whose vertices lie on the same number of faces. 1) Cube 2)Tetrahedron ± 4 faces which are triangles 3) Dodecahedron ± 12 sided pentagon 4) Icosahedron ± 20 sides b) Octagon All these have only 3,4,5 sides per face. Theorem: There are only 5 regular polyhedron Proof: Let n be the number of sides of a face and k, the number of faces on which a vertex lies. Note that k >= 3 For regular polynomial of n sides, angle is (n-2)π, so each angle is (n-2)π/n = π - 2π/n k(π - 2π/2) < 2π n>=3(Polygons have at least 3 sides). ⇒ 1/n <= 1/3 so ±1/n>= 1/3 π - 2π/n >= π - 2π/3 = π/3 k(π - 2π/n) >= kπ/3. ∴ kπ/3 < 2π ∴ k<6 ∴k =3,4, or 5 k =3... 3(π - 2π/2) < 2π 3(n-2/n)π < 2π 2n ± 6 <2 n<6. If k =3; n =3 tetrahedron k=3 & n =4 cube k =3, n=5 dodehedron. .
Load more

Recommended publications
  • Field Theory Pete L. Clark
    Field Theory Pete L. Clark Thanks to Asvin Gothandaraman and David Krumm for pointing out errors in these notes. Contents About These Notes 7 Some Conventions 9 Chapter 1. Introduction to Fields 11 Chapter 2. Some Examples of Fields 13 1. Examples From Undergraduate Mathematics 13 2. Fields of Fractions 14 3. Fields of Functions 17 4. Completion 18 Chapter 3. Field Extensions 23 1. Introduction 23 2. Some Impossible Constructions 26 3. Subfields of Algebraic Numbers 27 4. Distinguished Classes 29 Chapter 4. Normal Extensions 31 1. Algebraically closed fields 31 2. Existence of algebraic closures 32 3. The Magic Mapping Theorem 35 4. Conjugates 36 5. Splitting Fields 37 6. Normal Extensions 37 7. The Extension Theorem 40 8. Isaacs' Theorem 40 Chapter 5. Separable Algebraic Extensions 41 1. Separable Polynomials 41 2. Separable Algebraic Field Extensions 44 3. Purely Inseparable Extensions 46 4. Structural Results on Algebraic Extensions 47 Chapter 6. Norms, Traces and Discriminants 51 1. Dedekind's Lemma on Linear Independence of Characters 51 2. The Characteristic Polynomial, the Trace and the Norm 51 3. The Trace Form and the Discriminant 54 Chapter 7. The Primitive Element Theorem 57 1. The Alon-Tarsi Lemma 57 2. The Primitive Element Theorem and its Corollary 57 3 4 CONTENTS Chapter 8. Galois Extensions 61 1. Introduction 61 2. Finite Galois Extensions 63 3. An Abstract Galois Correspondence 65 4. The Finite Galois Correspondence 68 5. The Normal Basis Theorem 70 6. Hilbert's Theorem 90 72 7. Infinite Algebraic Galois Theory 74 8. A Characterization of Normal Extensions 75 Chapter 9.
    [Show full text]
  • Växjö University
    School of Mathematics and System Engineering Reports from MSI - Rapporter från MSI Växjö University Geometrical Constructions Tanveer Sabir Aamir Muneer June MSI Report 09021 2009 Växjö University ISSN 1650-2647 SE-351 95 VÄXJÖ ISRN VXU/MSI/MA/E/--09021/--SE Tanveer Sabir Aamir Muneer Trisecting the Angle, Doubling the Cube, Squaring the Circle and Construction of n-gons Master thesis Mathematics 2009 Växjö University Abstract In this thesis, we are dealing with following four problems (i) Trisecting the angle; (ii) Doubling the cube; (iii) Squaring the circle; (iv) Construction of all regular polygons; With the help of field extensions, a part of the theory of abstract algebra, these problems seems to be impossible by using unmarked ruler and compass. First two problems, trisecting the angle and doubling the cube are solved by using marked ruler and compass, because when we use marked ruler more points are possible to con- struct and with the help of these points more figures are possible to construct. The problems, squaring the circle and Construction of all regular polygons are still im- possible to solve. iii Key-words: iv Acknowledgments We are obliged to our supervisor Per-Anders Svensson for accepting and giving us chance to do our thesis under his kind supervision. We are also thankful to our Programme Man- ager Marcus Nilsson for his work that set up a road map for us. We wish to thank Astrid Hilbert for being in Växjö and teaching us, She is really a cool, calm and knowledgeable, as an educator should. We also want to thank of our head of department and teachers who time to time supported in different subjects.
    [Show full text]
  • The Art of the Intelligible
    JOHN L. BELL Department of Philosophy, University of Western Ontario THE ART OF THE INTELLIGIBLE An Elementary Survey of Mathematics in its Conceptual Development To my dear wife Mimi The purpose of geometry is to draw us away from the sensible and the perishable to the intelligible and eternal. Plutarch TABLE OF CONTENTS FOREWORD page xi ACKNOWLEDGEMENTS xiii CHAPTER 1 NUMERALS AND NOTATION 1 CHAPTER 2 THE MATHEMATICS OF ANCIENT GREECE 9 CHAPTER 3 THE DEVELOPMENT OF THE NUMBER CONCEPT 28 THE THEORY OF NUMBERS Perfect Numbers. Prime Numbers. Sums of Powers. Fermat’s Last Theorem. The Number π . WHAT ARE NUMBERS? CHAPTER 4 THE EVOLUTION OF ALGEBRA, I 53 Greek Algebra Chinese Algebra Hindu Algebra Arabic Algebra Algebra in Europe The Solution of the General Equation of Degrees 3 and 4 The Algebraic Insolubility of the General Equation of Degree Greater than 4 Early Abstract Algebra 70 CHAPTER 5 THE EVOLUTION OF ALGEBRA, II 72 Hamilton and Quaternions. Grassmann’s “Calculus of Extension”. Finite Dimensional Linear Algebras. Matrices. Lie Algebras. CHAPTER 6 THE EVOLUTION OF ALGEBRA, III 89 Algebraic Numbers and Ideals. ABSTRACT ALGEBRA Groups. Rings and Fields. Ordered Sets. Lattices and Boolean Algebras. Category Theory. CHAPTER 7 THE DEVELOPMENT OF GEOMETRY, I 111 COORDINATE/ALGEBRAIC/ANALYTIC GEOMETRY Algebraic Curves. Cubic Curves. Geometric Construction Problems. Higher Dimensional Spaces. NONEUCLIDEAN GEOMETRY CHAPTER 8 THE DEVELOPMENT OF GEOMETRY, II 129 PROJECTIVE GEOMETRY DIFFERENTIAL GEOMETRY The Theory of Surfaces. Riemann’s Conception of Geometry. TOPOLOGY Combinatorial Topology. Point-set topology. CHAPTER 9 THE CALCULUS AND MATHEMATICAL ANALYSIS 151 THE ORIGINS AND BASIC NOTIONS OF THE CALCULUS MATHEMATICAL ANALYSIS Infinite Series.
    [Show full text]
  • Advanced Mathematics for Secondary Teachers: a Capstone Experience
    Advanced Mathematics for Secondary Teachers: A Capstone Experience Curtis Bennett David Meel c 2000 January 3, 2001 2 Contents 1 Introduction 5 1.1 A Description of the Course . 5 1.2 What is Mathematics? . 6 1.3 Background . 9 1.3.1 GCDs and the Fundamental Theorem of Arithmetic . 10 1.3.2 Abstract Algebra and polynomials . 11 1.4 Problems . 14 2 Rational and Irrational Numbers 15 2.1 Decimal Representations . 18 2.2 Irrationality Proofs . 23 2.3 Irrationality of e and π ...................... 26 2.4 Problems . 30 3 Constructible Numbers 35 3.1 The Number Line . 37 3.2 Construction of Products and Sums . 39 3.3 Number Fields and Vector Spaces . 44 3.4 Impossibility Theorems . 54 3.5 Regular n-gons . 58 3.6 Problems . 60 4 Solving Equations by Radicals 63 4.1 Solving Simple Cubic Equations . 66 4.2 The General Cubic Equation . 72 4.3 The Complex Plane . 74 4.4 Algebraic Numbers . 78 4.5 Transcendental Numbers . 80 3 4 CONTENTS 5 Dedekind Cuts 87 5.1 Axioms for the Real Numbers . 87 5.2 Dedekind Cuts . 91 6 Classical Numbers 105 6.1 The Logarithmic Function . 105 7 Cardinality Questions 109 8 Finite Difference Methods 111 Chapter 1 Introduction 1.1 A Description of the Course This course is intended as a capstone experience for mathematics students planning on becoming high school teachers. This is not to say that this is a course in material from the high school curriculum. Nor is this a course on how to teach topics from the high school curriculum.
    [Show full text]
  • FROM GEOMETRY to NUMBER the Arithmetic Field Implicit in Geometry
    FROM GEOMETRY TO NUMBER The Arithmetic Field implicit in Geometry Johan Alm Contents Preface v Chapter 1. Kinds of Number in Early Geometry 1 1. Pythagorean natural philosophy 1 2. Plato and Euclid 4 Chapter 2. Fields and Geometric Construction 7 1. Line segment arithmetic 7 2. The Hilbert field 12 3. The constructible field 13 4. The real field 15 Chapter 3. The main theorems 17 1. The Cartesian space over a field 17 2. Faithful models 17 3. Three theorems 18 4. Proof of the first theorem 19 5. Proof of the second theorem 22 6. Proof of the third theorem 26 7. The main theorem 26 Chapter 4. Some applications 29 1. Duplicating the cube 29 2. Trisecting the angle 32 3. Squaring the circle 33 Appendix A. Hilbert’s axioms 35 Appendix B. Axioms and properties of groups and fields 39 Appendix C. Field extensions 41 Appendix. Bibliography 43 iii Preface This is a short monograph on geometry, written by the author as a supplement to David C. Kay’s ”College Geometry” while taking a course based on that book. The presentation in this monograph is very indebted to Robin Hartshorne’s book ”Geometry: Euclid and Beyond.” To be more precise we might say this is a text on classical, synthetic, or ax- iomatic geometry, but we consider this the heart of geometry and prefer to refer to the subject simply as ”geometry.” Most texts on geometry focus on the parallel axiom. Little fuzz is made about the other axioms, and one easily gets the idea that they serve only to provide a sensible foundation.
    [Show full text]
  • Ruler and Compass Constructions and Abstract Algebra
    Ruler and Compass Constructions and Abstract Algebra Introduction Around 300 BC, Euclid wrote a series of 13 books on geometry and number theory. These books are collectively called the Elements and are some of the most famous books ever written about any subject. In the Elements, Euclid described several “ruler and compass” constructions. By ruler, we mean a straightedge with no marks at all (so it does not look like the rulers with centimeters or inches that you get at the store). The ruler allows you to draw the (unique) line between two (distinct) given points. The compass allows you to draw a circle with a given point as its center and with radius equal to the distance between two given points. But there are three famous constructions that the Greeks could not perform using ruler and compass: • Doubling the cube: constructing a cube having twice the volume of a given cube. • Trisecting the angle: constructing an angle 1/3 the measure of a given angle. • Squaring the circle: constructing a square with area equal to that of a given circle. The Greeks were able to construct several regular polygons, but another famous problem was also beyond their reach: • Determine which regular polygons are constructible with ruler and compass. These famous problems were open (unsolved) for 2000 years! Thanks to the modern tools of abstract algebra, we now know the solutions: • It is impossible to double the cube, trisect the angle, or square the circle using only ruler (straightedge) and compass. • We also know precisely which regular polygons can be constructed and which ones cannot.
    [Show full text]
  • Constructible Numbers Weeks 7-8 UCSB 2014
    Math/CS 120: Intro. to Math Professor: Padraic Bartlett Lecture 6: Constructible Numbers Weeks 7-8 UCSB 2014 As the geometer who sets himself To square the circle and who cannot find, For all his thought, the principle he needs, Just so was I on seeing this new vision. Dante Alighieri, Paradise, canto XXXIII 1 Constructible Numbers Construction, in a sense, was the unofficial theme of the first five weeks of this class. We started our lectures by first defining how to make proofs; from there, we used our language of \proof" to construct N; Z; Q; R and C. However, the method in which we constructed these objects was remarkably abstract; we started with nothing,1 and in the words of J´anosBolyai, \Out of nothing [we] have created a strange new universe." While this was beautiful in its simplicity | given absolutely nothing but a small handful of logical axioms, we were able to build our way to the foundations of modern mathematics in a few weeks | it also makes one wonder if there are perhaps different or easier ways to approach the same constructions. That is: when we conceive of the number 4, we usually don't think of f;; f;g; f;; f;gg; f;; f;g; f;; f;gggg: These notes are meant to offer a different construction for many of the numbers we care about: the constructible numbers! We define them as follows: 2 Definition. Suppose that you are given an infinite piece of paper (think of this as R ,) and two tools: 1.A compass.
    [Show full text]
  • Transcendental Numbers
    Bachelor Thesis DEGREE IN MATHEMATICS Faculty of Mathematics University of Barcelona TRANSCENDENTAL NUMBERS Author: Adina Nedelea Director: Dr. Ricardo Garc´ıaL´opez Department: Algebra and Geometry Barcelona, June 27, 2016 Summary Transcendental numbers are a relatively recent finding in mathematics and they provide, togheter with the algebraic numbers, a classification of complex numbers. In the present work the aim is to characterize these numbers in order to see the way from they differ the algebraic ones. Going back to ancient times we will observe how, since the earliest history mathematicians worked with transcendental numbers even if they were not aware of it at that time. Finally we will describe some of the consequences and new horizons of mathematics since the apparition of the transcendental numbers. Agradecimientos El trabajo final de grado es un punto final de una etapa y me gustar´aagradecer a todas las personas que me han dado apoyo durante estos a~nosde estudios y hicieron posible que llegue hasta aqu´ı. Empezar´epor agradecer a quien hizo posible este trabajo, a mi tutor Ricardo Garc´ıa ya que sin su paciencia, acompa~namiento y ayuda me hubiera sido mucho mas dif´ıcil llevar a cabo esta tarea. Sigo con los agradecimientos hacia mi familia, sobretodo hacia mi madre quien me ofreci´ola libertad y el sustento aunque no le fuera f´acily a mi "familia adoptiva" Carles, Marilena y Ernest, quienes me ayudaron a sostener y pasar por todo el pro- ceso de maduraci´onque conlleva la carrera y no rendirme en los momentos dif´ıciles. Gracias por la paciencia y el amor de todos ellos.
    [Show full text]
  • Constructible Numbers Definition 13.1 (A) a Complex Number R Is Algebraic If P(R) = 0 for Some P(X) ∈ Q[X]
    Math 4030-001/Foundations of Algebra/Fall 2017 Ruler/Compass at the Foundations: Constructible Numbers Definition 13.1 (a) A complex number r is algebraic if p(r) = 0 for some p(x) 2 Q[x]. Otherwise r is transcendental. (b) Let A ⊂ C be the set of algebraic numbers. (c) Given r 2 A, there is a unique prime polynomial of the form: d d−1 f(x) = x + cd−1x + ··· + c0 2 Q[x] with r as a root. This is the minimal polynomial of r. p n 2π Examples. (a) All rational numbers, i; 2; cos( n ) are algebraic. (b) π; e are transcendental (this isn't easy to prove!). Remark. If r 2 A and v 2 Q[r], then we saw in the previous section that v 2 A, since the vectors 1; v; v2; :::: 2 Q[r] are linearly dependent. Proposition 13.2. The set of algebraic numbers A ⊂ C is a field. Proof. Let r 2 A. Then −r; 1=r 2 Q[r] are also algebraic by the previous remark (or else one can easily find their minimal polynomials). Let s 2 A be another algebraic number. It suffices to show that both r + s and r · s are algebraic numbers to conclude that A is a field. To do this, we make a tower of number fields. Let F = Q[r] and consider F [x], the commutative ring of polynomials with coefficients in F . One such polynomial is the minimal polynomial f(x) 2 Q[x] of s 2 A. This may not be prime in F [x], but it will have a prime factor g(x) with g(s) = 0.
    [Show full text]
  • The Three Classical Geometric Problems 1 Constructible Numbers
    The three classical geometric problems 1 Constructible numbers Suppose that you are given a line segment of length 1, and the Euclidean tools of compass and (unmarked) straightedge. What other lengths can you construct? It's easy to construct other integer lengths n, by attaching n copies of your unit length back to back along a line. A little more generally, if you can construct lengths x and y, then you can construct x + y and x − y. What about multiplication and division? The trick is to use similar triangles. Given lengths a and b, you can construct ab as follows. First, draw a triangle ∆XYZ with XY = 1 and XZ = a. Then, draw a segment X0Y 0 of length b, and construct a new triangle ∆X0Y 0Z0 similar to the first one. Then X0Z0 = ab. Z’ ab Z a X’ X 1 Y b Y’ If instead you start by making X0Z0 = b, then the new triangle will have X0Y 0 = b=a. So the set K of all constructible numbers is closed under the four arithmetic operations of addition, 1 subtraction, multiplication and division. In particular, every rational number is constructible.p But there are certainly non-rational numbers that are constructible. For example, 2 can be constructed as the length of the hypotenusep ofp an isosceles right triangle with sides of length 1. We can then construct segments with lengths like 5 2, 2=3 − 7, . p In fact, if k is any constructible number then k is also constructible. The idea is that p k k p = k 1 so we'd like to construct a pair of similar triangles ∆ABC, ∆A0B0C0 such that p AB k k A0B0 = p = = BC k 1 B0C0 p Since both of these triangles are to have sides of length k, it makes sense to arrange the construction so that BC and A0B0 are the same segment, and AB has length k and B0C0 has length 1.
    [Show full text]
  • 3 Hrs MAT246H1S - Concepts in Abstract Mathematics Examiners: D
    University of Toronto Faculty of Arts and Science Solutions to Final Examination, April 2019 Duration: 3 hrs MAT246H1S - Concepts in Abstract Mathematics Examiners: D. Burbulla, S. Homayouni Time: 9 AM-12 Noon, April 22, 2019. Aids permitted: None. General Comments: • Since it was only required to do 12 questions, there were many pages left blank. In particular, many students ended up skipping one or two of Questions 3, 5, 6, 7, 8, 9, 10, 11, 12 and 14. The breakdown of the number of zero's by question is: Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 3 0 28 4 99 44 86 49 55 31 35 98 5 130 • On the whole, the proof questions were done very poorly. Note that Question 12 was actually a very simple question, but many students didn't even try it. • There were only six questions with passing averages: Questions 1, 2, 3, 9, 10 and 13. For the most part these questions were either definitions or computations. It seems as if proofs are still a challenge to most students. Breakdown of Results: 216 students wrote this exam. The marks ranged from 2/140 to 129/140, and the average was 67.2/140. However, since we ended up taking your score on the exam as a mark out of 100, the average on the exam is in effect 67.2%. (It turned out that fifteen students had a mark greater than 100.) A histogram of the results by decade is to the right: 1. [course avg: 7.33/10] Define the following.
    [Show full text]
  • A Brief History of Impossibility
    VOL. 81, NO. 1, FEBRUARY 2008 27 A Brief History of Impossibility JEFF SUZUKI Brooklyn College Brooklyn, NY jeff [email protected] Sooner or later every student of geometry learns of three “impossible” problems: 1. Trisecting the angle: Given an arbitrary angle, construct an angle exactly one-third as great. 2. Duplicating the cube: Given a cube of arbitrary volume, find a cube with exactly twice the volume. 3. Squaring the circle: Given an arbitrary circle, find a square with the same area. These problems originated around 430 BC at a time when Greek geometry was advanc- ing rapidly. We might add a fourth problem: inscribing a regular heptagon in a circle. Within two centuries, all these problems had been solved (see [3, Vol. I, p. 218–270] and [1] for some of these solutions). So if these problems were all solved, why are they said to be impossible? The “im- possibility” stems from a restriction, allegedly imposed by Plato (427–347 BC), that geometers use no instruments besides the compass and straightedge. This restriction requires further explanation. For that, we turn to Euclid (fl. 300 BC), who collected and systematized much of the plane geometry of the Greeks in his Elements. Euclid’s goal was to develop geometry in a deductive manner from as few basic assumptions as possible. The first three postulates in the Elements are (in modernized form): 1. Between any two points, there exists a unique straight line. 2. A straight line may be extended indefinitely. 3. Given any point and any length, a circle may be constructed centered at the point with radius equal to the given length.
    [Show full text]